43:
12073:
242:
2442:
9525:. Types are inductively defined; given two types δ and σ there is also a type σ → δ that represents functions from objects of type σ to objects of type δ. A structure for a typed language (in the ordinary first-order semantics) must include a separate set of objects of each type, and for a function type the structure must have complete information about the function represented by each object of that type.
9547:. When using full higher-order semantics, a structure need only have a universe for objects of type 0, and the T-schema is extended so that a quantifier over a higher-order type is satisfied by the model if and only if it is disquotationally true. When using first-order semantics, an additional sort is added for each higher-order type, as in the case of a many sorted first order language.
2095:
2224:
1908:
7046:
Structures are sometimes referred to as "first-order structures". This is misleading, as nothing in their definition ties them to any specific logic, and in fact they are suitable as semantic objects both for very restricted fragments of first-order logic such as that used in universal algebra, and
9433:
Both universal algebra and model theory study classes of (structures or) algebras that are defined by a signature and a set of axioms. In the case of model theory these axioms have the form of first-order sentences. The formalism of universal algebra is much more restrictive; essentially it only
7027:
on a database can be described by another structure in the same signature as the database model. A homomorphism from the relational model to the structure representing the query is the same thing as a solution to the query. This shows that the conjunctive query problem is also equivalent to the
4507:
The set of integers gives an even smaller substructure of the real numbers which is not a field. Indeed, the integers are the substructure of the real numbers generated by the empty set, using this signature. The notion in abstract algebra that corresponds to a substructure of a field, in this
2437:{\displaystyle {\begin{alignedat}{3}\operatorname {ar} _{f}&(+)&&=2,\\\operatorname {ar} _{f}&(\times )&&=2,\\\operatorname {ar} _{f}&(-)&&=1,\\\operatorname {ar} _{f}&(0)&&=0,\\\operatorname {ar} _{f}&(1)&&=0.\\\end{alignedat}}}
9509:
0 × 0 = 1 is not true.) Therefore, one is naturally led to allow partial functions, i.e., functions that are defined only on a subset of their domain. However, there are several obvious ways to generalize notions such as substructure, homomorphism and identity.
9508:
In the case of fields this strategy works only for addition. For multiplication it fails because 0 does not have a multiplicative inverse. An ad hoc attempt to deal with this would be to define 0 = 0. (This attempt fails, essentially because with this definition
5494:
9396:
Many-sorted structures are often used as a convenient tool even when they could be avoided with a little effort. But they are rarely defined in a rigorous way, because it is straightforward and tedious (hence unrewarding) to carry out the generalization explicitly.
6681:
9391:
7849:
9497:). One consequence is that the choice of a signature is more significant in universal algebra than it is in model theory. For example, the class of groups, in the signature consisting of the binary function symbol × and the constant symbol 1, is an
2090:{\displaystyle {\begin{alignedat}{3}{\mathcal {Q}}&=(\mathbb {Q} ,\sigma _{f},I_{\mathcal {Q}})\\{\mathcal {R}}&=(\mathbb {R} ,\sigma _{f},I_{\mathcal {R}})\\{\mathcal {C}}&=(\mathbb {C} ,\sigma _{f},I_{\mathcal {C}})\\\end{alignedat}}}
8858:
also prescribe on which sorts the functions and relations of a many-sorted structure are defined. Therefore, the arities of function symbols or relation symbols must be more complicated objects such as tuples of sorts rather than natural numbers.
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2602:
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1568:
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2156:
762:
6990:
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5324:
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2823:
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6511:
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8342:
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3704:
9286:
8337:
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7109:
4223:
346:
4470:
3935:
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9806:
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1804:
together with two binary functions, that can be enhanced with a unary function, and two distinguished elements; but there is no requirement that it satisfy any of the field axioms. The
2219:
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576:
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603:
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7292:
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to distinguish them from the "set models" discussed above. When the domain is a proper class, each function and relation symbol may also be represented by a proper class.
4249:
2932:
1855:
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1694:
1672:
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3048:
8167:
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9452:
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400:
7163:
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4344:
1903:
463:
443:
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1395:
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1340:
1285:
2704:
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941:
374:
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8238:
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8144:
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7391:
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7268:
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1802:
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782:
703:
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is the same thing as a homomorphism between the two structures coding the graph. In the example of the previous section, even though the subgraph
2102:
11127:
6240:
1494:
4408:
of two subsets is the