8993:
the
Boolean prime ideal theorem. Other results in model theory depend on set-theoretic axioms beyond the standard ZFC framework. For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated (in its own cardinality). Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension. Neither of these results are provable in ZFC alone. Finally, some questions arising from model theory (such as compactness for infinitary logics) have been shown to be equivalent to large cardinal axioms.
2466:. According to the Löwenheim-Skolem Theorem, every infinite structure in a countable signature has a countable elementary substructure. Conversely, for any infinite cardinal Îș every infinite structure in a countable signature that is of cardinality less than Îș can be elementarily embedded in another structure of cardinality Îș (There is a straightforward generalisation to uncountable signatures). In particular, the Löwenheim-Skolem Theorem implies that any theory in a countable signature with infinite models has a countable model as well as arbitrarily large models.
13360:
11564:
8700:. Quantifier elimination allowed Tarski to show that the first-order theories of real-closed and algebraically closed fields as well as the first-order theory of Boolean algebras are decidable, classify the Boolean algebras up to elementary equivalence and show that the theories of real-closed fields and algebraically closed fields of a given characteristic are unique. Furthermore, quantifier elimination provided a precise description of definable relations on algebraically closed fields as
11576:
8864:. His work around stability changed the complexion of model theory, giving rise to a whole new class of concepts. This is known as the paradigm shift. Over the next decades, it became clear that the resulting stability hierarchy is closely connected to the geometry of sets that are definable in those models; this gave rise to the subdiscipline now known as geometric stability theory. An example of an influential proof from geometric model theory is
804:
11600:
11588:
517:
8647:, stating roughly that first-order logic is essentially the strongest logic in which both the Löwenheim-Skolem theorems and compactness hold. However, model theoretic techniques have been developed extensively for these logics too. It turns out, however, that much of the model theory of more expressive logical languages is independent of
3590:
quantifier-free formula in one variable. Quantifier-free formulas in one variable express
Boolean combinations of polynomial equations in one variable, and since a nontrivial polynomial equation in one variable has only a finite number of solutions, the theory of algebraically closed fields is strongly minimal.
799:{\displaystyle {\begin{array}{lcl}\varphi &=&\forall u\forall v(\exists w(x\times w=u\times v)\rightarrow (\exists w(x\times w=u)\lor \exists w(x\times w=v)))\land x\neq 0\land x\neq 1,\\\psi &=&\forall u\forall v((u\times v=x)\rightarrow (u=x)\lor (v=x))\land x\neq 0\land x\neq 1.\end{array}}}
7098:
Therefore, ultraproducts provide a way to talk about elementary equivalence that avoids mentioning first-order theories at all. Basic theorems of model theory such as the compactness theorem have alternative proofs using ultraproducts, and they can be used to construct saturated elementary extensions
8992:
In the other direction, model theory is itself formalised within
Zermelo-Fraenkel set theory. For instance, the development of the fundamentals of model theory (such as the compactness theorem) rely on the axiom of choice, and is in fact equivalent over Zermelo-Fraenkel set theory without choice to
3180:
If a theory does not have quantifier elimination, one can add additional symbols to its signature so that it does. Axiomatisability and quantifier elimination results for specific theories, especially in algebra, were among the early landmark results of model theory. But often instead of quantifier
3984:
One can even go one step further, and move beyond immediate substructures. Given a mathematical structure, there are very often associated structures which can be constructed as a quotient of part of the original structure via an equivalence relation. An important example is a quotient group of a
2671:
2433:
Similarly, if Ï' is a signature that extends another signature Ï, then a complete Ï'-theory can be restricted to Ï by intersecting the set of its sentences with the set of Ï-formulas. Conversely, a complete Ï-theory can be regarded as a Ï'-theory, and one can extend it (in more than one way) to a
2454:
is trivial, since every proof can have only a finite number of antecedents used in the proof. The completeness theorem allows us to transfer this to satisfiability. However, there are also several direct (semantic) proofs of the compactness theorem. As a corollary (i.e., its contrapositive), the
58:
in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be
8675:
relation generalising that of an elementary substructure. Even though its definition is purely semantic, every abstract elementary class can be presented as the models of a first-order theory which omit certain types. Generalising stability-theoretic notions to abstract elementary classes is an
7686:
Morley's proof revealed deep connections between uncountable categoricity and the internal structure of the models, which became the starting point of classification theory and stability theory. Uncountably categorical theories are from many points of view the most well-behaved theories. In
3825:
Particularly important are those definable sets that are also substructures, i. e. contain all constants and are closed under function application. For instance, one can study the definable subgroups of a certain group. However, there is no need to limit oneself to substructures in the same
3589:
if the theory of that structure is strongly minimal. Equivalently, a structure is strongly minimal if every elementary extension is minimal. Since the theory of algebraically closed fields has quantifier elimination, every definable subset of an algebraically closed field is definable by a
1828:
is a subset of its domain, closed under all functions in its signature Ï, which is regarded as a Ï-structure by restricting all functions and relations in Ï to the subset. This generalises the analogous concepts from algebra; for instance, a subgroup is a substructure in the signature with
8894:, which concentrates on finite structures, diverges significantly from the study of infinite structures in both the problems studied and the techniques used. In particular, many central results of classical model theory that fail when restricted to finite structures. This includes the
172:
Nonetheless, the interplay of classes of models and the sets definable in them has been crucial to the development of model theory throughout its history. For instance, while stability was originally introduced to classify theories by their numbers of models in a given
7687:
particular, complete strongly minimal theories are uncountably categorical. This shows that the theory of algebraically closed fields of a given characteristic is uncountably categorical, with the transcendence degree of the field determining its isomorphism type.
131:
The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within a model fluctuated in the history of the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively:
3173:. If the theory of a structure has quantifier elimination, every set definable in a structure is definable by a quantifier-free formula over the same parameters as the original definition. For example, the theory of algebraically closed fields in the signature Ï
3167:
8847:
In the further history of the discipline, different strands began to emerge, and the focus of the subject shifted. In the 1960s, techniques around ultraproducts became a popular tool in model theory. At the same time, researchers such as
5203:, there is no real number larger than every integer. However, a compactness argument shows that there is an elementary extension of the real number line in which there is an element larger than any integer. Therefore, the set of formulas
3345:
8659:, which studies the class of substructures of arbitrarily large homogeneous models. Fundamental results of stability theory and geometric stability theory generalise to this setting. As a generalisation of strongly minimal theories,
8304:
Many construction in model theory are easier when restricted to stable theories; for instance, every model of a stable theory has a saturated elementary extension, regardless of whether the generalised continuum hypothesis is true.
3985:
group. One might say that to understand the full structure one must understand these quotients. When the equivalence relation is definable, we can give the previous sentence a precise meaning. We say that these structures are
2518:
8615:
generalise o-minimal structures. They are related to stability since a theory is stable if and only if it is NIP and simple, and various aspects of stability theory have been generalised to theories in one of these classes.
7117:
if it determines a structure up to isomorphism. It turns out that this definition is not useful, due to serious restrictions in the expressivity of first-order logic. The LöwenheimâSkolem theorem implies that if a theory
5514:
While not every type is realised in every structure, every structure realises its isolated types. If the only types over the empty set that are realised in a structure are the isolated types, then the structure is called
8654:
More recently, alongside the shift in focus to complete stable and categorical theories, there has been work on classes of models defined semantically rather than axiomatised by a logical theory. One example is
9502:
8564:-stable. Morley Rank can be extended to types by setting the Morley Rank of a type to be the minimum of the Morley ranks of the formulas in the type. Thus, one can also speak of the Morley rank of an element
2039:
1968:
8800:, is often regarded as being of a model-theoretical nature in retrospect. The first significant result in what is now model theory was a special case of the downward LöwenheimâSkolem theorem, published by
7044:
8950:, since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model. Particularly the proof of the independence of the
8776:
in machine learning theory. This has led to several interactions between these separate areas. In 2018, the correspondence was extended as Hunter and Chase showed that stable theories correspond to
996:
9715:
3677:
8308:
Shelah's original motivation for studying stable theories was to decide how many models a countable theory has of any uncountable cardinality. If a theory is uncountably categorical, then it is
1045:
1303:
5279:
4784:
8101:
325:
3478:
2459:
says that every unsatisfiable first-order theory has a finite unsatisfiable subset. This theorem is of central importance in model theory, where the words "by compactness" are commonplace.
2171:
2141:
8205:
5249:
4207:
3041:
1609:
2206:
5140:
3785:
3543:
6646:
7823:
5712:
4966:
1110:
8139:
3417:
2721:
7475:
7216:
6414:
3177:= (Ă,+,â,0,1) has quantifier elimination. This means that in an algebraically closed field, every formula is equivalent to a Boolean combination of equations between polynomials.
7412:
7378:
7054:
In particular, any ultraproduct of models of a theory is itself a model of that theory, and thus if two models have isomorphic ultrapowers, they are elementarily equivalent. The
6265:
5626:
3989:. A key fact is that one can translate sentences from the language of the interpreted structures to the language of the original structure. Thus one can show that if a structure
3893:
6945:
6833:
6149:
6040:
5972:
5839:
5793:
5366:
5335:
4525:
4479:
4433:
4387:
4306:
4257:
4100:
3859:
2759:
6361:
4582:
2936:
1565:
1494:
8239:
98:
reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory is
8915:
7789:
6744:
6696:
6672:
6593:
6538:
6514:
6467:
5895:
5867:
5736:
5678:
5650:
5592:
5568:
5544:
5114:
5014:
4903:
4879:
4843:
4815:
4754:
4674:
4337:
4152:
4124:
4035:
4011:
3950:
3809:
3751:
3567:
3513:
2409:
2385:
2349:
2325:
2111:
2087:
1894:
1826:
1802:
1724:
1700:
1636:
835:
8918:, where the infinite models of a generic theory of a class of structures provide information on the distribution of finite models. Prominent application areas of FMT are
8383:
7564:
6110:
6088:
6062:
5994:
5301:
5197:
4554:
3723:
3613:
2981:
2895:
2510:
2294:
2272:
2250:
2228:
1516:
1445:
7946:
7593:
4714:
4634:
3979:
3922:
2794:
420:
252:
6914:
371:
348:
8403:
8352:
8165:
8061:
8035:
8011:
7919:
7899:
7863:
7843:
7765:
7743:
6285:
5090:
5050:
3701:
2063:
1854:
946:
878:
509:
460:
391:
8602:
8562:
8435:
8326:
8299:
8279:
7967:
7526:
2862:
1383:
8660:
6991:
5404:. The theory has quantifier elimination . This allows us to show that a type is determined exactly by the polynomial equations it contains. Thus the set of complete
7502:
6716:
6613:
6558:
6487:
6291:. This topology explains some of the terminology used in model theory: The compactness theorem says that the Stone space is a compact topological space, and a type
5160:
2829:
2430:, regarded as a structure in the signature {+,0} can be expanded to a field with the signature {×,+,1,0} or to an ordered group with the signature {+,0,<}.
918:
480:
284:
8745:
More recently, the connection between stability and the geometry of definable sets led to several applications from algebraic and diophantine geometry, including
5922:
2959:
2418:
A field or a vector space can be regarded as a (commutative) group by simply ignoring some of its structure. The corresponding notion in model theory is that of a
1536:
1423:
5680:. Therefore, a weaker notion has been introduced that captures the idea of a structure realising all types it could be expected to realise. A structure is called
3234:
6965:
6221:
6201:
6169:
5504:
5442:
5422:
5398:
5070:
4990:
1772:
1752:
1676:
1656:
1465:
1403:
1363:
1343:
1323:
1221:
1201:
1130:
1069:
898:
855:
440:
6176:
2991:
In general, definable sets without quantifiers are easy to describe, while definable sets involving possibly nested quantifiers can be much more complicated.
3981:
is true. In this way, one can study definable groups and fields in general structures, for instance, which has been important in geometric stability theory.
2666:{\displaystyle \forall u\forall v(\exists w(x\times w=u\times v)\rightarrow (\exists w(x\times w=u)\lor \exists w(x\times w=v)))\land x\neq 0\land x\neq 1}
11739:
7528:-categorical. This follows from the fact that in all those fields, any of the infinitely many natural numbers can be defined by a formula of the form
6433:
Constructing models that realise certain types and do not realise others is an important task in model theory. Not realising a type is referred to as
809:(Note that the equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the Ï
8684:
Among the early successes of model theory are Tarski's proofs of quantifier elimination for various algebraically interesting classes, such as the
8608:
12414:
3484:
Since we can negate this formula, every cofinite subset (which includes all but finitely many elements of the domain) is also always definable.
11024:
6564:
This implies that if a theory in a countable signature has only countably many types over the empty set, then this theory has an atomic model.
10062:
Jonathan Pila, Rational points of definable sets and results of AndrĂ©âOortâManinâMumford type, O-minimality and the AndrĂ©âOort conjecture for
3355:, which is a related model-complete theory that is not, in general, an extension of the original theory. A more general notion is that of a
82:, model theory is often less concerned with formal rigour and closer in spirit to classical mathematics. This has prompted the comment that
12497:
11638:
10050:
373:. A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are
6746:, this does not imply that every theory has a saturated model. In fact, whether every theory has a saturated model is independent of the
3200:
is an elementary substructure. There is a useful criterion for testing whether a substructure is an elementary substructure, called the
63:
in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to
7137:. Since two models of different sizes cannot possibly be isomorphic, only finite structures can be described by a categorical theory.
2411:. If it can be written as an isomorphism with an elementary substructure, it is called an elementary embedding. Every embedding is an
10981:
1973:
1902:
6302:
While types in algebraically closed fields correspond to the spectrum of the polynomial ring, the topology on the type space is the
7228:
is the cardinality of the signature). For finite or countable signatures this means that there is a fundamental difference between
1155:, i.e. no contradiction is proved by the theory. Therefore, model theorists often use "consistent" as a synonym for "satisfiable".
12811:
8773:
8624:
Model-theoretic results have been generalised beyond elementary classes, that is, classes axiomatisable by a first-order theory.
67:, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by
6996:
2473:, first-order logic is the most expressive logic for which both the LöwenheimâSkolem theorem and the compactness theorem hold.
8753:
in all characteristics In 2001, similar methods were used to prove a generalisation of the Manin-Mumford conjecture. In 2011,
3703:
to define arbitrary intervals on the real number line. It turns out that these suffice to represent every definable subset of
1179:. Note that in some literature, constant symbols are considered as function symbols with zero arity, and hence are omitted. A
12969:
10890:
10818:
10585:
10559:
10529:
10480:
10436:
10411:
10386:
10351:
10316:
10232:
9656:
8663:
classes are those in which every definable set is either countable or co-countable. They are key to the model theory of the
6567:
On the other hand, there is always an elementary extension in which any set of types over a fixed parameter set is realised:
954:
11757:
4556:, viewed as a structure with only the order relation {<}, will serve as a running example in this section. Every element
3621:
1148:
12824:
12147:
11387:
3865:-ary relations can also be definable. Functions are definable if the function graph is a definable relation, and constants
7638:, which are obtained as the limit of amalgamating all possible configurations of a class of finite relational structures.
1003:
19:
This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see
8820:. The LöwenheimâSkolem theorem and the compactness theorem received their respective general forms in 1936 and 1941 from
2446:
states that a set of sentences S is satisfiable if every finite subset of S is satisfiable. The analogous statement with
1236:
5254:
4759:
12829:
12819:
12556:
12409:
11762:
11532:
11017:
8841:
11753:
8899:
8817:
8636:
8417:
The stability hierarchy is also crucial for analysing the geometry of definable sets within a model of a theory. In
2415:
homomorphism, but the converse holds only if the signature contains no relation symbols, such as in groups or fields.
12965:
11604:
10867:
10841:
10780:
10757:
10731:
10701:
10678:
10651:
10621:
10273:
10194:"All three commentators agree that both the completeness and compactness theorems were implicit in Skolem 1923...."
10029:
9972:
9879:
9841:
9693:
9613:
9367:
8982:
8788:
Model theory as a subject has existed since approximately the middle of the 20th century, and the name was coined by
8066:
2871:, that is, the defining formulas don't mention any fixed domain elements. However, one can also consider definitions
292:
12307:
3425:
3162:{\displaystyle \forall x_{1}\dots \forall x_{n}(\phi (x_{1},\dots ,x_{n})\leftrightarrow \psi (x_{1},\dots ,x_{n}))}
2146:
2116:
13062:
12806:
11631:
11068:
10802:
10670:
7657:
6751:
8170:
5206:
4157:
12367:
12060:
11482:
7508:
is order-isomorphic to the rational number line. On the other hand, the theories of â, â and â as fields are not
1570:
11801:
9053:
8856:
were extending the concepts and results of first-order model theory to other logical systems. Then, inspired by
3683:
This defines the subset of non-negative real numbers, which is neither finite nor cofinite. One can in fact use
2182:
1425:
is a binary relation symbol. Then, when these symbols are interpreted to correspond with their usual meaning on
13323:
13025:
12788:
12783:
12608:
12029:
11713:
11580:
5119:
3760:
3518:
1180:
110:
55:
51:
47:
6618:
13318:
13101:
13018:
12731:
12662:
12539:
11781:
11010:
10948:
10613:
8919:
7798:
5687:
4928:
3820:
1727:
1224:
1133:
1085:
39:
8110:
3370:
2682:
13394:
13243:
13069:
12755:
12389:
11988:
11592:
9033:
8664:
8580:. There are also analogues of Morley rank which are well-defined if and only if a theory is superstable (
7425:
7178:
6366:
8824:. The development of model theory as an independent discipline was brought on by Alfred Tarski during the
13389:
13121:
13116:
12726:
12465:
12394:
11723:
11624:
11507:
11063:
10943:
8689:
8648:
8607:
More recently, stability has been decomposed into simplicity and "not the independence property" (NIP).
8604:-rank). Those dimension notions can be used to define notions of independence and of generic extensions.
7383:
7349:
6230:
6172:
5601:
3868:
1072:
35:
9126:
8768:
In a separate strand of inquiries that also grew around stable theories, Laskowski showed in 1992 that
6919:
6807:
6123:
5999:
5931:
5798:
5752:
5340:
5309:
4484:
4438:
4392:
4346:
4265:
4216:
4059:
3833:
2733:
2463:
13050:
12640:
12034:
12002:
11693:
11078:
10882:
10859:
10723:
10577:
9043:
8978:
8693:
6313:
5401:
4559:
1778:
7620:
3725:. This generalisation of minimality has been very useful in the model theory of ordered structures. A
2903:
2173:. Thus, an elementary substructure is a model of a theory exactly when the superstructure is a model.
1541:
1470:
13340:
13289:
13186:
12684:
12645:
12122:
11767:
11492:
11464:
11101:
8668:
8210:
1175:
such that each symbol is either a constant symbol, or a function or relation symbol with a specified
11796:
9866:. University Lecture Series. Vol. 50. Providence, Rhode Island: American Mathematical Society.
8793:
7770:
6725:
6677:
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5848:
5717:
5659:
5631:
5573:
5549:
5525:
5095:
4995:
4884:
4860:
4824:
4796:
4719:
4639:
4318:
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4105:
4016:
3992:
3931:
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3732:
3548:
3494:
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2366:
2330:
2306:
2092:
2068:
1875:
1807:
1783:
1705:
1681:
1617:
1203:
together with interpretations of each of the symbols of the signature as relations and functions on
816:
13181:
13111:
12650:
12502:
12485:
12208:
11688:
11537:
8837:
8697:
7251:
6782:
2434:
complete Ï'-theory. The terms reduct and expansion are sometimes applied to this relation as well.
103:
10938:
8762:
8361:
7531:
6093:
6071:
6045:
5977:
5522:
On the other hand, no structure realises every type over every parameter set; if one takes all of
5284:
5180:
4537:
3706:
3596:
2964:
2878:
2493:
2277:
2255:
2233:
2211:
1499:
1428:
1136:
or its negation. The complete theory of all sentences satisfied by a structure is also called the
13013:
12990:
12951:
12837:
12778:
12424:
12344:
12188:
12132:
11745:
11422:
11412:
11382:
11316:
11051:
10749:
10016:, Studies in Logic and the Foundations of Mathematics, vol. 69, Elsevier, pp. 261â279,
9959:, Studies in Logic and the Foundations of Mathematics, vol. 90, Elsevier, pp. 105â137,
9058:
8911:
8869:
8852:
were investigating the first-order model theory of various algebraic classes, and others such as
8840:, among other topics. His semantic methods culminated in the model theory he and a number of his
8777:
8750:
8104:
7924:
7712:
A key factor in the structure of the class of models of a first-order theory is its place in the
7571:
6747:
6112:
is itself countable and therefore only has to realise types over finite subsets to be saturated.
4679:
4599:
3955:
3898:
3578:
2767:
396:
207:
10925:
6858:
13303:
13030:
13008:
12975:
12868:
12714:
12699:
12672:
12623:
12507:
12442:
12267:
12233:
12228:
12102:
11933:
11910:
11520:
11417:
11397:
11392:
11321:
11046:
9028:
9023:
8927:
8644:
8242:
6303:
3201:
2995:
2252:
is not, as we can express "There is a square root of 2" as a first-order sentence satisfied by
1112:, i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set
353:
330:
287:
150:
where universal algebra stands for mathematical structures and logic for logical theories; and
9354:, Studies in Logic and the Foundations of Mathematics, vol. 13, Elsevier, pp. 1â34,
8388:
8337:
8150:
8046:
8020:
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7904:
7884:
7848:
7828:
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7728:
6270:
5075:
5019:
3686:
2048:
1839:
931:
863:
485:
445:
376:
13384:
13233:
13086:
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12596:
12332:
12238:
12097:
12082:
11963:
11938:
11547:
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11278:
11217:
11202:
11197:
11174:
11056:
10543:
10517:
10067:
9003:
8974:
8587:
8547:
8448:
8420:
8311:
8284:
8264:
7952:
7511:
6722:
However, since the parameter set is fixed and there is no mention here of the cardinality of
2834:
1368:
9140:
9094:
6970:
6042:. It is not saturated, however, since it does not realise any 1-type over the countable set
3340:{\displaystyle \exists v_{1}\dots \exists v_{m}\psi (x_{1},\dots ,x_{n},v_{1},\dots ,v_{m})}
13206:
13168:
13045:
12849:
12689:
12613:
12591:
12419:
12377:
12276:
12243:
12107:
11895:
11806:
11527:
11407:
11402:
11326:
11227:
9638:
9511:
8951:
8735:
8475:
8330:
7480:
6701:
6598:
6543:
6472:
5145:
3569:
is either finite or cofinite. The corresponding concept at the level of theories is called
2802:
1144:
903:
465:
257:
169:
where logical formulas are to definable sets what equations are to varieties over a field.
95:
8801:
5907:
2944:
1521:
1408:
8:
13335:
13226:
13211:
13191:
13148:
13035:
12985:
12911:
12856:
12793:
12586:
12581:
12529:
12297:
12286:
11958:
11858:
11786:
11777:
11773:
11708:
11703:
11542:
11452:
11374:
11273:
11207:
11164:
11154:
11134:
10790:
10181:
9018:
9013:
8947:
8895:
8891:
8886:
8857:
8829:
8805:
8701:
8640:
4046:
2456:
2443:
2422:
of a structure to a subset of the original signature. The opposite relation is called an
1172:
921:
162:
10911:
9515:
8611:
are those theories in which a well-behaved notion of independence can be defined, while
13364:
13133:
13096:
13081:
13074:
13057:
12861:
12843:
12709:
12635:
12618:
12571:
12384:
12293:
12127:
12112:
12072:
12024:
12009:
11997:
11953:
11928:
11698:
11647:
11568:
11487:
11427:
11359:
11349:
11288:
11263:
11139:
11096:
11091:
10833:
10769:
10663:
10643:
10551:
10486:
10238:
10119:
10093:
9933:
9802:
9753:
9727:
9271:
9073:
8797:
8769:
8719:
8705:
8628:
8612:
7981:
7635:
7596:
7505:
7108:
6950:
6765:
are used as a general technique for constructing models that realise certain types. An
6206:
6186:
6154:
5454:
5427:
5407:
5383:
5055:
4975:
4584:
satisfies the same 1-type over the empty set. This is clear since any two real numbers
3726:
1757:
1737:
1661:
1641:
1450:
1388:
1348:
1328:
1308:
1206:
1186:
1115:
1054:
883:
840:
425:
158:
91:
27:
20:
12317:
10907:
10150:
10021:
9964:
9534:
9497:
9359:
13359:
13299:
13106:
12916:
12906:
12798:
12679:
12514:
12490:
12271:
12255:
12160:
12137:
12014:
11983:
11948:
11843:
11678:
11563:
11283:
11268:
11212:
11159:
10886:
10863:
10837:
10814:
10776:
10753:
10727:
10697:
10674:
10647:
10617:
10581:
10555:
10525:
10490:
10476:
10432:
10407:
10382:
10347:
10312:
10279:
10269:
10242:
10228:
10172:
Wilfrid Hodges (2018-05-24). "Historical
Appendix: A short history of model theory".
10154:
10123:
10111:
10025:
9968:
9925:
9875:
9837:
9794:
9757:
9745:
9689:
9652:
9619:
9609:
9539:
9363:
9263:
8907:
8853:
8727:
8685:
8467:
7646:
6180:
5200:
1168:
1164:
522:
191:
140:
125:
10972:
9806:
8954:
requires considering sets in models which appear to be uncountable when viewed from
13313:
13308:
13201:
13158:
12980:
12941:
12936:
12921:
12747:
12704:
12601:
12399:
12349:
11923:
11885:
11497:
11472:
11344:
11192:
11129:
10992:
10806:
10693:
10539:
10513:
10468:
10374:
10339:
10304:
10261:
10220:
10177:
10146:
10103:
10017:
10009:
9960:
9952:
9917:
9867:
9829:
9784:
9737:
9681:
9644:
9529:
9519:
9355:
9347:
9255:
9106:
9048:
9038:
9008:
8833:
8731:
8632:
6417:
3352:
2427:
177:, stability theory proved crucial to understanding the geometry of definable sets.
10960:
10741:
10716:
9821:
9789:
9772:
13294:
13284:
13238:
13221:
13176:
13138:
13040:
12960:
12767:
12694:
12667:
12655:
12561:
12475:
12449:
12404:
12372:
12173:
11975:
11918:
11868:
11833:
11791:
11437:
11364:
11293:
11086:
10600:
9833:
9068:
8986:
8923:
8865:
8825:
8821:
8746:
8723:
7131:
7123:
6224:
5449:
3356:
43:
10999:, Perspectives in Mathematical Logic, Volume 8, New York: Springer-Verlag, 1985.
10996:
10524:. Studies in Logic and the Foundations of Mathematics (3rd ed.). Elsevier.
13279:
13258:
13216:
13196:
13091:
12946:
12544:
12534:
12524:
12519:
12453:
12327:
12203:
12092:
12087:
12065:
11666:
11515:
11442:
11149:
10967:
10955:
10711:
10569:
10368:
10333:
10298:
9993:
Ax, James; Kochen, Simon (1965). "Diophantine
Problems Over Local Fields: I.".
9673:
9503:
Proceedings of the
National Academy of Sciences of the United States of America
8809:
8739:
7624:
6770:
3189:
201:
99:
68:
10810:
10378:
10343:
10308:
9685:
9648:
109:
The most prominent scholarly organization in the field of model theory is the
13378:
13253:
12931:
12438:
12223:
12213:
12183:
12168:
11838:
11303:
11235:
11187:
10798:
10158:
10115:
9929:
9798:
9749:
9623:
9267:
8939:
8903:
8861:
8789:
8754:
8726:
proof of as special case of Artin's conjecture on diophantine equations, the
8501:
7707:
7616:
3228:
to an existential first-order formula, i.e. a formula of the following form:
2487:
1702:
with respect to the interpretation of the signature previously specified for
72:
64:
60:
10283:
9741:
9123:
Dirk van Dalen, (1980; Fifth revision 2013) "Logic and
Structure" Springer.
8970:
8813:
7595:-categorical theories and their countable models also have strong ties with
7265:
in a finite or countable signature the following conditions are equivalent:
5904:
The real number line is atomic in the language that contains only the order
13153:
12901:
12893:
12773:
12721:
12630:
12566:
12549:
12480:
12339:
12198:
11900:
11683:
11245:
11240:
11144:
10462:
10214:
9543:
9063:
8758:
8711:
6762:
4312:
3351:
where Ï is quantifier free. A theory that is not model-complete may have a
85:
79:
11002:
10472:
10224:
9524:
8910:. At the interface of finite and infinite model theory are algorithmic or
7615:
The equivalent characterisations of this subsection, due independently to
7477:
over the empty set is isolated by the pairwise order relation between the
3615:
of real numbers is not minimal: Consider, for instance, the definable set
13263:
13143:
12322:
12312:
12259:
11943:
11863:
11848:
11728:
11673:
11447:
11111:
11034:
10988:
10851:
10081:
8439:
5445:
5052:
is commonly used for the set of types over the empty set consistent with
1152:
174:
10107:
9871:
8946:
language), if it is consistent, has a countable model; this is known as
8875:
8667:. The most general semantic framework in which stability is studied are
12193:
12048:
12019:
11825:
11432:
11311:
11106:
9937:
9275:
8973:'s work on the constructible universe, which, along with the method of
8966:
4596:. The complete 2-type over the empty set realised by a pair of numbers
10049:
Ehud
Hrushovski, The Mordell-Lang Conjecture for Function Fields.
9716:"On what I do not understand and have something to say (model theory)"
8489:
has infinitely many disjoint definable subsets, each of rank at least
13345:
13248:
12301:
12218:
12178:
12142:
12078:
11890:
11880:
11853:
11616:
10985:. Notes of an introductory course for postgraduates (with exercises).
9905:
9732:
9603:
9243:
8943:
3754:
2412:
2300:
10660:
9921:
9259:
8722:, proving that the theory of finite fields is decidable, and Ax and
8528:
in which every definable set has well-defined Morley Rank is called
6777:
by identifying those tuples that agree on almost all entries, where
2726:
defines the subset of even numbers. In a similar way, formulas with
13330:
13128:
12576:
12281:
11875:
11336:
11255:
11182:
10137:
Tarski, Alfred (1954). "Contributions to the Theory of Models. I".
10098:
8849:
8715:
7611:-categorical if and only if its automorphism group is oligomorphic.
6307:
4592:
are connected by the order automorphism that shifts all numbers by
3753:
in a signature including a symbol for the order relation is called
925:
122:
8989:
and the continuum hypothesis from the other axioms of set theory.
8643:
do not in general hold for these logics. This is made concrete by
1075:, which takes the sentences in the set as its axioms. A theory is
180:
12926:
11718:
11121:
10265:
6792:. An ultraproduct of copies of the same structure is known as an
5845:
can be mapped to each other by such an automorphism. A structure
4972:
is the empty set, then the type space only depends on the theory
7989:
in a countable signature falls in one of the following classes:
5506:, and the type-definable sets are exactly the affine varieties.
10661:
Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang (1994).
8714:
construction led to new applications in algebra. This includes
8581:
2437:
2034:{\displaystyle {\mathcal {B}}\models \varphi (a_{1},...,a_{n})}
1963:{\displaystyle {\mathcal {A}}\models \varphi (a_{1},...,a_{n})}
10461:
Baldwin, John T. (2018-01-19). "Model theory and set theory".
8981:
can be shown to prove the (again philosophically interesting)
2363:
between the domains which can be written as an isomorphism of
2208:
is an elementary substructure of the field of complex numbers
12470:
11816:
11661:
10070:
173:3 (2011), pp. 1779â1840. doi=10.4007/annals.2011.173.3.11
9605:
Classification theory and the number of non-isomorphic models
3208:
is model-complete if and only if every first-order formula Ï(
1345:
are 0-ary function symbols (also known as constant symbols),
1176:
144:
8836:, the algebra of logic, the theory of definability, and the
7627:, are sometimes referred to as the Ryll-Nardzewski theorem.
6489:
be a countable set of non-isolated types over the empty set.
4857:
free variables that are realised in an elementary extension
88:
is about the sacred, then model theory is about the profane"
1726:. (Again, not to be confused with the formal notion of an "
10976:. The Stanford Encyclopedia Of Philosophy, E. Zalta (ed.).
10964:. The Stanford Encyclopedia Of Philosophy, E. Zalta (ed.).
9828:, Cambridge: Cambridge University Press, pp. 97â112,
4790:
depends on its value rounded down to the nearest integer.
3002:
has quantifier elimination if every first-order formula Ï(
10332:
Ebbinghaus, Heinz-Dieter; Flum, Jörg (1995). "0-1 Laws".
10010:"Ultrafilters and Ultraproducts in Non-Standard Analysis"
7039:{\displaystyle {\mathcal {M}}_{i}\models \varphi (a_{i})}
6068:
to be larger than any integer. The rational number line
3814:
10464:
Model Theory and the
Philosophy of Mathematical Practice
10216:
Model Theory and the
Philosophy of Mathematical Practice
9680:, Cambridge: Cambridge University Press, pp. 3â24,
8796:, in 1954. However some earlier research, especially in
8704:
and of the definable relations on real-closed fields as
7257:
can be characterised by properties of their type space:
4154:, one can consider the set of all first-order formulas
2998:
a crucial tool for analysing definable sets: A theory
2676:
defines the subset of prime numbers, while the formula
2462:
Another cornerstone of first-order model theory is the
991:{\displaystyle {\mathcal {N}}\models \varphi (n)\iff n}
9113:. Metaphysics Research Lab, Stanford University. 2020.
6615:
be a set of complete types over a given parameter set
4786:
of integers, the 1-type of a non-integer real number
3672:{\displaystyle \varphi (x)\;=\;\exists y(y\times y=x)}
924:. Tarski gave a rigorous definition, sometimes called
78:
Compared to other areas of mathematical logic such as
8876:
Connections to related branches of mathematical logic
8590:
8550:
8423:
8391:
8364:
8340:
8314:
8287:
8281:-stable, it is stable in every infinite cardinal, so
8267:
8213:
8173:
8153:
8113:
8069:
8049:
8023:
7999:
7955:
7927:
7907:
7887:
7851:
7831:
7801:
7773:
7753:
7731:
7574:
7534:
7514:
7483:
7428:
7386:
7352:
7181:
7175:
is bigger than the cardinality of the language (i.e.
6999:
6973:
6953:
6922:
6861:
6810:
6728:
6704:
6680:
6656:
6621:
6601:
6577:
6546:
6522:
6498:
6475:
6451:
6369:
6316:
6273:
6233:
6209:
6189:
6157:
6126:
6096:
6074:
6048:
6002:
5980:
5934:
5910:
5879:
5851:
5801:
5755:
5720:
5690:
5662:
5634:
5604:
5576:
5552:
5528:
5457:
5430:
5410:
5386:
5343:
5312:
5287:
5257:
5209:
5183:
5148:
5122:
5098:
5078:
5058:
5022:
4998:
4978:
4931:
4887:
4863:
4827:
4799:
4762:
4722:
4682:
4642:
4602:
4562:
4540:
4487:
4441:
4395:
4349:
4321:
4268:
4219:
4160:
4136:
4108:
4062:
4019:
4013:
interprets another whose theory is undecidable, then
3995:
3958:
3934:
3901:
3871:
3836:
3793:
3763:
3735:
3709:
3689:
3624:
3599:
3551:
3521:
3497:
3428:
3419:
is definable with parameters: Simply use the formula
3373:
3237:
3044:
2967:
2947:
2906:
2881:
2837:
2805:
2770:
2736:
2685:
2521:
2496:
2393:
2369:
2333:
2309:
2280:
2258:
2236:
2214:
2185:
2149:
2119:
2095:
2071:
2051:
1976:
1905:
1878:
1842:
1810:
1786:
1760:
1740:
1708:
1684:
1664:
1644:
1620:
1573:
1544:
1524:
1502:
1473:
1453:
1431:
1411:
1391:
1371:
1351:
1331:
1311:
1239:
1209:
1189:
1132:. A complete theory is a theory that contains every
1118:
1088:
1057:
1006:
957:
934:
906:
886:
866:
843:
819:
520:
488:
468:
448:
428:
399:
379:
356:
333:
295:
260:
210:
10637:
9672:
Barwise, J. (2016), Barwise, J; Feferman, S (eds.),
8812:, but it was first published in 1930, as a lemma in
8443:
is an important dimension notion for definable sets
7678:
is κ-categorical for all uncountable cardinals
6796:. The key to using ultraproducts in model theory is
1040:{\displaystyle {\mathcal {N}}\models \psi (n)\iff n}
10850:
7694:-categorical and uncountably categorical is called
7148:has become a key concept in model theory. A theory
6177:
Stone's representation theorem for Boolean algebras
5337:that can be expressed as exactly those elements of
1298:{\displaystyle \sigma _{or}=(0,1,+,\times ,-,<)}
10982:An introduction to Good old fashioned model theory
10768:
10715:
10662:
9771:Buechler, Steven; Lessmann, Olivier (2002-10-08).
8772:describe exactly those definable classes that are
8596:
8556:
8429:
8397:
8377:
8346:
8320:
8293:
8273:
8233:
8199:
8159:
8133:
8095:
8055:
8029:
8005:
7961:
7940:
7913:
7893:
7857:
7837:
7817:
7783:
7759:
7737:
7587:
7558:
7520:
7496:
7469:
7406:
7372:
7210:
7167:are isomorphic. It turns out that the question of
7038:
6985:
6959:
6939:
6908:
6827:
6738:
6710:
6690:
6666:
6640:
6607:
6587:
6552:
6532:
6508:
6481:
6461:
6408:
6355:
6279:
6259:
6215:
6195:
6163:
6143:
6104:
6082:
6056:
6034:
5988:
5966:
5916:
5889:
5861:
5833:
5787:
5749:, it is generally not true that any two sequences
5730:
5706:
5672:
5644:
5620:
5586:
5562:
5538:
5498:
5436:
5416:
5392:
5360:
5329:
5295:
5274:{\displaystyle \mathbb {Z} \subseteq \mathbb {R} }
5273:
5243:
5191:
5154:
5134:
5108:
5084:
5064:
5044:
5008:
4984:
4960:
4909:contains every such formula or its negation, then
4897:
4873:
4837:
4809:
4779:{\displaystyle \mathbb {Z} \subseteq \mathbb {R} }
4778:
4748:
4708:
4668:
4628:
4576:
4548:
4519:
4473:
4427:
4381:
4331:
4300:
4251:
4201:
4146:
4118:
4094:
4029:
4005:
3973:
3944:
3916:
3887:
3853:
3803:
3779:
3745:
3717:
3695:
3671:
3607:
3561:
3537:
3507:
3472:
3411:
3339:
3161:
2975:
2953:
2930:
2889:
2856:
2823:
2788:
2753:
2715:
2665:
2504:
2403:
2379:
2343:
2319:
2288:
2266:
2244:
2222:
2200:
2165:
2135:
2105:
2081:
2057:
2033:
1962:
1888:
1848:
1820:
1796:
1766:
1746:
1718:
1694:
1670:
1650:
1630:
1603:
1559:
1530:
1510:
1488:
1459:
1439:
1417:
1397:
1377:
1357:
1337:
1317:
1297:
1223:(not to be confused with the formal notion of an "
1215:
1195:
1124:
1104:
1063:
1039:
990:
940:
912:
892:
872:
849:
829:
798:
503:
474:
454:
434:
414:
385:
365:
342:
319:
278:
246:
10303:. Perspectives in Mathematical Logic. p. v.
10007:
9348:"I: A General Method in Proofs of Undecidability"
8965:The model-theoretic viewpoint has been useful in
8334:implies that if there is an uncountable cardinal
5996:are isolated by the order relations between the
2490:are important objects of study. For instance, in
1158:
1151:) that a theory has a model if and only if it is
102:in nature, in contrast to model theory, which is
13376:
10687:
9770:
7979:A fundamental result in stability theory is the
7649:showed in 1963 that there is only one notion of
7630:In combinatorial signatures, a common source of
1638:is said to model a set of first-order sentences
837:of the natural numbers, for example, an element
10876:
10402:Kunen, Kenneth (2011). "Models of set theory".
9498:"On theories categorical in uncountable powers"
8738:, which aims to provide a rigorous calculus of
8096:{\displaystyle \lambda ^{\aleph _{0}}=\lambda }
7070:are elementary equivalent, then there is a set
6428:
5684:if it realises every type over a parameter set
3204:. It follows from this criterion that a theory
320:{\displaystyle \neg ,\land ,\lor ,\rightarrow }
181:Fundamental notions of first-order model theory
10366:
10331:
10296:
10171:
8619:
5628:. However, any proper elementary extension of
3473:{\displaystyle x=a_{1}\vee \dots \vee x=a_{n}}
2166:{\displaystyle {\mathcal {B}}\models \varphi }
2136:{\displaystyle {\mathcal {A}}\models \varphi }
11632:
11018:
10550:. Dover Books on Mathematics (3rd ed.).
10367:Ebbinghaus, Heinz-Dieter; Flum, Jörg (1995).
10297:Ebbinghaus, Heinz-Dieter; Flum, Jörg (1995).
9674:"Model-Theoretic Logics: Background and Aims"
8412:
6469:be a theory in a countable signature and let
5546:as the parameter set, then every 1-type over
5281:that is not realised in the real number line
10795:A Concise Introduction to Mathematical Logic
10538:
10512:
10176:. By Button, Tim; Walsh, Sean. p. 439.
10080:CHASE, HUNTER; FREITAG, JAMES (2019-02-15).
10079:
10051:Journal of the American Mathematical Society
9777:Journal of the American Mathematical Society
8730:. The ultraproduct construction also led to
8572:, defined as the Morley rank of the type of
8301:-stability is stronger than superstability.
8200:{\displaystyle \lambda \geq 2^{\aleph _{0}}}
7171:-categoricity depends critically on whether
6403:
6370:
6350:
6317:
6299:is an isolated point in the Stone topology.
6254:
6234:
5244:{\displaystyle \{n<x|n\in \mathbb {Z} \}}
5238:
5210:
4202:{\displaystyle \varphi (x_{1},\dots ,x_{n})}
3406:
3374:
2438:Compactness and the Löwenheim-Skolem theorem
11032:
10906:
9904:Doner, John; Hodges, Wilfrid (March 1988).
9903:
9241:
7985:, which implies that every complete theory
7670:-categorical for some uncountable cardinal
7641:
3811:is a finite union of points and intervals.
2867:Both of the definitions mentioned here are
1604:{\displaystyle (\mathbb {Q} ,\sigma _{or})}
11824:
11639:
11625:
11025:
11011:
10789:
8844:students developed in the 1950s and '60s.
7974:
5741:While an automorphism that is constant on
3641:
3637:
3224:) over its signature is equivalent modulo
3018:) over its signature is equivalent modulo
2986:
2201:{\displaystyle {\overline {\mathbb {Q} }}}
1658:in the given language if each sentence in
1405:is a unary (= 1-ary) function symbol, and
1033:
1029:
984:
980:
10827:
10640:Models and Ultraproducts: An Introduction
10638:Bell, John L.; Slomson, Alan B. (2006) .
10594:
10097:
10008:Cherlin, Greg; Hirschfeld, Joram (1972),
9788:
9731:
9533:
9523:
8540:is totally transcendental if and only if
8257:and a theory of the third type is called
7391:
7357:
7336:free variables, up to equivalence modulo
6773:of a set of structures over an index set
6183:, which consists exactly of the complete
6098:
6076:
6050:
5982:
5869:in which this converse does hold for all
5289:
5267:
5259:
5234:
5185:
5135:{\displaystyle \varphi \rightarrow \psi }
4772:
4764:
4570:
4542:
3780:{\displaystyle A\subseteq {\mathcal {M}}}
3711:
3601:
3538:{\displaystyle A\subseteq {\mathcal {M}}}
2969:
2883:
2498:
2282:
2260:
2238:
2216:
2189:
1578:
1547:
1504:
1476:
1433:
34:is the study of the relationship between
9992:
9601:
8958:the model, but are countable to someone
8679:
8253:, a theory of the second type is called
7122:has an infinite model for some infinite
6641:{\displaystyle A\subset {\mathcal {M}}.}
3367:In every structure, every finite subset
3181:elimination a weaker property suffices:
1233:A common signature for ordered rings is
10740:
10460:
10212:
9861:
9671:
9111:The Stanford Encyclopedia of Philosophy
7818:{\displaystyle A\subset {\mathcal {M}}}
7607:in a finite or countable signature is
5928:-types over the empty set realised by
5707:{\displaystyle A\subset {\mathcal {M}}}
5509:
4961:{\displaystyle S_{n}^{\mathcal {M}}(A)}
2761:. For example, in a field, the formula
1385:are binary (= 2-ary) function symbols,
1105:{\displaystyle {\mathcal {M}}\models T}
1071:of sentences is called a (first-order)
13377:
11646:
10923:
10766:
10710:
10607:
10568:
10506:
10373:. Perspectives in Mathematical Logic.
10338:. Perspectives in Mathematical Logic.
10258:Mathematical logic in the 20th century
10136:
10014:Contributions to Non-Standard Analysis
9906:"Alfred Tarski and Decidable Theories"
9819:
9713:
9636:
9580:
9568:
9556:
9495:
9420:
9399:
9387:
9345:
9324:
9312:
9300:
9288:
9244:"Alfred Tarski and Decidable Theories"
9229:
9217:
9205:
9193:
9172:
9160:
8880:
8710:In the 1960s, the introduction of the
8134:{\displaystyle \lambda ^{\aleph _{0}}}
7666:in a finite or countable signature is
6650:Then there is an elementary extension
6423:
3815:Definable and interpretable structures
3412:{\displaystyle \{a_{1},\dots ,a_{n}\}}
2716:{\displaystyle \exists y(2\times y=x)}
90:. The applications of model theory to
11620:
11006:
10426:
10401:
10255:
9950:
9242:Doner, John; Hodges, Wilfrid (1988).
8249:A theory of the first type is called
7653:for theories in countable languages.
7470:{\displaystyle p(x_{1},\dots ,x_{n})}
7211:{\displaystyle \aleph _{0}+|\sigma |}
6437:it, and is generally possible by the
6409:{\displaystyle \{p:f(x)\neq 0\in p\}}
5594:is isolated by a formula of the form
2179:While the field of algebraic numbers
128:model theory of infinite structures.
11587:
10900:
6839:-structures indexed by an index set
5714:that is of smaller cardinality than
5380:. For an algebraic example, suppose
4527:realise the same complete type over
3895:are definable if there is a formula
3192:if every substructure of a model of
185:
11599:
10082:"Model Theory and Machine Learning"
9826:Lectures on Infinitary Model Theory
7921:. Traditionally, theories that are
7901:-stable for some infinite cardinal
7701:
7407:{\displaystyle (\mathbb {R} ,<)}
7373:{\displaystyle (\mathbb {Q} ,<)}
6916:is true in the ultraproduct of the
6260:{\displaystyle \{p|\varphi \in p\}}
5621:{\displaystyle a\in {\mathcal {M}}}
3888:{\displaystyle a\in {\mathcal {M}}}
2469:In a certain sense made precise by
2426:- e.g. the (additive) group of the
13:
10690:Fundamentals of Mathematical Logic
10631:
10182:10.1093/oso/9780198790396.003.0018
8591:
8220:
8186:
8120:
8076:
7929:
7810:
7776:
7576:
7504:. This means that every countable
7261:For a complete first-order theory
7183:
7003:
6940:{\displaystyle {\mathcal {M}}_{i}}
6926:
6828:{\displaystyle {\mathcal {M}}_{i}}
6814:
6750:of set theory, and is true if the
6731:
6705:
6683:
6659:
6630:
6602:
6580:
6547:
6525:
6501:
6476:
6454:
6439:(Countable) Omitting types theorem
6144:{\displaystyle {\mathcal {M}}^{n}}
6130:
6090:is saturated, in contrast, since
6035:{\displaystyle a_{1},\dots ,a_{n}}
5967:{\displaystyle a_{1},\dots ,a_{n}}
5882:
5854:
5834:{\displaystyle b_{1},\dots ,b_{n}}
5788:{\displaystyle a_{1},\dots ,a_{n}}
5723:
5699:
5665:
5637:
5613:
5579:
5555:
5531:
5361:{\displaystyle {\mathcal {M}}^{n}}
5347:
5330:{\displaystyle {\mathcal {M}}^{n}}
5316:
5101:
5001:
4943:
4890:
4866:
4830:
4802:
4520:{\displaystyle b_{1},\dots ,b_{n}}
4474:{\displaystyle a_{1},\dots ,a_{n}}
4428:{\displaystyle b_{1},\dots ,b_{n}}
4382:{\displaystyle a_{1},\dots ,a_{n}}
4324:
4301:{\displaystyle a_{1},\dots ,a_{n}}
4252:{\displaystyle a_{1},\dots ,a_{n}}
4139:
4111:
4095:{\displaystyle a_{1},\dots ,a_{n}}
4022:
3998:
3937:
3880:
3854:{\displaystyle {\mathcal {M}}^{n}}
3840:
3830:free variables define subsets of
3796:
3772:
3738:
3642:
3585:is minimal. A structure is called
3554:
3530:
3515:is called minimal if every subset
3500:
3254:
3238:
3061:
3045:
2754:{\displaystyle {\mathcal {M}}^{n}}
2740:
2686:
2606:
2576:
2537:
2528:
2522:
2396:
2372:
2336:
2312:
2152:
2122:
2098:
2074:
1979:
1908:
1881:
1813:
1789:
1711:
1687:
1623:
1091:
1009:
960:
822:
699:
693:
619:
589:
550:
541:
535:
357:
334:
296:
14:
13406:
10597:Model Theory and Its Applications
8455:The Morley rank is at least 0 if
8447:within a model. It is defined by
7243:
6356:{\displaystyle \{p:f(x)=0\in p\}}
5072:. If there is a single formula
4577:{\displaystyle a\in \mathbb {R} }
2730:free variables define subsets of
2481:
1730:" of one structure in another) A
13358:
11598:
11586:
11575:
11574:
11562:
8765:for products of Modular curves.
6757:
6752:generalised continuum hypothesis
6120:The set of definable subsets of
5841:that satisfy the same type over
5745:will always preserve types over
4051:
2931:{\displaystyle y=x\times x+\pi }
1560:{\displaystyle \mathbb {Q} ^{2}}
1489:{\displaystyle \mathbb {Q} ^{2}}
1227:" of one structure in another).
928:, for the satisfaction relation
11483:Computational complexity theory
10803:Springer Science+Business Media
10454:
10445:
10420:
10395:
10360:
10325:
10290:
10249:
10213:Baldwin, John T. (2018-01-19).
10206:
10197:
10188:
10165:
10130:
10073:
10056:
10043:
10001:
9995:American Journal of Mathematics
9986:
9953:"Ultraproducts for Algebraists"
9944:
9897:
9888:
9855:
9813:
9764:
9707:
9665:
9630:
9595:
9586:
9574:
9562:
9550:
9489:
9480:
9471:
9462:
9453:
9444:
9435:
9426:
9414:
9405:
9393:
9381:
9339:
9330:
9318:
9306:
9294:
9282:
9235:
9223:
9211:
9199:
8749:'s 1996 proof of the geometric
8234:{\displaystyle 2^{\aleph _{0}}}
7113:A theory was originally called
7102:
6115:
4636:depends on their order: either
3826:signature. Since formulas with
3787:definable with parameters from
3545:definable with parameters from
3487:This leads to the concept of a
2476:
1836:if for any first-order formula
900:is a prime number. The formula
10467:. Cambridge University Press.
10260:. Singapore University Press.
10219:. Cambridge University Press.
10086:The Bulletin of Symbolic Logic
9957:HANDBOOK OF MATHEMATICAL LOGIC
9187:
9178:
9166:
9154:
9145:
9133:
9117:
9099:
9087:
8504:, the Morley rank is at least
8470:, the Morley rank is at least
8261:. Furthermore, if a theory is
7784:{\displaystyle {\mathcal {M}}}
7603:A complete first-order theory
7464:
7432:
7401:
7387:
7380:, which is also the theory of
7367:
7353:
7204:
7196:
7140:However, the weaker notion of
7126:, then it has a model of size
7033:
7020:
6903:
6900:
6885:
6871:
6868:
6865:
6739:{\displaystyle {\mathcal {N}}}
6691:{\displaystyle {\mathcal {M}}}
6667:{\displaystyle {\mathcal {N}}}
6588:{\displaystyle {\mathcal {M}}}
6533:{\displaystyle {\mathcal {T}}}
6509:{\displaystyle {\mathcal {M}}}
6462:{\displaystyle {\mathcal {T}}}
6388:
6382:
6335:
6329:
6241:
5890:{\displaystyle {\mathcal {M}}}
5862:{\displaystyle {\mathcal {M}}}
5731:{\displaystyle {\mathcal {M}}}
5673:{\displaystyle {\mathcal {M}}}
5645:{\displaystyle {\mathcal {M}}}
5587:{\displaystyle {\mathcal {M}}}
5563:{\displaystyle {\mathcal {M}}}
5539:{\displaystyle {\mathcal {M}}}
5493:
5461:
5368:realising a certain type over
5223:
5126:
5109:{\displaystyle {\mathcal {M}}}
5039:
5033:
5009:{\displaystyle {\mathcal {M}}}
4955:
4949:
4898:{\displaystyle {\mathcal {M}}}
4874:{\displaystyle {\mathcal {N}}}
4838:{\displaystyle {\mathcal {M}}}
4810:{\displaystyle {\mathcal {M}}}
4749:{\displaystyle a_{2}<a_{1}}
4669:{\displaystyle a_{1}<a_{2}}
4332:{\displaystyle {\mathcal {M}}}
4196:
4164:
4147:{\displaystyle {\mathcal {M}}}
4119:{\displaystyle {\mathcal {M}}}
4030:{\displaystyle {\mathcal {M}}}
4006:{\displaystyle {\mathcal {M}}}
3968:
3962:
3945:{\displaystyle {\mathcal {M}}}
3911:
3905:
3804:{\displaystyle {\mathcal {M}}}
3746:{\displaystyle {\mathcal {M}}}
3666:
3648:
3634:
3628:
3562:{\displaystyle {\mathcal {M}}}
3508:{\displaystyle {\mathcal {M}}}
3334:
3270:
3156:
3153:
3121:
3115:
3112:
3080:
3074:
2873:with parameters from the model
2818:
2806:
2710:
2692:
2636:
2633:
2630:
2612:
2600:
2582:
2573:
2570:
2567:
2543:
2534:
2404:{\displaystyle {\mathcal {B}}}
2380:{\displaystyle {\mathcal {A}}}
2344:{\displaystyle {\mathcal {B}}}
2320:{\displaystyle {\mathcal {A}}}
2106:{\displaystyle {\mathcal {B}}}
2089:an elementary substructure of
2082:{\displaystyle {\mathcal {A}}}
2028:
1990:
1957:
1919:
1889:{\displaystyle {\mathcal {A}}}
1821:{\displaystyle {\mathcal {B}}}
1797:{\displaystyle {\mathcal {A}}}
1719:{\displaystyle {\mathcal {N}}}
1695:{\displaystyle {\mathcal {N}}}
1631:{\displaystyle {\mathcal {N}}}
1598:
1574:
1292:
1256:
1159:Basic model-theoretic concepts
1143:It's a consequence of Gödel's
1030:
1026:
1020:
981:
977:
971:
926:"Tarski's definition of truth"
830:{\displaystyle {\mathcal {N}}}
765:
762:
750:
744:
732:
729:
726:
708:
705:
649:
646:
643:
625:
613:
595:
586:
583:
580:
556:
547:
498:
492:
409:
403:
314:
241:
232:
220:
214:
111:Association for Symbolic Logic
46:expressing statements about a
1:
13319:History of mathematical logic
10879:An Invitation to Model Theory
10614:Graduate Texts in Mathematics
10610:Model Theory: An Introduction
10501:
10151:10.1016/S1385-7258(54)50074-0
10022:10.1016/s0049-237x(08)71563-5
9965:10.1016/s0049-237x(08)71099-1
9910:The Journal of Symbolic Logic
9790:10.1090/s0894-0347-02-00407-1
9360:10.1016/s0049-237x(09)70292-7
9248:The Journal of Symbolic Logic
8933:
8920:descriptive complexity theory
8635:is hampered by the fact that
8584:) or merely stable (Shelah's
8328:-stable. More generally, the
7825:of cardinality not exceeding
7658:Morley's categoricity theorem
7236:-cardinality for uncountable
7144:-categoricity for a cardinal
7082:such that the ultrapowers by
6698:which realises every type in
6289:Stone space of n-types over A
4261:complete (n-)type realised by
3821:Interpretation (model theory)
3593:On the other hand, the field
3362:
1832:A substructure is said to be
1147:(not to be confused with his
948:, so that one easily proves:
327:and prefixing of quantifiers
13244:Primitive recursive function
10927:Lecture Notes â Model Theory
10913:Introduction to Model Theory
10830:Introduction to Model Theory
10642:(reprint of 1974 ed.).
9862:Baldwin, John (2009-07-24).
9834:10.1017/cbo9781316855560.009
8900:Gödel's completeness theorem
8838:semantic definition of truth
8665:complex exponential function
8378:{\displaystyle 2^{\lambda }}
7559:{\displaystyle x=1+\dots +1}
6429:Realising and omitting types
6105:{\displaystyle \mathbb {Q} }
6083:{\displaystyle \mathbb {Q} }
6057:{\displaystyle \mathbb {Z} }
5989:{\displaystyle \mathbb {R} }
5873:of smaller cardinality than
5652:contains an element that is
5296:{\displaystyle \mathbb {R} }
5192:{\displaystyle \mathbb {R} }
4549:{\displaystyle \mathbb {R} }
3718:{\displaystyle \mathbb {R} }
3608:{\displaystyle \mathbb {R} }
3038:) without quantifiers, i.e.
2976:{\displaystyle \mathbb {R} }
2890:{\displaystyle \mathbb {R} }
2505:{\displaystyle \mathbb {N} }
2289:{\displaystyle \mathbb {Q} }
2267:{\displaystyle \mathbb {C} }
2245:{\displaystyle \mathbb {Q} }
2223:{\displaystyle \mathbb {C} }
2193:
1829:multiplication and inverse.
1511:{\displaystyle \mathbb {Q} }
1440:{\displaystyle \mathbb {Q} }
7:
10944:Encyclopedia of Mathematics
10828:Rothmaler, Philipp (2000).
10174:Philosophy and model theory
9773:"Simple homogeneous models"
9184:Barwise and Feferman, p. 43
8996:
8694:algebraically closed fields
8669:abstract elementary classes
8649:Zermelo-Fraenkel set theory
8620:Non-elementary model theory
7941:{\displaystyle \aleph _{0}}
7588:{\displaystyle \aleph _{0}}
7130:for any sufficiently large
6295:is isolated if and only if
4709:{\displaystyle a_{1}=a_{2}}
4629:{\displaystyle a_{1},a_{2}}
4056:For a sequence of elements
3974:{\displaystyle \varphi (a)}
3917:{\displaystyle \varphi (x)}
3196:which is itself a model of
3022:to a first-order formula Ï(
2789:{\displaystyle y=x\times x}
1754:is a structure that models
1567:), one obtains a structure
442:is the unbound variable in
415:{\displaystyle \varphi (x)}
247:{\displaystyle R(f(x,y),z)}
116:
10:
13413:
12308:SchröderâBernstein theorem
12035:Monadic predicate calculus
11694:Foundations of mathematics
11533:Films about mathematicians
10883:Cambridge University Press
10860:Cambridge University Press
10854:; Ziegler, Martin (2012).
10724:Cambridge University Press
10578:Cambridge University Press
9044:Institutional model theory
8884:
8783:
8757:applied techniques around
8676:ongoing research program.
8413:Geometric stability theory
8241:is the cardinality of the
7982:stability spectrum theorem
7705:
7634:-categorical theories are
7106:
6909:{\displaystyle \varphi ()}
6540:which omits every type in
6306:: a set of types is basic
5444:corresponds to the set of
5402:algebraically closed field
4044:
3818:
189:
18:
16:Area of mathematical logic
13354:
13341:Philosophy of mathematics
13290:Automated theorem proving
13272:
13167:
12999:
12892:
12744:
12461:
12437:
12415:Von NeumannâBernaysâGödel
12360:
12254:
12158:
12056:
12047:
11974:
11909:
11815:
11737:
11654:
11556:
11506:
11463:
11373:
11335:
11302:
11254:
11226:
11173:
11120:
11102:Philosophy of mathematics
11077:
11042:
10811:10.1007/978-1-4419-1221-3
10688:Hinman, Peter G. (2005).
10379:10.1007/978-3-662-03182-7
10344:10.1007/978-3-662-03182-7
10309:10.1007/978-3-662-03182-7
10139:Indagationes Mathematicae
9822:"Quasiminimal excellence"
9686:10.1017/9781316717158.004
9649:10.1007/978-94-017-3002-0
9602:Saharon., Shelah (1990).
9336:Hodges (1993), pp. 31, 92
8942:(which is expressed in a
8828:. Tarski's work included
8671:, which are defined by a
7314:, the number of formulas
7310:For every natural number
7292:For every natural number
6416:. This is finer than the
4793:More generally, whenever
2799:defines the curve of all
2327:into another Ï-structure
366:{\displaystyle \exists v}
343:{\displaystyle \forall v}
11538:Recreational mathematics
10979:Simmons, Harold (2004),
10973:First-order Model theory
10877:Kirby, Jonathan (2019).
10856:A Course in Model Theory
10771:A Course in Model Theory
10431:. College Publications.
10406:. College Publications.
10053:9:3 (1996), pp. 667-690.
9714:Shelah, Saharon (2000).
9496:Morley, Michael (1963).
9477:Bell and Slomson, p. 102
9080:
9054:LöwenheimâSkolem theorem
8808:was implicit in work by
8778:online learnable classes
8661:quasiminimally excellent
8657:homogeneous model theory
8398:{\displaystyle \lambda }
8347:{\displaystyle \lambda }
8160:{\displaystyle \lambda }
8056:{\displaystyle \lambda }
8030:{\displaystyle \lambda }
8006:{\displaystyle \lambda }
7914:{\displaystyle \lambda }
7894:{\displaystyle \lambda }
7858:{\displaystyle \lambda }
7838:{\displaystyle \lambda }
7760:{\displaystyle \lambda }
7738:{\displaystyle \lambda }
7662:If a first-order theory
7651:uncountable categoricity
7642:Uncountable categoricity
7163:that are of cardinality
6280:{\displaystyle \varphi }
6179:there is a natural dual
5092:such that the theory of
5085:{\displaystyle \varphi }
5045:{\displaystyle S_{n}(T)}
4040:
3696:{\displaystyle \varphi }
2464:Löwenheim-Skolem theorem
2058:{\displaystyle \varphi }
1849:{\displaystyle \varphi }
1138:theory of that structure
941:{\displaystyle \models }
873:{\displaystyle \varphi }
504:{\displaystyle \psi (x)}
455:{\displaystyle \varphi }
386:{\displaystyle \varphi }
12991:Self-verifying theories
12812:Tarski's axiomatization
11763:Tarski's undefinability
11758:incompleteness theorems
11423:Mathematical statistics
11413:Mathematical psychology
11383:Engineering mathematics
11317:Algebraic number theory
10750:Oxford University Press
10427:Kunen, Kenneth (2011).
9951:Eklof, Paul C. (1977),
9894:Hodges (1993), p. 68-69
9742:10.4064/fm-166-1-2-1-82
9720:Fundamenta Mathematicae
9346:Tarski, Alfred (1953),
9059:Model-theoretic grammar
8912:computable model theory
8870:MordellâLang conjecture
8751:Mordell-Lang conjecture
8597:{\displaystyle \infty }
8557:{\displaystyle \omega }
8430:{\displaystyle \omega }
8321:{\displaystyle \omega }
8294:{\displaystyle \omega }
8274:{\displaystyle \omega }
8105:Cardinal exponentiation
8063:-stable if and only if
7993:There are no cardinals
7975:The stability hierarchy
7962:{\displaystyle \omega }
7521:{\displaystyle \omega }
7418:-categorical, as every
6748:Zermelo-Fraenkel axioms
6595:be a structure and let
5424:-types over a subfield
5177:Since the real numbers
4037:itself is undecidable.
3928:is the only element of
3727:densely totally ordered
3169:holds in all models of
2987:Eliminating quantifiers
2857:{\displaystyle y=x^{2}}
2387:with a substructure of
1378:{\displaystyle \times }
1149:incompleteness theorems
511:), defined as follows:
13365:Mathematics portal
12976:Proof of impossibility
12624:propositional variable
11934:Propositional calculus
11569:Mathematics portal
11418:Mathematical sociology
11398:Mathematical economics
11393:Mathematical chemistry
11322:Analytic number theory
11203:Differential equations
10997:Model-Theoretic Logics
10924:Pillay, Anand (2002).
10767:Poizat, Bruno (2000).
10608:Marker, David (2002).
10595:Kopperman, R. (1972).
10574:A shorter model theory
10256:Sacks, Gerald (2003).
9820:Marker, David (2016),
9678:Model-Theoretic Logics
9637:Wagner, Frank (2011).
9029:Elementary equivalence
9024:Descriptive complexity
8928:formal language theory
8598:
8558:
8530:totally transcendental
8431:
8399:
8385:models of cardinality
8379:
8348:
8322:
8295:
8275:
8235:
8201:
8161:
8135:
8107:for an explanation of
8097:
8057:
8031:
8007:
7963:
7942:
7915:
7895:
7859:
7839:
7819:
7795:and any parameter set
7785:
7761:
7739:
7690:A theory that is both
7589:
7560:
7522:
7498:
7471:
7408:
7374:
7212:
7056:Keisler-Shelah theorem
7040:
6987:
6986:{\displaystyle i\in I}
6961:
6941:
6910:
6829:
6781:is made precise by an
6740:
6712:
6692:
6668:
6642:
6609:
6589:
6554:
6534:
6510:
6492:Then there is a model
6483:
6463:
6410:
6357:
6310:iff it is of the form
6304:constructible topology
6281:
6261:
6217:
6197:
6165:
6145:
6106:
6084:
6058:
6036:
5990:
5968:
5918:
5891:
5863:
5835:
5789:
5732:
5708:
5674:
5646:
5622:
5588:
5564:
5540:
5500:
5438:
5418:
5394:
5362:
5331:
5297:
5275:
5245:
5193:
5156:
5136:
5110:
5086:
5066:
5046:
5010:
4986:
4962:
4917:. The set of complete
4899:
4875:
4839:
4811:
4780:
4750:
4710:
4670:
4630:
4578:
4550:
4521:
4475:
4429:
4383:
4333:
4302:
4253:
4213:that are satisfied by
4203:
4148:
4120:
4096:
4031:
4007:
3975:
3946:
3918:
3889:
3855:
3805:
3781:
3747:
3719:
3697:
3673:
3609:
3563:
3539:
3509:
3474:
3413:
3341:
3163:
2996:quantifier elimination
2977:
2955:
2932:
2891:
2858:
2825:
2790:
2755:
2717:
2667:
2506:
2405:
2381:
2345:
2321:
2290:
2268:
2246:
2224:
2202:
2167:
2137:
2107:
2083:
2059:
2035:
1964:
1890:
1850:
1822:
1798:
1768:
1748:
1720:
1696:
1672:
1652:
1632:
1605:
1561:
1532:
1512:
1490:
1461:
1441:
1419:
1399:
1379:
1359:
1339:
1319:
1299:
1217:
1197:
1126:
1106:
1065:
1041:
992:
942:
914:
894:
874:
851:
831:
800:
505:
476:
456:
436:
416:
387:
367:
344:
321:
280:
248:
48:mathematical structure
13234:Kolmogorov complexity
13187:Computably enumerable
13087:Model complete theory
12879:Principia Mathematica
11939:Propositional formula
11768:BanachâTarski paradox
11548:Mathematics education
11478:Theory of computation
11198:Hypercomplex analysis
10473:10.1017/9781316987216
10451:Hodges (1993), p. 272
10225:10.1017/9781316987216
10203:Hodges (1993), p. 475
10068:Annals of Mathematics
9592:Hodges (1993), p. 494
9525:10.1073/pnas.49.2.213
9486:Hodges (1993), p. 492
9468:Hodges (1993), p. 452
9459:Hodges (1993), p. 450
9441:Hodges (1993), p. 451
9432:Hodges (1993), p. 333
9411:Hodges (1993), p. 280
9151:Hodges (1997), p. vii
9004:Abstract model theory
8872:for function fields.
8763:André-Oort conjecture
8680:Selected applications
8599:
8568:over a parameter set
8559:
8493: − 1.
8449:transfinite induction
8432:
8400:
8380:
8349:
8323:
8296:
8276:
8236:
8202:
8162:
8136:
8098:
8058:
8032:
8008:
7964:
7943:
7916:
7896:
7860:
7840:
7820:
7786:
7762:
7740:
7590:
7561:
7523:
7499:
7497:{\displaystyle x_{i}}
7472:
7409:
7375:
7255:-categorical theories
7213:
7159:if any two models of
7058:provides a converse:
7041:
6988:
6962:
6942:
6911:
6830:
6769:is obtained from the
6741:
6713:
6711:{\displaystyle \Phi }
6693:
6669:
6643:
6610:
6608:{\displaystyle \Phi }
6590:
6555:
6553:{\displaystyle \Phi }
6535:
6511:
6484:
6482:{\displaystyle \Phi }
6464:
6411:
6358:
6287:. This is called the
6282:
6262:
6218:
6198:
6166:
6151:over some parameters
6146:
6107:
6085:
6059:
6037:
5991:
5969:
5919:
5892:
5864:
5836:
5790:
5733:
5709:
5675:
5647:
5623:
5589:
5565:
5541:
5501:
5439:
5419:
5395:
5363:
5332:
5298:
5276:
5246:
5194:
5157:
5155:{\displaystyle \psi }
5137:
5111:
5087:
5067:
5047:
5011:
4987:
4963:
4900:
4876:
4849:is a set of formulas
4840:
4812:
4781:
4751:
4711:
4671:
4631:
4579:
4551:
4534:The real number line
4522:
4476:
4430:
4384:
4334:
4303:
4259:. This is called the
4254:
4204:
4149:
4121:
4097:
4032:
4008:
3976:
3947:
3919:
3890:
3856:
3806:
3782:
3748:
3720:
3698:
3674:
3610:
3564:
3540:
3510:
3475:
3414:
3342:
3164:
2978:
2956:
2933:
2892:
2859:
2826:
2824:{\displaystyle (x,y)}
2791:
2756:
2718:
2668:
2507:
2406:
2382:
2346:
2322:
2291:
2269:
2247:
2230:, the rational field
2225:
2203:
2168:
2138:
2108:
2084:
2060:
2036:
1965:
1891:
1851:
1823:
1799:
1769:
1749:
1721:
1697:
1673:
1653:
1633:
1606:
1562:
1533:
1513:
1491:
1462:
1442:
1420:
1400:
1380:
1360:
1340:
1320:
1300:
1218:
1198:
1127:
1107:
1066:
1042:
993:
943:
915:
913:{\displaystyle \psi }
895:
875:
852:
832:
801:
506:
477:
475:{\displaystyle \psi }
457:
437:
417:
388:
368:
345:
322:
281:
279:{\displaystyle y=x+1}
249:
121:This page focuses on
13182:ChurchâTuring thesis
13169:Computability theory
12378:continuum hypothesis
11896:Square of opposition
11754:Gödel's completeness
11528:Informal mathematics
11408:Mathematical physics
11403:Mathematical finance
11388:Mathematical biology
11327:Diophantine geometry
10933:. pp. 61 pages.
10919:. pp. 26 pages.
10834:Taylor & Francis
10791:Rautenberg, Wolfgang
9352:Undecidable Theories
9034:First-order theories
8952:continuum hypothesis
8902:, and the method of
8818:completeness theorem
8736:nonstandard analysis
8588:
8548:
8476:elementary extension
8421:
8389:
8362:
8338:
8312:
8285:
8265:
8211:
8171:
8151:
8111:
8067:
8047:
8021:
7997:
7953:
7925:
7905:
7885:
7849:
7845:, there are at most
7829:
7799:
7771:
7751:
7729:
7572:
7532:
7512:
7481:
7426:
7384:
7350:
7179:
6997:
6971:
6951:
6920:
6859:
6808:
6726:
6702:
6678:
6654:
6619:
6599:
6575:
6544:
6520:
6496:
6473:
6449:
6367:
6314:
6271:
6267:for single formulas
6231:
6227:by sets of the form
6207:
6187:
6155:
6124:
6094:
6072:
6046:
6000:
5978:
5932:
5917:{\displaystyle <}
5908:
5877:
5849:
5799:
5753:
5718:
5688:
5660:
5632:
5602:
5574:
5550:
5526:
5510:Structures and types
5455:
5428:
5408:
5384:
5341:
5310:
5285:
5255:
5207:
5181:
5146:
5120:
5096:
5076:
5056:
5020:
4996:
4976:
4929:
4925:is often written as
4885:
4861:
4825:
4797:
4760:
4720:
4680:
4640:
4600:
4560:
4538:
4485:
4439:
4393:
4347:
4339:that is constant on
4319:
4266:
4217:
4158:
4134:
4106:
4060:
4017:
3993:
3956:
3932:
3899:
3869:
3834:
3791:
3761:
3733:
3707:
3687:
3622:
3597:
3549:
3519:
3495:
3426:
3371:
3235:
3042:
2965:
2954:{\displaystyle \pi }
2945:
2904:
2879:
2835:
2803:
2768:
2734:
2683:
2519:
2494:
2391:
2367:
2331:
2307:
2278:
2256:
2234:
2212:
2183:
2147:
2117:
2093:
2069:
2049:
1974:
1903:
1876:
1840:
1808:
1784:
1758:
1738:
1706:
1682:
1662:
1642:
1618:
1571:
1542:
1531:{\displaystyle <}
1522:
1500:
1471:
1451:
1429:
1418:{\displaystyle <}
1409:
1389:
1369:
1349:
1329:
1309:
1237:
1207:
1187:
1145:completeness theorem
1116:
1086:
1055:
1004:
955:
932:
904:
884:
864:
841:
817:
518:
486:
466:
446:
426:
397:
377:
354:
331:
293:
258:
208:
96:Diophantine geometry
13395:Mathematical proofs
13336:Mathematical object
13227:P versus NP problem
13192:Computable function
12986:Reverse mathematics
12912:Logical consequence
12789:primitive recursive
12784:elementary function
12557:Free/bound variable
12410:TarskiâGrothendieck
11929:Logical connectives
11859:Logical equivalence
11709:Logical consequence
11543:Mathematics and art
11453:Operations research
11208:Functional analysis
10507:Canonical textbooks
10370:Finite Model Theory
10335:Finite Model Theory
10300:Finite Model Theory
10108:10.1017/bsl.2018.71
9516:1963PNAS...49..213M
9139:Chang and Keisler,
9093:Chang and Keisler,
9019:Compactness theorem
9014:Axiomatizable class
8896:compactness theorem
8892:Finite model theory
8887:Finite model theory
8881:Finite model theory
8860:, Shelah developed
8830:logical consequence
8806:compactness theorem
8720:pseudofinite fields
8702:algebraic varieties
8673:strong substructure
8645:Lindstrom's theorem
8629:higher-order logics
8536:is countable, then
8354:such that a theory
7948:-stable are called
7877:A theory is called
7714:stability hierarchy
7696:totally categorical
7597:oligomorphic groups
7074:and an ultrafilter
6424:Constructing models
4948:
4817:is a structure and
4435:respectively, then
4209:with parameters in
4047:Type (model theory)
2983:to define a curve.
2941:uses the parameter
2875:. For instance, in
2471:Lindström's theorem
2457:compactness theorem
2444:compactness theorem
1467:is a function from
1173:non-logical symbols
288:Boolean connectives
13390:Mathematical logic
13134:Transfer principle
13097:Semantics of logic
13082:Categorical theory
13058:Non-standard model
12572:Logical connective
11699:Information theory
11648:Mathematical logic
11488:Numerical analysis
11097:Mathematical logic
11092:Information theory
10665:Mathematical Logic
10644:Dover Publications
10552:Dover Publications
10544:Keisler, H. Jerome
10518:Keisler, H. Jerome
9997:. 87pages=605-630.
9450:Hodges (1993), 492
9423:, pp. 124â125
9402:, pp. 125â155
9390:, pp. 115â124
9074:Skolem normal form
8798:mathematical logic
8794:LwĂłwâWarsaw school
8792:, a member of the
8734:'s development of
8706:semialgebraic sets
8686:real closed fields
8594:
8554:
8508:if it is at least
8437:-stable theories,
8427:
8395:
8375:
8344:
8318:
8291:
8271:
8231:
8197:
8157:
8131:
8093:
8053:
8027:
8003:
7959:
7938:
7911:
7891:
7855:
7835:
7815:
7781:
7757:
7735:
7720:A complete theory
7585:
7556:
7518:
7506:dense linear order
7494:
7467:
7404:
7370:
7208:
7109:Categorical theory
7036:
6983:
6967:if the set of all
6957:
6937:
6906:
6847:an ultrafilter on
6825:
6736:
6708:
6688:
6664:
6638:
6605:
6585:
6550:
6530:
6506:
6479:
6459:
6406:
6353:
6277:
6257:
6213:
6193:
6161:
6141:
6102:
6080:
6054:
6032:
5986:
5964:
5914:
5887:
5859:
5831:
5785:
5728:
5704:
5670:
5642:
5618:
5584:
5560:
5536:
5496:
5434:
5414:
5390:
5358:
5327:
5293:
5271:
5251:is a 1-type over
5241:
5189:
5152:
5142:for every formula
5132:
5106:
5082:
5062:
5042:
5006:
4982:
4958:
4932:
4895:
4871:
4835:
4807:
4776:
4756:. Over the subset
4746:
4706:
4666:
4626:
4574:
4546:
4517:
4471:
4425:
4379:
4329:
4298:
4249:
4199:
4144:
4116:
4092:
4027:
4003:
3971:
3942:
3914:
3885:
3851:
3801:
3777:
3743:
3715:
3693:
3669:
3605:
3581:if every model of
3559:
3535:
3505:
3470:
3409:
3337:
3202:TarskiâVaught test
3159:
2973:
2951:
2928:
2887:
2854:
2821:
2786:
2751:
2713:
2663:
2502:
2401:
2377:
2341:
2317:
2286:
2264:
2242:
2220:
2198:
2163:
2133:
2103:
2079:
2065:is a sentence and
2055:
2045:In particular, if
2031:
1960:
1886:
1846:
1818:
1794:
1764:
1744:
1716:
1692:
1668:
1648:
1628:
1601:
1557:
1528:
1508:
1486:
1457:
1437:
1415:
1395:
1375:
1355:
1335:
1315:
1295:
1213:
1193:
1122:
1102:
1061:
1037:
998:is a prime number.
988:
938:
920:similarly defines
910:
890:
870:
847:
827:
796:
794:
501:
472:
452:
432:
412:
383:
363:
340:
317:
276:
244:
159:algebraic geometry
28:mathematical logic
21:Mathematical model
13372:
13371:
13304:Abstract category
13107:Theories of truth
12917:Rule of inference
12907:Natural deduction
12888:
12887:
12433:
12432:
12138:Cartesian product
12043:
12042:
11949:Many-valued logic
11924:Boolean functions
11807:Russell's paradox
11782:diagonal argument
11679:First-order logic
11614:
11613:
11213:Harmonic analysis
10901:Free online texts
10892:978-1-107-16388-1
10820:978-1-4419-1220-6
10587:978-0-521-58713-6
10561:978-0-486-48821-9
10540:Chang, Chen Chung
10531:978-0-444-88054-3
10514:Chang, Chen Chung
10482:978-1-107-18921-8
10438:978-1-84890-050-9
10413:978-1-84890-050-9
10388:978-3-662-03184-1
10353:978-3-662-03184-1
10318:978-3-662-03184-1
10234:978-1-107-18921-8
9872:10.1090/ulect/050
9658:978-90-481-5417-3
9608:. North-Holland.
8969:; for example in
8914:and the study of
8908:first-order logic
8854:H. Jerome Keisler
8834:deductive systems
8802:Leopold Löwenheim
8728:Ax-Kochen theorem
8633:infinitary logics
8468:successor ordinal
7767:if for any model
7232:-cardinality and
6960:{\displaystyle U}
6216:{\displaystyle A}
6196:{\displaystyle n}
6181:topological space
6164:{\displaystyle A}
5499:{\displaystyle A}
5437:{\displaystyle A}
5417:{\displaystyle n}
5393:{\displaystyle M}
5065:{\displaystyle T}
4985:{\displaystyle T}
4311:. If there is an
3571:strong minimality
3489:minimal structure
2486:In model theory,
2303:of a Ï-structure
2196:
1856:and any elements
1804:of a Ï-structure
1767:{\displaystyle T}
1747:{\displaystyle T}
1671:{\displaystyle T}
1651:{\displaystyle T}
1460:{\displaystyle +}
1398:{\displaystyle -}
1358:{\displaystyle +}
1338:{\displaystyle 1}
1318:{\displaystyle 0}
1216:{\displaystyle M}
1196:{\displaystyle M}
1125:{\displaystyle T}
1064:{\displaystyle T}
893:{\displaystyle n}
850:{\displaystyle n}
435:{\displaystyle x}
192:First-order logic
186:First-order logic
141:universal algebra
38:(a collection of
13402:
13363:
13362:
13314:History of logic
13309:Category of sets
13202:Decision problem
12981:Ordinal analysis
12922:Sequent calculus
12820:Boolean algebras
12760:
12759:
12734:
12705:logical/constant
12459:
12458:
12445:
12368:ZermeloâFraenkel
12119:Set operations:
12054:
12053:
11991:
11822:
11821:
11802:LöwenheimâSkolem
11689:Formal semantics
11641:
11634:
11627:
11618:
11617:
11602:
11601:
11590:
11589:
11578:
11577:
11567:
11566:
11498:Computer algebra
11473:Computer science
11193:Complex analysis
11027:
11020:
11013:
11004:
11003:
10952:
10934:
10932:
10920:
10918:
10908:Chatzidakis, Zoé
10896:
10873:
10847:
10832:(new ed.).
10824:
10797:(3rd ed.).
10786:
10774:
10763:
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9221:
9215:
9209:
9203:
9197:
9191:
9185:
9182:
9176:
9170:
9164:
9158:
9152:
9149:
9143:
9137:
9131:
9121:
9115:
9114:
9103:
9097:
9091:
9049:Kripke semantics
9039:Hyperreal number
9009:Algebraic theory
8948:Skolem's paradox
8868:'s proof of the
8862:stability theory
8858:Morley's problem
8816:'s proof of his
8732:Abraham Robinson
8690:Boolean algebras
8627:Model theory in
8603:
8601:
8600:
8595:
8563:
8561:
8560:
8555:
8436:
8434:
8433:
8428:
8409:is superstable.
8404:
8402:
8401:
8396:
8384:
8382:
8381:
8376:
8374:
8373:
8353:
8351:
8350:
8345:
8331:Main gap theorem
8327:
8325:
8324:
8319:
8300:
8298:
8297:
8292:
8280:
8278:
8277:
8272:
8240:
8238:
8237:
8232:
8230:
8229:
8228:
8227:
8206:
8204:
8203:
8198:
8196:
8195:
8194:
8193:
8167:-stable for any
8166:
8164:
8163:
8158:
8140:
8138:
8137:
8132:
8130:
8129:
8128:
8127:
8102:
8100:
8099:
8094:
8086:
8085:
8084:
8083:
8062:
8060:
8059:
8054:
8036:
8034:
8033:
8028:
8012:
8010:
8009:
8004:
7968:
7966:
7965:
7960:
7947:
7945:
7944:
7939:
7937:
7936:
7920:
7918:
7917:
7912:
7900:
7898:
7897:
7892:
7864:
7862:
7861:
7856:
7844:
7842:
7841:
7836:
7824:
7822:
7821:
7816:
7814:
7813:
7790:
7788:
7787:
7782:
7780:
7779:
7766:
7764:
7763:
7758:
7744:
7742:
7741:
7736:
7702:Stability theory
7693:
7681:
7673:
7669:
7633:
7610:
7594:
7592:
7591:
7586:
7584:
7583:
7565:
7563:
7562:
7557:
7527:
7525:
7524:
7519:
7503:
7501:
7500:
7495:
7493:
7492:
7476:
7474:
7473:
7468:
7463:
7462:
7444:
7443:
7417:
7413:
7411:
7410:
7405:
7394:
7379:
7377:
7376:
7371:
7360:
7274:
7254:
7246:
7239:
7235:
7231:
7227:
7225:
7217:
7215:
7214:
7209:
7207:
7199:
7191:
7190:
7174:
7170:
7166:
7156:
7147:
7143:
7136:
7129:
7093:
7089:
7085:
7081:
7077:
7073:
7069:
7065:
7049:
7045:
7043:
7042:
7037:
7032:
7031:
7013:
7012:
7007:
7006:
6992:
6990:
6989:
6984:
6966:
6964:
6963:
6958:
6946:
6944:
6943:
6938:
6936:
6935:
6930:
6929:
6915:
6913:
6912:
6907:
6899:
6898:
6883:
6882:
6854:
6850:
6846:
6842:
6838:
6834:
6832:
6831:
6826:
6824:
6823:
6818:
6817:
6791:
6787:
6776:
6745:
6743:
6742:
6737:
6735:
6734:
6717:
6715:
6714:
6709:
6697:
6695:
6694:
6689:
6687:
6686:
6673:
6671:
6670:
6665:
6663:
6662:
6647:
6645:
6644:
6639:
6634:
6633:
6614:
6612:
6611:
6606:
6594:
6592:
6591:
6586:
6584:
6583:
6559:
6557:
6556:
6551:
6539:
6537:
6536:
6531:
6529:
6528:
6515:
6513:
6512:
6507:
6505:
6504:
6488:
6486:
6485:
6480:
6468:
6466:
6465:
6460:
6458:
6457:
6418:Zariski topology
6415:
6413:
6412:
6407:
6362:
6360:
6359:
6354:
6286:
6284:
6283:
6278:
6266:
6264:
6263:
6258:
6244:
6222:
6220:
6219:
6214:
6202:
6200:
6199:
6194:
6170:
6168:
6167:
6162:
6150:
6148:
6147:
6142:
6140:
6139:
6134:
6133:
6111:
6109:
6108:
6103:
6101:
6089:
6087:
6086:
6081:
6079:
6063:
6061:
6060:
6055:
6053:
6041:
6039:
6038:
6033:
6031:
6030:
6012:
6011:
5995:
5993:
5992:
5987:
5985:
5973:
5971:
5970:
5965:
5963:
5962:
5944:
5943:
5923:
5921:
5920:
5915:
5896:
5894:
5893:
5888:
5886:
5885:
5868:
5866:
5865:
5860:
5858:
5857:
5840:
5838:
5837:
5832:
5830:
5829:
5811:
5810:
5794:
5792:
5791:
5786:
5784:
5783:
5765:
5764:
5737:
5735:
5734:
5729:
5727:
5726:
5713:
5711:
5710:
5705:
5703:
5702:
5679:
5677:
5676:
5671:
5669:
5668:
5651:
5649:
5648:
5643:
5641:
5640:
5627:
5625:
5624:
5619:
5617:
5616:
5593:
5591:
5590:
5585:
5583:
5582:
5569:
5567:
5566:
5561:
5559:
5558:
5545:
5543:
5542:
5537:
5535:
5534:
5505:
5503:
5502:
5497:
5492:
5491:
5473:
5472:
5443:
5441:
5440:
5435:
5423:
5421:
5420:
5415:
5399:
5397:
5396:
5391:
5367:
5365:
5364:
5359:
5357:
5356:
5351:
5350:
5336:
5334:
5333:
5328:
5326:
5325:
5320:
5319:
5302:
5300:
5299:
5294:
5292:
5280:
5278:
5277:
5272:
5270:
5262:
5250:
5248:
5247:
5242:
5237:
5226:
5198:
5196:
5195:
5190:
5188:
5161:
5159:
5158:
5153:
5141:
5139:
5138:
5133:
5115:
5113:
5112:
5107:
5105:
5104:
5091:
5089:
5088:
5083:
5071:
5069:
5068:
5063:
5051:
5049:
5048:
5043:
5032:
5031:
5015:
5013:
5012:
5007:
5005:
5004:
4991:
4989:
4988:
4983:
4967:
4965:
4964:
4959:
4947:
4946:
4940:
4904:
4902:
4901:
4896:
4894:
4893:
4880:
4878:
4877:
4872:
4870:
4869:
4844:
4842:
4841:
4836:
4834:
4833:
4816:
4814:
4813:
4808:
4806:
4805:
4785:
4783:
4782:
4777:
4775:
4767:
4755:
4753:
4752:
4747:
4745:
4744:
4732:
4731:
4715:
4713:
4712:
4707:
4705:
4704:
4692:
4691:
4675:
4673:
4672:
4667:
4665:
4664:
4652:
4651:
4635:
4633:
4632:
4627:
4625:
4624:
4612:
4611:
4583:
4581:
4580:
4575:
4573:
4555:
4553:
4552:
4547:
4545:
4526:
4524:
4523:
4518:
4516:
4515:
4497:
4496:
4480:
4478:
4477:
4472:
4470:
4469:
4451:
4450:
4434:
4432:
4431:
4426:
4424:
4423:
4405:
4404:
4388:
4386:
4385:
4380:
4378:
4377:
4359:
4358:
4338:
4336:
4335:
4330:
4328:
4327:
4307:
4305:
4304:
4299:
4297:
4296:
4278:
4277:
4258:
4256:
4255:
4250:
4248:
4247:
4229:
4228:
4208:
4206:
4205:
4200:
4195:
4194:
4176:
4175:
4153:
4151:
4150:
4145:
4143:
4142:
4125:
4123:
4122:
4117:
4115:
4114:
4101:
4099:
4098:
4093:
4091:
4090:
4072:
4071:
4036:
4034:
4033:
4028:
4026:
4025:
4012:
4010:
4009:
4004:
4002:
4001:
3980:
3978:
3977:
3972:
3951:
3949:
3948:
3943:
3941:
3940:
3923:
3921:
3920:
3915:
3894:
3892:
3891:
3886:
3884:
3883:
3860:
3858:
3857:
3852:
3850:
3849:
3844:
3843:
3810:
3808:
3807:
3802:
3800:
3799:
3786:
3784:
3783:
3778:
3776:
3775:
3757:if every subset
3752:
3750:
3749:
3744:
3742:
3741:
3724:
3722:
3721:
3716:
3714:
3702:
3700:
3699:
3694:
3678:
3676:
3675:
3670:
3614:
3612:
3611:
3606:
3604:
3587:strongly minimal
3579:strongly minimal
3568:
3566:
3565:
3560:
3558:
3557:
3544:
3542:
3541:
3536:
3534:
3533:
3514:
3512:
3511:
3506:
3504:
3503:
3479:
3477:
3476:
3471:
3469:
3468:
3444:
3443:
3418:
3416:
3415:
3410:
3405:
3404:
3386:
3385:
3353:model completion
3346:
3344:
3343:
3338:
3333:
3332:
3314:
3313:
3301:
3300:
3282:
3281:
3266:
3265:
3250:
3249:
3168:
3166:
3165:
3160:
3152:
3151:
3133:
3132:
3111:
3110:
3092:
3091:
3073:
3072:
3057:
3056:
2982:
2980:
2979:
2974:
2972:
2960:
2958:
2957:
2952:
2937:
2935:
2934:
2929:
2896:
2894:
2893:
2888:
2886:
2863:
2861:
2860:
2855:
2853:
2852:
2830:
2828:
2827:
2822:
2795:
2793:
2792:
2787:
2760:
2758:
2757:
2752:
2750:
2749:
2744:
2743:
2722:
2720:
2719:
2714:
2672:
2670:
2669:
2664:
2511:
2509:
2508:
2503:
2501:
2428:rational numbers
2410:
2408:
2407:
2402:
2400:
2399:
2386:
2384:
2383:
2378:
2376:
2375:
2350:
2348:
2347:
2342:
2340:
2339:
2326:
2324:
2323:
2318:
2316:
2315:
2295:
2293:
2292:
2287:
2285:
2273:
2271:
2270:
2265:
2263:
2251:
2249:
2248:
2243:
2241:
2229:
2227:
2226:
2221:
2219:
2207:
2205:
2204:
2199:
2197:
2192:
2187:
2172:
2170:
2169:
2164:
2156:
2155:
2143:if and only if
2142:
2140:
2139:
2134:
2126:
2125:
2112:
2110:
2109:
2104:
2102:
2101:
2088:
2086:
2085:
2080:
2078:
2077:
2064:
2062:
2061:
2056:
2040:
2038:
2037:
2032:
2027:
2026:
2002:
2001:
1983:
1982:
1969:
1967:
1966:
1961:
1956:
1955:
1931:
1930:
1912:
1911:
1895:
1893:
1892:
1887:
1885:
1884:
1855:
1853:
1852:
1847:
1827:
1825:
1824:
1819:
1817:
1816:
1803:
1801:
1800:
1795:
1793:
1792:
1773:
1771:
1770:
1765:
1753:
1751:
1750:
1745:
1725:
1723:
1722:
1717:
1715:
1714:
1701:
1699:
1698:
1693:
1691:
1690:
1677:
1675:
1674:
1669:
1657:
1655:
1654:
1649:
1637:
1635:
1634:
1629:
1627:
1626:
1610:
1608:
1607:
1602:
1597:
1596:
1581:
1566:
1564:
1563:
1558:
1556:
1555:
1550:
1537:
1535:
1534:
1529:
1517:
1515:
1514:
1509:
1507:
1495:
1493:
1492:
1487:
1485:
1484:
1479:
1466:
1464:
1463:
1458:
1446:
1444:
1443:
1438:
1436:
1424:
1422:
1421:
1416:
1404:
1402:
1401:
1396:
1384:
1382:
1381:
1376:
1364:
1362:
1361:
1356:
1344:
1342:
1341:
1336:
1324:
1322:
1321:
1316:
1304:
1302:
1301:
1296:
1252:
1251:
1222:
1220:
1219:
1214:
1202:
1200:
1199:
1194:
1131:
1129:
1128:
1123:
1111:
1109:
1108:
1103:
1095:
1094:
1070:
1068:
1067:
1062:
1046:
1044:
1043:
1038:
1013:
1012:
997:
995:
994:
989:
964:
963:
947:
945:
944:
939:
919:
917:
916:
911:
899:
897:
896:
891:
879:
877:
876:
871:
856:
854:
853:
848:
836:
834:
833:
828:
826:
825:
805:
803:
802:
797:
795:
510:
508:
507:
502:
481:
479:
478:
473:
461:
459:
458:
453:
441:
439:
438:
433:
421:
419:
418:
413:
392:
390:
389:
384:
372:
370:
369:
364:
349:
347:
346:
341:
326:
324:
323:
318:
286:by means of the
285:
283:
282:
277:
253:
251:
250:
245:
200:is built out of
73:stability theory
13412:
13411:
13405:
13404:
13403:
13401:
13400:
13399:
13375:
13374:
13373:
13368:
13357:
13350:
13295:Category theory
13285:Algebraic logic
13268:
13239:Lambda calculus
13177:Church encoding
13163:
13139:Truth predicate
12995:
12961:Complete theory
12884:
12753:
12749:
12745:
12740:
12732:
12452: and
12448:
12443:
12429:
12405:New Foundations
12373:axiom of choice
12356:
12318:Gödel numbering
12258: and
12250:
12154:
12039:
11989:
11970:
11919:Boolean algebra
11905:
11869:Equiconsistency
11834:Classical logic
11811:
11792:Halting problem
11780: and
11756: and
11744: and
11743:
11738:Theorems (
11733:
11650:
11645:
11615:
11610:
11561:
11552:
11502:
11459:
11438:Systems science
11369:
11365:Homotopy theory
11331:
11298:
11250:
11222:
11169:
11116:
11087:Category theory
11073:
11038:
11031:
10968:Hodges, Wilfrid
10956:Hodges, Wilfrid
10937:
10930:
10916:
10903:
10893:
10870:
10844:
10821:
10783:
10760:
10734:
10712:Hodges, Wilfrid
10704:
10681:
10654:
10634:
10632:Other textbooks
10624:
10616:217. Springer.
10601:Allyn and Bacon
10588:
10570:Hodges, Wilfrid
10562:
10554:. p. 672.
10532:
10509:
10504:
10499:
10498:
10483:
10459:
10455:
10450:
10446:
10439:
10425:
10421:
10414:
10400:
10396:
10389:
10365:
10361:
10354:
10330:
10326:
10319:
10295:
10291:
10276:
10254:
10250:
10235:
10211:
10207:
10202:
10198:
10193:
10189:
10170:
10166:
10135:
10131:
10078:
10074:
10061:
10057:
10048:
10044:
10036:
10034:
10032:
10006:
10002:
9991:
9987:
9979:
9977:
9975:
9949:
9945:
9922:10.2307/2274425
9902:
9898:
9893:
9889:
9882:
9860:
9856:
9848:
9846:
9844:
9818:
9814:
9769:
9765:
9712:
9708:
9700:
9698:
9696:
9670:
9666:
9659:
9640:Simple theories
9635:
9631:
9616:
9600:
9596:
9591:
9587:
9579:
9575:
9567:
9563:
9555:
9551:
9494:
9490:
9485:
9481:
9476:
9472:
9467:
9463:
9458:
9454:
9449:
9445:
9440:
9436:
9431:
9427:
9419:
9415:
9410:
9406:
9398:
9394:
9386:
9382:
9374:
9372:
9370:
9344:
9340:
9335:
9331:
9323:
9319:
9311:
9307:
9299:
9295:
9287:
9283:
9260:10.2307/2274425
9240:
9236:
9228:
9224:
9216:
9212:
9204:
9200:
9192:
9188:
9183:
9179:
9171:
9167:
9159:
9155:
9150:
9146:
9138:
9134:
9122:
9118:
9105:
9104:
9100:
9092:
9088:
9083:
9078:
9069:Saturated model
8999:
8987:axiom of choice
8936:
8924:database theory
8889:
8883:
8878:
8822:Anatoly Maltsev
8786:
8747:Ehud Hrushovski
8682:
8622:
8609:Simple theories
8589:
8586:
8585:
8549:
8546:
8545:
8422:
8419:
8418:
8415:
8390:
8387:
8386:
8369:
8365:
8363:
8360:
8359:
8339:
8336:
8335:
8313:
8310:
8309:
8286:
8283:
8282:
8266:
8263:
8262:
8255:strictly stable
8223:
8219:
8218:
8214:
8212:
8209:
8208:
8189:
8185:
8184:
8180:
8172:
8169:
8168:
8152:
8149:
8148:
8123:
8119:
8118:
8114:
8112:
8109:
8108:
8079:
8075:
8074:
8070:
8068:
8065:
8064:
8048:
8045:
8044:
8022:
8019:
8018:
7998:
7995:
7994:
7977:
7954:
7951:
7950:
7932:
7928:
7926:
7923:
7922:
7906:
7903:
7902:
7886:
7883:
7882:
7850:
7847:
7846:
7830:
7827:
7826:
7809:
7808:
7800:
7797:
7796:
7775:
7774:
7772:
7769:
7768:
7752:
7749:
7748:
7747:for a cardinal
7730:
7727:
7726:
7710:
7704:
7691:
7679:
7671:
7667:
7644:
7631:
7621:Ryll-Nardzewski
7608:
7579:
7575:
7573:
7570:
7569:
7533:
7530:
7529:
7513:
7510:
7509:
7488:
7484:
7482:
7479:
7478:
7458:
7454:
7439:
7435:
7427:
7424:
7423:
7415:
7390:
7385:
7382:
7381:
7356:
7351:
7348:
7347:
7331:
7324:
7301:
7283:
7272:
7252:
7249:
7244:
7237:
7233:
7229:
7221:
7219:
7203:
7195:
7186:
7182:
7180:
7177:
7176:
7172:
7168:
7164:
7154:
7145:
7141:
7134:
7132:cardinal number
7127:
7124:cardinal number
7111:
7105:
7099:if they exist.
7094:are isomorphic.
7091:
7087:
7083:
7079:
7075:
7071:
7067:
7063:
7047:
7027:
7023:
7008:
7002:
7001:
7000:
6998:
6995:
6994:
6972:
6969:
6968:
6952:
6949:
6948:
6931:
6925:
6924:
6923:
6921:
6918:
6917:
6888:
6884:
6878:
6874:
6860:
6857:
6856:
6852:
6848:
6844:
6840:
6836:
6819:
6813:
6812:
6811:
6809:
6806:
6805:
6789:
6785:
6774:
6760:
6730:
6729:
6727:
6724:
6723:
6703:
6700:
6699:
6682:
6681:
6679:
6676:
6675:
6658:
6657:
6655:
6652:
6651:
6629:
6628:
6620:
6617:
6616:
6600:
6597:
6596:
6579:
6578:
6576:
6573:
6572:
6545:
6542:
6541:
6524:
6523:
6521:
6518:
6517:
6500:
6499:
6497:
6494:
6493:
6474:
6471:
6470:
6453:
6452:
6450:
6447:
6446:
6431:
6426:
6368:
6365:
6364:
6363:or of the form
6315:
6312:
6311:
6272:
6269:
6268:
6240:
6232:
6229:
6228:
6223:. The topology
6208:
6205:
6204:
6188:
6185:
6184:
6173:Boolean algebra
6156:
6153:
6152:
6135:
6129:
6128:
6127:
6125:
6122:
6121:
6118:
6097:
6095:
6092:
6091:
6075:
6073:
6070:
6069:
6049:
6047:
6044:
6043:
6026:
6022:
6007:
6003:
6001:
5998:
5997:
5981:
5979:
5976:
5975:
5958:
5954:
5939:
5935:
5933:
5930:
5929:
5909:
5906:
5905:
5881:
5880:
5878:
5875:
5874:
5853:
5852:
5850:
5847:
5846:
5825:
5821:
5806:
5802:
5800:
5797:
5796:
5779:
5775:
5760:
5756:
5754:
5751:
5750:
5722:
5721:
5719:
5716:
5715:
5698:
5697:
5689:
5686:
5685:
5664:
5663:
5661:
5658:
5657:
5636:
5635:
5633:
5630:
5629:
5612:
5611:
5603:
5600:
5599:
5578:
5577:
5575:
5572:
5571:
5554:
5553:
5551:
5548:
5547:
5530:
5529:
5527:
5524:
5523:
5512:
5487:
5483:
5468:
5464:
5456:
5453:
5452:
5450:polynomial ring
5429:
5426:
5425:
5409:
5406:
5405:
5385:
5382:
5381:
5352:
5346:
5345:
5344:
5342:
5339:
5338:
5321:
5315:
5314:
5313:
5311:
5308:
5307:
5288:
5286:
5283:
5282:
5266:
5258:
5256:
5253:
5252:
5233:
5222:
5208:
5205:
5204:
5184:
5182:
5179:
5178:
5147:
5144:
5143:
5121:
5118:
5117:
5100:
5099:
5097:
5094:
5093:
5077:
5074:
5073:
5057:
5054:
5053:
5027:
5023:
5021:
5018:
5017:
5016:. The notation
5000:
4999:
4997:
4994:
4993:
4977:
4974:
4973:
4942:
4941:
4936:
4930:
4927:
4926:
4889:
4888:
4886:
4883:
4882:
4865:
4864:
4862:
4859:
4858:
4829:
4828:
4826:
4823:
4822:
4801:
4800:
4798:
4795:
4794:
4771:
4763:
4761:
4758:
4757:
4740:
4736:
4727:
4723:
4721:
4718:
4717:
4700:
4696:
4687:
4683:
4681:
4678:
4677:
4660:
4656:
4647:
4643:
4641:
4638:
4637:
4620:
4616:
4607:
4603:
4601:
4598:
4597:
4569:
4561:
4558:
4557:
4541:
4539:
4536:
4535:
4511:
4507:
4492:
4488:
4486:
4483:
4482:
4465:
4461:
4446:
4442:
4440:
4437:
4436:
4419:
4415:
4400:
4396:
4394:
4391:
4390:
4373:
4369:
4354:
4350:
4348:
4345:
4344:
4323:
4322:
4320:
4317:
4316:
4292:
4288:
4273:
4269:
4267:
4264:
4263:
4243:
4239:
4224:
4220:
4218:
4215:
4214:
4190:
4186:
4171:
4167:
4159:
4156:
4155:
4138:
4137:
4135:
4132:
4131:
4110:
4109:
4107:
4104:
4103:
4102:of a structure
4086:
4082:
4067:
4063:
4061:
4058:
4057:
4054:
4049:
4043:
4021:
4020:
4018:
4015:
4014:
3997:
3996:
3994:
3991:
3990:
3957:
3954:
3953:
3936:
3935:
3933:
3930:
3929:
3900:
3897:
3896:
3879:
3878:
3870:
3867:
3866:
3845:
3839:
3838:
3837:
3835:
3832:
3831:
3823:
3817:
3795:
3794:
3792:
3789:
3788:
3771:
3770:
3762:
3759:
3758:
3737:
3736:
3734:
3731:
3730:
3710:
3708:
3705:
3704:
3688:
3685:
3684:
3623:
3620:
3619:
3600:
3598:
3595:
3594:
3553:
3552:
3550:
3547:
3546:
3529:
3528:
3520:
3517:
3516:
3499:
3498:
3496:
3493:
3492:
3491:. A structure
3464:
3460:
3439:
3435:
3427:
3424:
3423:
3400:
3396:
3381:
3377:
3372:
3369:
3368:
3365:
3357:model companion
3328:
3324:
3309:
3305:
3296:
3292:
3277:
3273:
3261:
3257:
3245:
3241:
3236:
3233:
3232:
3223:
3214:
3176:
3147:
3143:
3128:
3124:
3106:
3102:
3087:
3083:
3068:
3064:
3052:
3048:
3043:
3040:
3039:
3037:
3028:
3017:
3008:
2989:
2968:
2966:
2963:
2962:
2946:
2943:
2942:
2905:
2902:
2901:
2882:
2880:
2877:
2876:
2848:
2844:
2836:
2833:
2832:
2804:
2801:
2800:
2769:
2766:
2765:
2745:
2739:
2738:
2737:
2735:
2732:
2731:
2684:
2681:
2680:
2520:
2517:
2516:
2497:
2495:
2492:
2491:
2484:
2479:
2440:
2395:
2394:
2392:
2389:
2388:
2371:
2370:
2368:
2365:
2364:
2335:
2334:
2332:
2329:
2328:
2311:
2310:
2308:
2305:
2304:
2281:
2279:
2276:
2275:
2259:
2257:
2254:
2253:
2237:
2235:
2232:
2231:
2215:
2213:
2210:
2209:
2188:
2186:
2184:
2181:
2180:
2151:
2150:
2148:
2145:
2144:
2121:
2120:
2118:
2115:
2114:
2097:
2096:
2094:
2091:
2090:
2073:
2072:
2070:
2067:
2066:
2050:
2047:
2046:
2022:
2018:
1997:
1993:
1978:
1977:
1975:
1972:
1971:
1970:if and only if
1951:
1947:
1926:
1922:
1907:
1906:
1904:
1901:
1900:
1880:
1879:
1877:
1874:
1873:
1871:
1862:
1841:
1838:
1837:
1812:
1811:
1809:
1806:
1805:
1788:
1787:
1785:
1782:
1781:
1759:
1756:
1755:
1739:
1736:
1735:
1710:
1709:
1707:
1704:
1703:
1686:
1685:
1683:
1680:
1679:
1663:
1660:
1659:
1643:
1640:
1639:
1622:
1621:
1619:
1616:
1615:
1589:
1585:
1577:
1572:
1569:
1568:
1551:
1546:
1545:
1543:
1540:
1539:
1538:is a subset of
1523:
1520:
1519:
1503:
1501:
1498:
1497:
1480:
1475:
1474:
1472:
1469:
1468:
1452:
1449:
1448:
1432:
1430:
1427:
1426:
1410:
1407:
1406:
1390:
1387:
1386:
1370:
1367:
1366:
1350:
1347:
1346:
1330:
1327:
1326:
1310:
1307:
1306:
1244:
1240:
1238:
1235:
1234:
1208:
1205:
1204:
1188:
1185:
1184:
1161:
1117:
1114:
1113:
1090:
1089:
1087:
1084:
1083:
1056:
1053:
1052:
1047:is irreducible.
1008:
1007:
1005:
1002:
1001:
959:
958:
956:
953:
952:
933:
930:
929:
905:
902:
901:
885:
882:
881:
880:if and only if
865:
862:
861:
842:
839:
838:
821:
820:
818:
815:
814:
812:
793:
792:
691:
686:
680:
679:
533:
528:
521:
519:
516:
515:
487:
484:
483:
467:
464:
463:
447:
444:
443:
427:
424:
423:
398:
395:
394:
378:
375:
374:
355:
352:
351:
332:
329:
328:
294:
291:
290:
259:
256:
255:
209:
206:
205:
202:atomic formulas
194:
188:
183:
119:
44:formal language
36:formal theories
24:
17:
12:
11:
5:
13410:
13409:
13398:
13397:
13392:
13387:
13370:
13369:
13355:
13352:
13351:
13349:
13348:
13343:
13338:
13333:
13328:
13327:
13326:
13316:
13311:
13306:
13297:
13292:
13287:
13282:
13280:Abstract logic
13276:
13274:
13270:
13269:
13267:
13266:
13261:
13259:Turing machine
13256:
13251:
13246:
13241:
13236:
13231:
13230:
13229:
13224:
13219:
13214:
13209:
13199:
13197:Computable set
13194:
13189:
13184:
13179:
13173:
13171:
13165:
13164:
13162:
13161:
13156:
13151:
13146:
13141:
13136:
13131:
13126:
13125:
13124:
13119:
13114:
13104:
13099:
13094:
13092:Satisfiability
13089:
13084:
13079:
13078:
13077:
13067:
13066:
13065:
13055:
13054:
13053:
13048:
13043:
13038:
13033:
13023:
13022:
13021:
13016:
13009:Interpretation
13005:
13003:
12997:
12996:
12994:
12993:
12988:
12983:
12978:
12973:
12963:
12958:
12957:
12956:
12955:
12954:
12944:
12939:
12929:
12924:
12919:
12914:
12909:
12904:
12898:
12896:
12890:
12889:
12886:
12885:
12883:
12882:
12874:
12873:
12872:
12871:
12866:
12865:
12864:
12859:
12854:
12834:
12833:
12832:
12830:minimal axioms
12827:
12816:
12815:
12814:
12803:
12802:
12801:
12796:
12791:
12786:
12781:
12776:
12763:
12761:
12742:
12741:
12739:
12738:
12737:
12736:
12724:
12719:
12718:
12717:
12712:
12707:
12702:
12692:
12687:
12682:
12677:
12676:
12675:
12670:
12660:
12659:
12658:
12653:
12648:
12643:
12633:
12628:
12627:
12626:
12621:
12616:
12606:
12605:
12604:
12599:
12594:
12589:
12584:
12579:
12569:
12564:
12559:
12554:
12553:
12552:
12547:
12542:
12537:
12527:
12522:
12520:Formation rule
12517:
12512:
12511:
12510:
12505:
12495:
12494:
12493:
12483:
12478:
12473:
12468:
12462:
12456:
12439:Formal systems
12435:
12434:
12431:
12430:
12428:
12427:
12422:
12417:
12412:
12407:
12402:
12397:
12392:
12387:
12382:
12381:
12380:
12375:
12364:
12362:
12358:
12357:
12355:
12354:
12353:
12352:
12342:
12337:
12336:
12335:
12328:Large cardinal
12325:
12320:
12315:
12310:
12305:
12291:
12290:
12289:
12284:
12279:
12264:
12262:
12252:
12251:
12249:
12248:
12247:
12246:
12241:
12236:
12226:
12221:
12216:
12211:
12206:
12201:
12196:
12191:
12186:
12181:
12176:
12171:
12165:
12163:
12156:
12155:
12153:
12152:
12151:
12150:
12145:
12140:
12135:
12130:
12125:
12117:
12116:
12115:
12110:
12100:
12095:
12093:Extensionality
12090:
12088:Ordinal number
12085:
12075:
12070:
12069:
12068:
12057:
12051:
12045:
12044:
12041:
12040:
12038:
12037:
12032:
12027:
12022:
12017:
12012:
12007:
12006:
12005:
11995:
11994:
11993:
11980:
11978:
11972:
11971:
11969:
11968:
11967:
11966:
11961:
11956:
11946:
11941:
11936:
11931:
11926:
11921:
11915:
11913:
11907:
11906:
11904:
11903:
11898:
11893:
11888:
11883:
11878:
11873:
11872:
11871:
11861:
11856:
11851:
11846:
11841:
11836:
11830:
11828:
11819:
11813:
11812:
11810:
11809:
11804:
11799:
11794:
11789:
11784:
11772:Cantor's
11770:
11765:
11760:
11750:
11748:
11735:
11734:
11732:
11731:
11726:
11721:
11716:
11711:
11706:
11701:
11696:
11691:
11686:
11681:
11676:
11671:
11670:
11669:
11658:
11656:
11652:
11651:
11644:
11643:
11636:
11629:
11621:
11612:
11611:
11609:
11608:
11596:
11584:
11572:
11557:
11554:
11553:
11551:
11550:
11545:
11540:
11535:
11530:
11525:
11524:
11523:
11516:Mathematicians
11512:
11510:
11508:Related topics
11504:
11503:
11501:
11500:
11495:
11490:
11485:
11480:
11475:
11469:
11467:
11461:
11460:
11458:
11457:
11456:
11455:
11450:
11445:
11443:Control theory
11435:
11430:
11425:
11420:
11415:
11410:
11405:
11400:
11395:
11390:
11385:
11379:
11377:
11371:
11370:
11368:
11367:
11362:
11357:
11352:
11347:
11341:
11339:
11333:
11332:
11330:
11329:
11324:
11319:
11314:
11308:
11306:
11300:
11299:
11297:
11296:
11291:
11286:
11281:
11276:
11271:
11266:
11260:
11258:
11252:
11251:
11249:
11248:
11243:
11238:
11232:
11230:
11224:
11223:
11221:
11220:
11218:Measure theory
11215:
11210:
11205:
11200:
11195:
11190:
11185:
11179:
11177:
11171:
11170:
11168:
11167:
11162:
11157:
11152:
11147:
11142:
11137:
11132:
11126:
11124:
11118:
11117:
11115:
11114:
11109:
11104:
11099:
11094:
11089:
11083:
11081:
11075:
11074:
11072:
11071:
11066:
11061:
11060:
11059:
11054:
11043:
11040:
11039:
11030:
11029:
11022:
11015:
11007:
11001:
11000:
10986:
10977:
10965:
10953:
10939:"Model theory"
10935:
10921:
10902:
10899:
10898:
10897:
10891:
10874:
10868:
10848:
10842:
10825:
10819:
10787:
10781:
10764:
10758:
10742:Manzano, MarĂa
10738:
10732:
10708:
10702:
10685:
10679:
10658:
10652:
10633:
10630:
10629:
10628:
10622:
10605:
10592:
10586:
10566:
10560:
10536:
10530:
10508:
10505:
10503:
10500:
10497:
10496:
10481:
10453:
10444:
10437:
10419:
10412:
10394:
10387:
10359:
10352:
10324:
10317:
10289:
10274:
10248:
10233:
10205:
10196:
10187:
10164:
10129:
10092:(3): 319â332.
10072:
10055:
10042:
10030:
10000:
9985:
9973:
9943:
9896:
9887:
9880:
9854:
9842:
9812:
9763:
9706:
9694:
9664:
9657:
9629:
9614:
9594:
9585:
9573:
9561:
9549:
9510:(2): 213â216.
9488:
9479:
9470:
9461:
9452:
9443:
9434:
9425:
9413:
9404:
9392:
9380:
9368:
9338:
9329:
9317:
9305:
9293:
9281:
9234:
9222:
9210:
9198:
9186:
9177:
9165:
9153:
9144:
9132:
9116:
9107:"Model Theory"
9098:
9085:
9084:
9082:
9079:
9077:
9076:
9071:
9066:
9061:
9056:
9051:
9046:
9041:
9036:
9031:
9026:
9021:
9016:
9011:
9006:
9000:
8998:
8995:
8935:
8932:
8885:Main article:
8882:
8879:
8877:
8874:
8810:Thoralf Skolem
8785:
8782:
8761:to prove the
8740:infinitesimals
8698:characteristic
8681:
8678:
8621:
8618:
8593:
8553:
8522:
8521:
8494:
8460:
8426:
8414:
8411:
8394:
8372:
8368:
8358:has less than
8343:
8317:
8290:
8270:
8247:
8246:
8226:
8222:
8217:
8192:
8188:
8183:
8179:
8176:
8156:
8142:
8126:
8122:
8117:
8092:
8089:
8082:
8078:
8073:
8052:
8038:
8026:
8002:
7976:
7973:
7958:
7935:
7931:
7910:
7890:
7875:
7874:
7854:
7834:
7812:
7807:
7804:
7778:
7756:
7734:
7706:Main article:
7703:
7700:
7684:
7683:
7660:
7647:Michael Morley
7643:
7640:
7636:Fraïssé limits
7613:
7612:
7582:
7578:
7555:
7552:
7549:
7546:
7543:
7540:
7537:
7517:
7491:
7487:
7466:
7461:
7457:
7453:
7450:
7447:
7442:
7438:
7434:
7431:
7403:
7400:
7397:
7393:
7389:
7369:
7366:
7363:
7359:
7355:
7346:The theory of
7344:
7343:
7342:
7341:
7329:
7322:
7308:
7299:
7290:
7289:) is isolated.
7281:
7278:Every type in
7276:
7248:
7242:
7206:
7202:
7198:
7194:
7189:
7185:
7107:Main article:
7104:
7101:
7096:
7095:
7052:
7051:
7035:
7030:
7026:
7022:
7019:
7016:
7011:
7005:
6982:
6979:
6976:
6956:
6934:
6928:
6905:
6902:
6897:
6894:
6891:
6887:
6881:
6877:
6873:
6870:
6867:
6864:
6822:
6816:
6771:direct product
6759:
6756:
6733:
6720:
6719:
6707:
6685:
6661:
6648:
6637:
6632:
6627:
6624:
6604:
6582:
6562:
6561:
6549:
6527:
6503:
6490:
6478:
6456:
6430:
6427:
6425:
6422:
6405:
6402:
6399:
6396:
6393:
6390:
6387:
6384:
6381:
6378:
6375:
6372:
6352:
6349:
6346:
6343:
6340:
6337:
6334:
6331:
6328:
6325:
6322:
6319:
6276:
6256:
6253:
6250:
6247:
6243:
6239:
6236:
6212:
6192:
6160:
6138:
6132:
6117:
6114:
6100:
6078:
6052:
6029:
6025:
6021:
6018:
6015:
6010:
6006:
5984:
5961:
5957:
5953:
5950:
5947:
5942:
5938:
5913:
5884:
5856:
5828:
5824:
5820:
5817:
5814:
5809:
5805:
5782:
5778:
5774:
5771:
5768:
5763:
5759:
5725:
5701:
5696:
5693:
5667:
5639:
5615:
5610:
5607:
5581:
5557:
5533:
5511:
5508:
5495:
5490:
5486:
5482:
5479:
5476:
5471:
5467:
5463:
5460:
5433:
5413:
5389:
5374:type-definable
5355:
5349:
5324:
5318:
5291:
5269:
5265:
5261:
5240:
5236:
5232:
5229:
5225:
5221:
5218:
5215:
5212:
5187:
5151:
5131:
5128:
5125:
5103:
5081:
5061:
5041:
5038:
5035:
5030:
5026:
5003:
4981:
4957:
4954:
4951:
4945:
4939:
4935:
4892:
4868:
4845:, a (partial)
4832:
4804:
4774:
4770:
4766:
4743:
4739:
4735:
4730:
4726:
4703:
4699:
4695:
4690:
4686:
4663:
4659:
4655:
4650:
4646:
4623:
4619:
4615:
4610:
4606:
4572:
4568:
4565:
4544:
4514:
4510:
4506:
4503:
4500:
4495:
4491:
4468:
4464:
4460:
4457:
4454:
4449:
4445:
4422:
4418:
4414:
4411:
4408:
4403:
4399:
4376:
4372:
4368:
4365:
4362:
4357:
4353:
4326:
4295:
4291:
4287:
4284:
4281:
4276:
4272:
4246:
4242:
4238:
4235:
4232:
4227:
4223:
4198:
4193:
4189:
4185:
4182:
4179:
4174:
4170:
4166:
4163:
4141:
4113:
4089:
4085:
4081:
4078:
4075:
4070:
4066:
4053:
4050:
4045:Main article:
4042:
4039:
4024:
4000:
3970:
3967:
3964:
3961:
3939:
3913:
3910:
3907:
3904:
3882:
3877:
3874:
3848:
3842:
3819:Main article:
3816:
3813:
3798:
3774:
3769:
3766:
3740:
3713:
3692:
3681:
3680:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3640:
3636:
3633:
3630:
3627:
3603:
3556:
3532:
3527:
3524:
3502:
3482:
3481:
3467:
3463:
3459:
3456:
3453:
3450:
3447:
3442:
3438:
3434:
3431:
3408:
3403:
3399:
3395:
3392:
3389:
3384:
3380:
3376:
3364:
3361:
3349:
3348:
3336:
3331:
3327:
3323:
3320:
3317:
3312:
3308:
3304:
3299:
3295:
3291:
3288:
3285:
3280:
3276:
3272:
3269:
3264:
3260:
3256:
3253:
3248:
3244:
3240:
3219:
3212:
3190:model-complete
3174:
3158:
3155:
3150:
3146:
3142:
3139:
3136:
3131:
3127:
3123:
3120:
3117:
3114:
3109:
3105:
3101:
3098:
3095:
3090:
3086:
3082:
3079:
3076:
3071:
3067:
3063:
3060:
3055:
3051:
3047:
3033:
3026:
3013:
3006:
2988:
2985:
2971:
2950:
2939:
2938:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2897:, the formula
2885:
2869:parameter-free
2851:
2847:
2843:
2840:
2820:
2817:
2814:
2811:
2808:
2797:
2796:
2785:
2782:
2779:
2776:
2773:
2748:
2742:
2724:
2723:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2674:
2673:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2500:
2488:definable sets
2483:
2482:Definable sets
2480:
2478:
2475:
2439:
2436:
2398:
2374:
2338:
2314:
2284:
2262:
2240:
2218:
2195:
2191:
2162:
2159:
2154:
2132:
2129:
2124:
2100:
2076:
2054:
2043:
2042:
2030:
2025:
2021:
2017:
2014:
2011:
2008:
2005:
2000:
1996:
1992:
1989:
1986:
1981:
1959:
1954:
1950:
1946:
1943:
1940:
1937:
1934:
1929:
1925:
1921:
1918:
1915:
1910:
1883:
1867:
1860:
1845:
1815:
1791:
1763:
1743:
1728:interpretation
1713:
1689:
1667:
1647:
1625:
1600:
1595:
1592:
1588:
1584:
1580:
1576:
1554:
1549:
1527:
1506:
1483:
1478:
1456:
1447:(so that e.g.
1435:
1414:
1394:
1374:
1354:
1334:
1314:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1250:
1247:
1243:
1225:interpretation
1212:
1192:
1160:
1157:
1121:
1101:
1098:
1093:
1060:
1049:
1048:
1036:
1032:
1028:
1025:
1022:
1019:
1016:
1011:
999:
987:
983:
979:
976:
973:
970:
967:
962:
937:
922:irreducibility
909:
889:
869:
846:
824:
810:
807:
806:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
690:
687:
685:
682:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
532:
529:
527:
524:
523:
500:
497:
494:
491:
471:
451:
431:
411:
408:
405:
402:
382:
362:
359:
339:
336:
316:
313:
310:
307:
304:
301:
298:
275:
272:
269:
266:
263:
243:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
196:A first-order
190:Main article:
187:
184:
182:
179:
167:
166:
148:
147:
118:
115:
69:Saharon Shelah
15:
9:
6:
4:
3:
2:
13408:
13407:
13396:
13393:
13391:
13388:
13386:
13383:
13382:
13380:
13367:
13366:
13361:
13353:
13347:
13344:
13342:
13339:
13337:
13334:
13332:
13329:
13325:
13322:
13321:
13320:
13317:
13315:
13312:
13310:
13307:
13305:
13301:
13298:
13296:
13293:
13291:
13288:
13286:
13283:
13281:
13278:
13277:
13275:
13271:
13265:
13262:
13260:
13257:
13255:
13254:Recursive set
13252:
13250:
13247:
13245:
13242:
13240:
13237:
13235:
13232:
13228:
13225:
13223:
13220:
13218:
13215:
13213:
13210:
13208:
13205:
13204:
13203:
13200:
13198:
13195:
13193:
13190:
13188:
13185:
13183:
13180:
13178:
13175:
13174:
13172:
13170:
13166:
13160:
13157:
13155:
13152:
13150:
13147:
13145:
13142:
13140:
13137:
13135:
13132:
13130:
13127:
13123:
13120:
13118:
13115:
13113:
13110:
13109:
13108:
13105:
13103:
13100:
13098:
13095:
13093:
13090:
13088:
13085:
13083:
13080:
13076:
13073:
13072:
13071:
13068:
13064:
13063:of arithmetic
13061:
13060:
13059:
13056:
13052:
13049:
13047:
13044:
13042:
13039:
13037:
13034:
13032:
13029:
13028:
13027:
13024:
13020:
13017:
13015:
13012:
13011:
13010:
13007:
13006:
13004:
13002:
12998:
12992:
12989:
12987:
12984:
12982:
12979:
12977:
12974:
12971:
12970:from ZFC
12967:
12964:
12962:
12959:
12953:
12950:
12949:
12948:
12945:
12943:
12940:
12938:
12935:
12934:
12933:
12930:
12928:
12925:
12923:
12920:
12918:
12915:
12913:
12910:
12908:
12905:
12903:
12900:
12899:
12897:
12895:
12891:
12881:
12880:
12876:
12875:
12870:
12869:non-Euclidean
12867:
12863:
12860:
12858:
12855:
12853:
12852:
12848:
12847:
12845:
12842:
12841:
12839:
12835:
12831:
12828:
12826:
12823:
12822:
12821:
12817:
12813:
12810:
12809:
12808:
12804:
12800:
12797:
12795:
12792:
12790:
12787:
12785:
12782:
12780:
12777:
12775:
12772:
12771:
12769:
12765:
12764:
12762:
12757:
12751:
12746:Example
12743:
12735:
12730:
12729:
12728:
12725:
12723:
12720:
12716:
12713:
12711:
12708:
12706:
12703:
12701:
12698:
12697:
12696:
12693:
12691:
12688:
12686:
12683:
12681:
12678:
12674:
12671:
12669:
12666:
12665:
12664:
12661:
12657:
12654:
12652:
12649:
12647:
12644:
12642:
12639:
12638:
12637:
12634:
12632:
12629:
12625:
12622:
12620:
12617:
12615:
12612:
12611:
12610:
12607:
12603:
12600:
12598:
12595:
12593:
12590:
12588:
12585:
12583:
12580:
12578:
12575:
12574:
12573:
12570:
12568:
12565:
12563:
12560:
12558:
12555:
12551:
12548:
12546:
12543:
12541:
12538:
12536:
12533:
12532:
12531:
12528:
12526:
12523:
12521:
12518:
12516:
12513:
12509:
12506:
12504:
12503:by definition
12501:
12500:
12499:
12496:
12492:
12489:
12488:
12487:
12484:
12482:
12479:
12477:
12474:
12472:
12469:
12467:
12464:
12463:
12460:
12457:
12455:
12451:
12446:
12440:
12436:
12426:
12423:
12421:
12418:
12416:
12413:
12411:
12408:
12406:
12403:
12401:
12398:
12396:
12393:
12391:
12390:KripkeâPlatek
12388:
12386:
12383:
12379:
12376:
12374:
12371:
12370:
12369:
12366:
12365:
12363:
12359:
12351:
12348:
12347:
12346:
12343:
12341:
12338:
12334:
12331:
12330:
12329:
12326:
12324:
12321:
12319:
12316:
12314:
12311:
12309:
12306:
12303:
12299:
12295:
12292:
12288:
12285:
12283:
12280:
12278:
12275:
12274:
12273:
12269:
12266:
12265:
12263:
12261:
12257:
12253:
12245:
12242:
12240:
12237:
12235:
12234:constructible
12232:
12231:
12230:
12227:
12225:
12222:
12220:
12217:
12215:
12212:
12210:
12207:
12205:
12202:
12200:
12197:
12195:
12192:
12190:
12187:
12185:
12182:
12180:
12177:
12175:
12172:
12170:
12167:
12166:
12164:
12162:
12157:
12149:
12146:
12144:
12141:
12139:
12136:
12134:
12131:
12129:
12126:
12124:
12121:
12120:
12118:
12114:
12111:
12109:
12106:
12105:
12104:
12101:
12099:
12096:
12094:
12091:
12089:
12086:
12084:
12080:
12076:
12074:
12071:
12067:
12064:
12063:
12062:
12059:
12058:
12055:
12052:
12050:
12046:
12036:
12033:
12031:
12028:
12026:
12023:
12021:
12018:
12016:
12013:
12011:
12008:
12004:
12001:
12000:
11999:
11996:
11992:
11987:
11986:
11985:
11982:
11981:
11979:
11977:
11973:
11965:
11962:
11960:
11957:
11955:
11952:
11951:
11950:
11947:
11945:
11942:
11940:
11937:
11935:
11932:
11930:
11927:
11925:
11922:
11920:
11917:
11916:
11914:
11912:
11911:Propositional
11908:
11902:
11899:
11897:
11894:
11892:
11889:
11887:
11884:
11882:
11879:
11877:
11874:
11870:
11867:
11866:
11865:
11862:
11860:
11857:
11855:
11852:
11850:
11847:
11845:
11842:
11840:
11839:Logical truth
11837:
11835:
11832:
11831:
11829:
11827:
11823:
11820:
11818:
11814:
11808:
11805:
11803:
11800:
11798:
11795:
11793:
11790:
11788:
11785:
11783:
11779:
11775:
11771:
11769:
11766:
11764:
11761:
11759:
11755:
11752:
11751:
11749:
11747:
11741:
11736:
11730:
11727:
11725:
11722:
11720:
11717:
11715:
11712:
11710:
11707:
11705:
11702:
11700:
11697:
11695:
11692:
11690:
11687:
11685:
11682:
11680:
11677:
11675:
11672:
11668:
11665:
11664:
11663:
11660:
11659:
11657:
11653:
11649:
11642:
11637:
11635:
11630:
11628:
11623:
11622:
11619:
11607:
11606:
11597:
11595:
11594:
11585:
11583:
11582:
11573:
11571:
11570:
11565:
11559:
11558:
11555:
11549:
11546:
11544:
11541:
11539:
11536:
11534:
11531:
11529:
11526:
11522:
11519:
11518:
11517:
11514:
11513:
11511:
11509:
11505:
11499:
11496:
11494:
11491:
11489:
11486:
11484:
11481:
11479:
11476:
11474:
11471:
11470:
11468:
11466:
11465:Computational
11462:
11454:
11451:
11449:
11446:
11444:
11441:
11440:
11439:
11436:
11434:
11431:
11429:
11426:
11424:
11421:
11419:
11416:
11414:
11411:
11409:
11406:
11404:
11401:
11399:
11396:
11394:
11391:
11389:
11386:
11384:
11381:
11380:
11378:
11376:
11372:
11366:
11363:
11361:
11358:
11356:
11353:
11351:
11348:
11346:
11343:
11342:
11340:
11338:
11334:
11328:
11325:
11323:
11320:
11318:
11315:
11313:
11310:
11309:
11307:
11305:
11304:Number theory
11301:
11295:
11292:
11290:
11287:
11285:
11282:
11280:
11277:
11275:
11272:
11270:
11267:
11265:
11262:
11261:
11259:
11257:
11253:
11247:
11244:
11242:
11239:
11237:
11236:Combinatorics
11234:
11233:
11231:
11229:
11225:
11219:
11216:
11214:
11211:
11209:
11206:
11204:
11201:
11199:
11196:
11194:
11191:
11189:
11188:Real analysis
11186:
11184:
11181:
11180:
11178:
11176:
11172:
11166:
11163:
11161:
11158:
11156:
11153:
11151:
11148:
11146:
11143:
11141:
11138:
11136:
11133:
11131:
11128:
11127:
11125:
11123:
11119:
11113:
11110:
11108:
11105:
11103:
11100:
11098:
11095:
11093:
11090:
11088:
11085:
11084:
11082:
11080:
11076:
11070:
11067:
11065:
11062:
11058:
11055:
11053:
11050:
11049:
11048:
11045:
11044:
11041:
11036:
11028:
11023:
11021:
11016:
11014:
11009:
11008:
11005:
10998:
10994:
10990:
10987:
10984:
10983:
10978:
10975:
10974:
10969:
10966:
10963:
10962:
10957:
10954:
10950:
10946:
10945:
10940:
10936:
10929:
10928:
10922:
10915:
10914:
10909:
10905:
10904:
10894:
10888:
10884:
10880:
10875:
10871:
10869:9780521763240
10865:
10861:
10857:
10853:
10849:
10845:
10843:90-5699-313-5
10839:
10835:
10831:
10826:
10822:
10816:
10812:
10808:
10804:
10800:
10796:
10792:
10788:
10784:
10782:0-387-98655-3
10778:
10773:
10772:
10765:
10761:
10759:0-19-853851-0
10755:
10751:
10747:
10743:
10739:
10735:
10733:0-521-30442-3
10729:
10725:
10720:
10719:
10713:
10709:
10705:
10703:1-56881-262-0
10699:
10695:
10691:
10686:
10682:
10680:0-387-94258-0
10676:
10672:
10667:
10666:
10659:
10655:
10653:0-486-44979-3
10649:
10645:
10641:
10636:
10635:
10625:
10623:0-387-98760-6
10619:
10615:
10611:
10606:
10602:
10598:
10593:
10589:
10583:
10579:
10576:. Cambridge:
10575:
10571:
10567:
10563:
10557:
10553:
10549:
10545:
10541:
10537:
10533:
10527:
10523:
10519:
10515:
10511:
10510:
10492:
10488:
10484:
10478:
10474:
10470:
10466:
10465:
10457:
10448:
10440:
10434:
10430:
10423:
10415:
10409:
10405:
10398:
10390:
10384:
10380:
10376:
10372:
10371:
10363:
10355:
10349:
10345:
10341:
10337:
10336:
10328:
10320:
10314:
10310:
10306:
10302:
10301:
10293:
10285:
10281:
10277:
10275:981-256-489-6
10271:
10267:
10263:
10259:
10252:
10244:
10240:
10236:
10230:
10226:
10222:
10218:
10217:
10209:
10200:
10191:
10183:
10179:
10175:
10168:
10160:
10156:
10152:
10148:
10144:
10140:
10133:
10125:
10121:
10117:
10113:
10109:
10105:
10100:
10095:
10091:
10087:
10083:
10076:
10069:
10065:
10059:
10052:
10046:
10033:
10031:9780720420654
10027:
10023:
10019:
10015:
10011:
10004:
9996:
9989:
9976:
9974:9780444863881
9970:
9966:
9962:
9958:
9954:
9947:
9939:
9935:
9931:
9927:
9923:
9919:
9915:
9911:
9907:
9900:
9891:
9883:
9881:9780821848937
9877:
9873:
9869:
9865:
9858:
9845:
9843:9781316855560
9839:
9835:
9831:
9827:
9823:
9816:
9808:
9804:
9800:
9796:
9791:
9786:
9783:(1): 91â121.
9782:
9778:
9774:
9767:
9759:
9755:
9751:
9747:
9743:
9739:
9734:
9729:
9725:
9721:
9717:
9710:
9697:
9695:9781316717158
9691:
9687:
9683:
9679:
9675:
9668:
9660:
9654:
9650:
9646:
9642:
9641:
9633:
9625:
9621:
9617:
9615:0-444-70260-1
9611:
9607:
9606:
9598:
9589:
9583:, p. 136
9582:
9581:Marker (2002)
9577:
9571:, p. 172
9570:
9569:Marker (2002)
9565:
9559:, p. 135
9558:
9557:Marker (2002)
9553:
9545:
9541:
9536:
9531:
9526:
9521:
9517:
9513:
9509:
9505:
9504:
9499:
9492:
9483:
9474:
9465:
9456:
9447:
9438:
9429:
9422:
9421:Marker (2002)
9417:
9408:
9401:
9400:Marker (2002)
9396:
9389:
9388:Marker (2002)
9384:
9371:
9369:9780444533784
9365:
9361:
9357:
9353:
9349:
9342:
9333:
9326:
9325:Marker (2002)
9321:
9315:, p. 208
9314:
9313:Marker (2002)
9309:
9303:, p. 106
9302:
9301:Marker (2002)
9297:
9290:
9289:Marker (2002)
9285:
9277:
9273:
9269:
9265:
9261:
9257:
9253:
9249:
9245:
9238:
9231:
9230:Marker (2002)
9226:
9219:
9218:Marker (2002)
9214:
9207:
9206:Marker (2002)
9202:
9195:
9194:Marker (2002)
9190:
9181:
9174:
9173:Marker (2002)
9169:
9162:
9161:Marker (2002)
9157:
9148:
9142:
9136:
9129:
9128:
9120:
9112:
9108:
9102:
9096:
9090:
9086:
9075:
9072:
9070:
9067:
9065:
9062:
9060:
9057:
9055:
9052:
9050:
9047:
9045:
9042:
9040:
9037:
9035:
9032:
9030:
9027:
9025:
9022:
9020:
9017:
9015:
9012:
9010:
9007:
9005:
9002:
9001:
8994:
8990:
8988:
8984:
8980:
8977:developed by
8976:
8972:
8968:
8963:
8961:
8957:
8953:
8949:
8945:
8941:
8931:
8929:
8925:
8921:
8917:
8913:
8909:
8905:
8904:ultraproducts
8901:
8897:
8893:
8888:
8873:
8871:
8867:
8863:
8859:
8855:
8851:
8845:
8843:
8839:
8835:
8831:
8827:
8823:
8819:
8815:
8811:
8807:
8804:in 1915. The
8803:
8799:
8795:
8791:
8790:Alfred Tarski
8781:
8779:
8775:
8774:PAC-learnable
8771:
8766:
8764:
8760:
8756:
8755:Jonathan Pila
8752:
8748:
8743:
8741:
8737:
8733:
8729:
8725:
8721:
8717:
8713:
8708:
8707:
8703:
8699:
8695:
8691:
8687:
8677:
8674:
8670:
8666:
8662:
8658:
8652:
8650:
8646:
8642:
8638:
8634:
8630:
8625:
8617:
8614:
8610:
8605:
8583:
8579:
8575:
8571:
8567:
8551:
8543:
8539:
8535:
8531:
8527:
8519:
8515:
8511:
8507:
8503:
8502:limit ordinal
8499:
8495:
8492:
8488:
8484:
8480:
8477:
8473:
8469:
8465:
8461:
8459:is non-empty.
8458:
8454:
8453:
8452:
8450:
8446:
8442:
8441:
8424:
8410:
8408:
8392:
8370:
8366:
8357:
8341:
8333:
8332:
8315:
8306:
8302:
8288:
8268:
8260:
8256:
8252:
8244:
8224:
8215:
8190:
8181:
8177:
8174:
8154:
8146:
8143:
8124:
8115:
8106:
8090:
8087:
8080:
8071:
8050:
8042:
8039:
8024:
8016:
8000:
7992:
7991:
7990:
7988:
7984:
7983:
7972:
7970:
7956:
7933:
7908:
7888:
7880:
7872:
7868:
7852:
7832:
7805:
7802:
7794:
7754:
7746:
7732:
7723:
7719:
7718:
7717:
7715:
7709:
7708:Stable theory
7699:
7697:
7688:
7677:
7665:
7661:
7659:
7656:
7655:
7654:
7652:
7648:
7639:
7637:
7628:
7626:
7622:
7618:
7606:
7602:
7601:
7600:
7598:
7580:
7567:
7553:
7550:
7547:
7544:
7541:
7538:
7535:
7515:
7507:
7489:
7485:
7459:
7455:
7451:
7448:
7445:
7440:
7436:
7429:
7421:
7398:
7395:
7364:
7361:
7339:
7335:
7328:
7321:
7317:
7313:
7309:
7306:
7302:
7295:
7291:
7288:
7284:
7277:
7275:-categorical.
7270:
7267:
7266:
7264:
7260:
7259:
7258:
7256:
7247:-categoricity
7241:
7224:
7200:
7192:
7187:
7162:
7158:
7151:
7138:
7133:
7125:
7121:
7116:
7110:
7100:
7061:
7060:
7059:
7057:
7028:
7024:
7017:
7014:
7009:
6980:
6977:
6974:
6954:
6932:
6895:
6892:
6889:
6879:
6875:
6862:
6820:
6803:
6802:
6801:
6799:
6798:ĆoĆ's theorem
6795:
6784:
6780:
6772:
6768:
6764:
6763:Ultraproducts
6758:Ultraproducts
6755:
6753:
6749:
6649:
6635:
6625:
6622:
6570:
6569:
6568:
6565:
6491:
6444:
6443:
6442:
6440:
6436:
6421:
6419:
6400:
6397:
6394:
6391:
6385:
6379:
6376:
6373:
6347:
6344:
6341:
6338:
6332:
6326:
6323:
6320:
6309:
6305:
6300:
6298:
6294:
6290:
6274:
6251:
6248:
6245:
6237:
6226:
6210:
6190:
6182:
6178:
6174:
6158:
6136:
6113:
6067:
6064:that implies
6027:
6023:
6019:
6016:
6013:
6008:
6004:
5959:
5955:
5951:
5948:
5945:
5940:
5936:
5927:
5911:
5902:
5900:
5872:
5844:
5826:
5822:
5818:
5815:
5812:
5807:
5803:
5780:
5776:
5772:
5769:
5766:
5761:
5757:
5748:
5744:
5739:
5694:
5691:
5683:
5655:
5608:
5605:
5597:
5520:
5518:
5507:
5488:
5484:
5480:
5477:
5474:
5469:
5465:
5458:
5451:
5447:
5431:
5411:
5403:
5387:
5379:
5375:
5371:
5353:
5322:
5304:
5263:
5230:
5227:
5219:
5216:
5213:
5202:
5175:
5173:
5169:
5165:
5149:
5129:
5123:
5079:
5059:
5036:
5028:
5024:
4979:
4971:
4952:
4937:
4933:
4924:
4920:
4916:
4912:
4908:
4856:
4853:with at most
4852:
4848:
4847:n-type over A
4820:
4791:
4789:
4768:
4741:
4737:
4733:
4728:
4724:
4701:
4697:
4693:
4688:
4684:
4661:
4657:
4653:
4648:
4644:
4621:
4617:
4613:
4608:
4604:
4595:
4591:
4587:
4566:
4563:
4532:
4530:
4512:
4508:
4504:
4501:
4498:
4493:
4489:
4466:
4462:
4458:
4455:
4452:
4447:
4443:
4420:
4416:
4412:
4409:
4406:
4401:
4397:
4374:
4370:
4366:
4363:
4360:
4355:
4351:
4342:
4314:
4310:
4293:
4289:
4285:
4282:
4279:
4274:
4270:
4262:
4244:
4240:
4236:
4233:
4230:
4225:
4221:
4212:
4191:
4187:
4183:
4180:
4177:
4172:
4168:
4161:
4129:
4126:and a subset
4087:
4083:
4079:
4076:
4073:
4068:
4064:
4052:Basic notions
4048:
4038:
3988:
3987:interpretable
3982:
3965:
3959:
3927:
3908:
3902:
3875:
3872:
3864:
3846:
3829:
3822:
3812:
3767:
3764:
3756:
3728:
3690:
3663:
3660:
3657:
3654:
3651:
3645:
3638:
3631:
3625:
3618:
3617:
3616:
3591:
3588:
3584:
3580:
3576:
3572:
3525:
3522:
3490:
3485:
3465:
3461:
3457:
3454:
3451:
3448:
3445:
3440:
3436:
3432:
3429:
3422:
3421:
3420:
3401:
3397:
3393:
3390:
3387:
3382:
3378:
3360:
3358:
3354:
3329:
3325:
3321:
3318:
3315:
3310:
3306:
3302:
3297:
3293:
3289:
3286:
3283:
3278:
3274:
3267:
3262:
3258:
3251:
3246:
3242:
3231:
3230:
3229:
3227:
3222:
3218:
3211:
3207:
3203:
3199:
3195:
3191:
3187:
3182:
3178:
3172:
3148:
3144:
3140:
3137:
3134:
3129:
3125:
3118:
3107:
3103:
3099:
3096:
3093:
3088:
3084:
3077:
3069:
3065:
3058:
3053:
3049:
3036:
3032:
3025:
3021:
3016:
3012:
3005:
3001:
2997:
2992:
2984:
2948:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2900:
2899:
2898:
2874:
2870:
2865:
2849:
2845:
2841:
2838:
2815:
2812:
2809:
2783:
2780:
2777:
2774:
2771:
2764:
2763:
2762:
2746:
2729:
2707:
2704:
2701:
2698:
2695:
2689:
2679:
2678:
2677:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2627:
2624:
2621:
2618:
2615:
2609:
2603:
2597:
2594:
2591:
2588:
2585:
2579:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2540:
2531:
2525:
2515:
2514:
2513:
2489:
2474:
2472:
2467:
2465:
2460:
2458:
2453:
2449:
2445:
2435:
2431:
2429:
2425:
2421:
2416:
2414:
2362:
2358:
2354:
2302:
2297:
2178:
2174:
2160:
2157:
2130:
2127:
2052:
2023:
2019:
2015:
2012:
2009:
2006:
2003:
1998:
1994:
1987:
1984:
1952:
1948:
1944:
1941:
1938:
1935:
1932:
1927:
1923:
1916:
1913:
1899:
1898:
1897:
1870:
1866:
1859:
1843:
1835:
1830:
1780:
1775:
1761:
1741:
1733:
1729:
1665:
1645:
1612:
1593:
1590:
1586:
1582:
1552:
1525:
1481:
1454:
1412:
1392:
1372:
1352:
1332:
1312:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1253:
1248:
1245:
1241:
1232:
1228:
1226:
1210:
1190:
1182:
1178:
1174:
1170:
1166:
1156:
1154:
1150:
1146:
1141:
1139:
1135:
1119:
1099:
1096:
1082:
1078:
1074:
1058:
1034:
1023:
1017:
1014:
1000:
985:
974:
968:
965:
951:
950:
949:
935:
927:
923:
907:
887:
867:
859:
844:
789:
786:
783:
780:
777:
774:
771:
768:
759:
756:
753:
747:
741:
738:
735:
723:
720:
717:
714:
711:
702:
696:
688:
683:
676:
673:
670:
667:
664:
661:
658:
655:
652:
640:
637:
634:
631:
628:
622:
616:
610:
607:
604:
601:
598:
592:
577:
574:
571:
568:
565:
562:
559:
553:
544:
538:
530:
525:
514:
513:
512:
495:
489:
469:
449:
429:
406:
400:
380:
360:
337:
311:
308:
305:
302:
299:
289:
273:
270:
267:
264:
261:
238:
235:
229:
226:
223:
217:
211:
203:
199:
193:
178:
176:
170:
164:
160:
156:
153:
152:
151:
146:
142:
138:
135:
134:
133:
129:
127:
124:
114:
112:
107:
105:
101:
97:
93:
89:
87:
81:
76:
74:
70:
66:
65:Alfred Tarski
62:
57:
53:
50:), and their
49:
45:
41:
37:
33:
29:
22:
13385:Model theory
13356:
13154:Ultraproduct
13001:Model theory
13000:
12966:Independence
12902:Formal proof
12894:Proof theory
12877:
12850:
12807:real numbers
12779:second-order
12690:Substitution
12567:Metalanguage
12508:conservative
12481:Axiom schema
12425:Constructive
12395:MorseâKelley
12361:Set theories
12340:Aleph number
12333:inaccessible
12239:Grothendieck
12123:intersection
12010:Higher-order
11998:Second-order
11944:Truth tables
11901:Venn diagram
11684:Formal proof
11603:
11591:
11579:
11560:
11493:Optimization
11355:Differential
11279:Differential
11246:Order theory
11241:Graph theory
11145:Group theory
10980:
10971:
10961:Model theory
10959:
10942:
10926:
10912:
10878:
10855:
10852:Tent, Katrin
10829:
10794:
10775:. Springer.
10770:
10746:Model theory
10745:
10718:Model theory
10717:
10689:
10664:
10639:
10609:
10596:
10573:
10548:Model Theory
10547:
10522:Model Theory
10521:
10463:
10456:
10447:
10428:
10422:
10403:
10397:
10369:
10362:
10334:
10327:
10299:
10292:
10266:10.1142/4800
10257:
10251:
10215:
10208:
10199:
10190:
10173:
10167:
10142:
10138:
10132:
10089:
10085:
10075:
10063:
10058:
10045:
10035:, retrieved
10013:
10003:
9994:
9988:
9978:, retrieved
9956:
9946:
9913:
9909:
9899:
9890:
9864:Categoricity
9863:
9857:
9847:, retrieved
9825:
9815:
9780:
9776:
9766:
9733:math/9910158
9723:
9719:
9709:
9699:, retrieved
9677:
9667:
9643:. Springer.
9639:
9632:
9604:
9597:
9588:
9576:
9564:
9552:
9507:
9501:
9491:
9482:
9473:
9464:
9455:
9446:
9437:
9428:
9416:
9407:
9395:
9383:
9373:, retrieved
9351:
9341:
9332:
9327:, p. 97
9320:
9308:
9296:
9291:, p. 45
9284:
9251:
9247:
9237:
9232:, p. 85
9225:
9220:, p. 72
9213:
9208:, p. 71
9201:
9196:, p. 19
9189:
9180:
9175:, p. 45
9168:
9163:, p. 32
9156:
9147:
9135:
9124:
9119:
9110:
9101:
9089:
9064:Proof theory
8991:
8983:independence
8964:
8959:
8955:
8937:
8890:
8846:
8787:
8770:NIP theories
8767:
8759:o-minimality
8744:
8712:ultraproduct
8709:
8683:
8672:
8656:
8653:
8637:completeness
8626:
8623:
8613:NIP theories
8606:
8577:
8573:
8569:
8565:
8541:
8537:
8533:
8529:
8525:
8523:
8517:
8513:
8509:
8505:
8497:
8490:
8486:
8482:
8478:
8471:
8463:
8456:
8444:
8438:
8416:
8406:
8355:
8329:
8307:
8303:
8258:
8254:
8250:
8248:
8144:
8040:
8014:
7986:
7980:
7978:
7949:
7878:
7876:
7870:
7869:-types over
7866:
7792:
7725:
7721:
7713:
7711:
7695:
7689:
7685:
7675:
7663:
7650:
7645:
7629:
7614:
7604:
7568:
7419:
7345:
7340:, is finite.
7337:
7333:
7326:
7319:
7315:
7311:
7307:) is finite.
7304:
7297:
7293:
7286:
7279:
7268:
7262:
7250:
7222:
7160:
7157:-categorical
7153:
7149:
7139:
7119:
7114:
7112:
7103:Categoricity
7097:
7055:
7053:
6835:be a set of
6797:
6793:
6778:
6767:ultraproduct
6766:
6761:
6721:
6566:
6563:
6438:
6434:
6432:
6301:
6296:
6292:
6288:
6203:-types over
6119:
6116:Stone spaces
6065:
5925:
5924:, since all
5903:
5898:
5870:
5842:
5746:
5742:
5740:
5681:
5653:
5595:
5570:realised in
5521:
5516:
5513:
5446:prime ideals
5377:
5373:
5369:
5306:A subset of
5305:
5176:
5171:
5167:
5163:
4969:
4922:
4921:-types over
4918:
4914:
4910:
4906:
4854:
4850:
4846:
4821:a subset of
4818:
4792:
4787:
4593:
4589:
4585:
4533:
4528:
4340:
4313:automorphism
4308:
4260:
4210:
4127:
4055:
3986:
3983:
3925:
3862:
3827:
3824:
3682:
3592:
3586:
3582:
3574:
3570:
3488:
3486:
3483:
3366:
3350:
3225:
3220:
3216:
3209:
3205:
3197:
3193:
3185:
3183:
3179:
3170:
3034:
3030:
3023:
3019:
3014:
3010:
3003:
2999:
2993:
2990:
2940:
2872:
2868:
2866:
2798:
2727:
2725:
2675:
2512:the formula
2485:
2477:Definability
2468:
2461:
2451:
2447:
2441:
2432:
2423:
2419:
2417:
2360:
2356:
2352:
2298:
2176:
2175:
2044:
1868:
1864:
1857:
1833:
1831:
1779:substructure
1776:
1731:
1614:A structure
1613:
1230:
1229:
1171:is a set of
1162:
1142:
1137:
1080:
1079:if it has a
1076:
1050:
860:the formula
857:
808:
422:to indicate
197:
195:
171:
168:
155:model theory
154:
149:
137:model theory
136:
130:
120:
108:
86:proof theory
83:
80:proof theory
77:
32:model theory
31:
25:
13264:Type theory
13212:undecidable
13144:Truth value
13031:equivalence
12710:non-logical
12323:Enumeration
12313:Isomorphism
12260:cardinality
12244:Von Neumann
12209:Ultrafilter
12174:Uncountable
12108:equivalence
12025:Quantifiers
12015:Fixed-point
11984:First-order
11864:Consistency
11849:Proposition
11826:Traditional
11797:Lindström's
11787:Compactness
11729:Type theory
11674:Cardinality
11605:WikiProject
11448:Game theory
11428:Probability
11165:Homological
11155:Multilinear
11135:Commutative
11112:Type theory
11079:Foundations
11035:mathematics
10995:(editors),
10993:S. Feferman
10145:: 572â581.
9726:(1): 1â82.
8962:the model.
8826:interbellum
8696:of a given
8641:compactness
8500:a non-zero
8474:if in some
8440:Morley rank
8259:superstable
7115:categorical
6851:. Then any
6783:ultrafilter
5899:homogeneous
5201:Archimedean
3952:such that
3573:: A theory
2994:This makes
2452:satisfiable
2450:instead of
2274:but not by
1678:is true in
1077:satisfiable
813:-structure
175:cardinality
126:first order
106:in nature.
13379:Categories
13075:elementary
12768:arithmetic
12636:Quantifier
12614:functional
12486:Expression
12204:Transitive
12148:identities
12133:complement
12066:hereditary
12049:Set theory
11433:Statistics
11312:Arithmetic
11274:Arithmetic
11140:Elementary
11107:Set theory
10989:J. Barwise
10694:A K Peters
10599:. Boston:
10502:References
10429:Set Theory
10404:Set Theory
10099:1801.06566
10037:2022-01-23
9980:2022-01-23
9849:2022-01-23
9701:2022-01-15
9375:2022-01-26
8979:Paul Cohen
8971:Kurt Gödel
8967:set theory
8940:set theory
8934:Set theory
8866:Hrushovski
8814:Kurt Gödel
8516:less than
8485:, the set
8013:such that
7724:is called
7152:is called
7090:and :
6993:for which
6893:∈ :
6794:ultrapower
6779:almost all
5897:is called
5372:is called
5170:is called
4343:and sends
3924:such that
3729:structure
3577:is called
3363:Minimality
3188:is called
2831:such that
2448:consistent
1834:elementary
1153:consistent
56:structures
13346:Supertask
13249:Recursion
13207:decidable
13041:saturated
13019:of models
12942:deductive
12937:axiomatic
12857:Hilbert's
12844:Euclidean
12825:canonical
12748:axiomatic
12680:Signature
12609:Predicate
12498:Extension
12420:Ackermann
12345:Operation
12224:Universal
12214:Recursive
12189:Singleton
12184:Inhabited
12169:Countable
12159:Types of
12143:power set
12113:partition
12030:Predicate
11976:Predicate
11891:Syllogism
11881:Soundness
11854:Inference
11844:Tautology
11746:paradoxes
11360:Geometric
11350:Algebraic
11289:Euclidean
11264:Algebraic
11160:Universal
10949:EMS Press
10546:(2012) .
10520:(1990) .
10491:126311148
10243:126311148
10159:1385-7258
10124:119689419
10116:1079-8986
9930:0022-4812
9916:(1): 20.
9799:0894-0347
9758:116922041
9750:0016-2736
9624:800472113
9268:0022-4812
9254:(1): 20.
8944:countable
8592:∞
8552:ω
8524:A theory
8425:ω
8393:λ
8371:λ
8342:λ
8316:ω
8289:ω
8269:ω
8243:continuum
8221:ℵ
8187:ℵ
8178:≥
8175:λ
8155:λ
8121:ℵ
8116:λ
8091:λ
8077:ℵ
8072:λ
8051:λ
8025:λ
8001:λ
7957:ω
7930:ℵ
7909:λ
7889:λ
7881:if it is
7865:complete
7853:λ
7833:λ
7806:⊂
7755:λ
7733:λ
7625:Svenonius
7577:ℵ
7548:⋯
7516:ω
7449:…
7201:σ
7184:ℵ
7018:φ
7015:⊨
6978:∈
6863:φ
6855:-formula
6706:Φ
6626:⊂
6603:Φ
6548:Φ
6477:Φ
6398:∈
6392:≠
6345:∈
6275:φ
6249:∈
6246:φ
6225:generated
6017:…
5949:…
5816:…
5770:…
5695:⊂
5682:saturated
5609:∈
5478:…
5264:⊆
5231:∈
5150:ψ
5130:ψ
5127:→
5124:φ
5080:φ
4769:⊆
4567:∈
4502:…
4456:…
4410:…
4364:…
4283:…
4234:…
4181:…
4162:φ
4077:…
3960:φ
3903:φ
3876:∈
3768:⊆
3755:o-minimal
3691:φ
3655:×
3643:∃
3626:φ
3526:⊆
3452:∨
3449:⋯
3446:∨
3391:…
3319:…
3287:…
3268:ψ
3255:∃
3252:…
3239:∃
3184:A theory
3138:…
3119:ψ
3116:↔
3097:…
3078:ϕ
3062:∀
3059:…
3046:∀
2949:π
2926:π
2917:×
2781:×
2699:×
2687:∃
2658:≠
2652:∧
2646:≠
2640:∧
2619:×
2607:∃
2604:∨
2589:×
2577:∃
2571:→
2562:×
2550:×
2538:∃
2529:∀
2523:∀
2424:expansion
2413:injective
2351:is a map
2301:embedding
2194:¯
2161:φ
2158:⊨
2131:φ
2128:⊨
2053:φ
1988:φ
1985:⊨
1917:φ
1914:⊨
1844:φ
1587:σ
1393:−
1373:×
1284:−
1278:×
1242:σ
1183:is a set
1181:structure
1165:signature
1097:⊨
1031:⟺
1018:ψ
1015:⊨
982:⟺
969:φ
966:⊨
936:⊨
908:ψ
868:φ
858:satisfies
787:≠
781:∧
775:≠
769:∧
748:∨
730:→
715:×
700:∀
694:∀
684:ψ
671:≠
665:∧
659:≠
653:∧
632:×
620:∃
617:∨
602:×
590:∃
584:→
575:×
563:×
551:∃
542:∀
536:∀
526:φ
490:ψ
470:ψ
450:φ
401:φ
381:φ
358:∃
335:∀
315:→
309:∨
303:∧
297:¬
100:syntactic
92:algebraic
40:sentences
13331:Logicism
13324:timeline
13300:Concrete
13159:Validity
13129:T-schema
13122:Kripke's
13117:Tarski's
13112:semantic
13102:Strength
13051:submodel
13046:spectrum
13014:function
12862:Tarski's
12851:Elements
12838:geometry
12794:Robinson
12715:variable
12700:function
12673:spectrum
12663:Sentence
12619:variable
12562:Language
12515:Relation
12476:Automata
12466:Alphabet
12450:language
12304:-jection
12282:codomain
12268:Function
12229:Universe
12199:Infinite
12103:Relation
11886:Validity
11876:Argument
11774:theorem,
11581:Category
11337:Topology
11284:Discrete
11269:Analytic
11256:Geometry
11228:Discrete
11183:Calculus
11175:Analysis
11130:Abstract
11069:Glossary
11052:Timeline
10910:(2001).
10799:New York
10793:(2010).
10744:(1999).
10714:(1993).
10671:Springer
10572:(1997).
10284:62715985
9807:12044966
9544:16591050
8997:See also
8916:0-1 laws
8850:James Ax
8842:Berkeley
8724:Kochen's
8718:work on
8512:for all
8251:unstable
8037:-stable.
7218:, where
7046:lies in
6435:omitting
5738:itself.
5172:isolated
5116:implies
4915:complete
2177:Example:
2113:, then
1305:, where
1231:Example:
1169:language
1134:sentence
204:such as
123:finitary
117:Overview
104:semantic
13273:Related
13070:Diagram
12968: (
12947:Hilbert
12932:Systems
12927:Theorem
12805:of the
12750:systems
12530:Formula
12525:Grammar
12441: (
12385:General
12098:Forcing
12083:Element
12003:Monadic
11778:paradox
11719:Theorem
11655:General
11593:Commons
11375:Applied
11345:General
11122:Algebra
11047:History
10951:, 2001
9938:2274425
9512:Bibcode
9276:2274425
9127:page 1.
8985:of the
8975:forcing
8960:outside
8784:History
8405:, then
8207:(where
7969:-stable
7745:-stable
7674:, then
7617:Engeler
7325:, ...,
6754:holds.
5598:for an
5448:of the
5166:, then
4905:. If
3215:, ...,
3029:, ...,
3009:, ...,
1863:, ...,
198:formula
61:defined
54:(those
13036:finite
12799:Skolem
12752:
12727:Theory
12695:Symbol
12685:String
12668:atomic
12545:ground
12540:closed
12535:atomic
12491:ground
12454:syntax
12350:binary
12277:domain
12194:Finite
11959:finite
11817:Logics
11776:
11724:Theory
11294:Finite
11150:Linear
11057:Future
11033:Major
10889:
10866:
10840:
10817:
10779:
10756:
10730:
10700:
10677:
10650:
10620:
10584:
10558:
10528:
10489:
10479:
10435:
10410:
10385:
10350:
10315:
10282:
10272:
10241:
10231:
10157:
10122:
10114:
10028:
9971:
9936:
9928:
9878:
9840:
9805:
9797:
9756:
9748:
9692:
9655:
9622:
9612:
9542:
9535:299780
9532:
9366:
9274:
9266:
8956:within
8582:U-rank
7879:stable
7692:ω
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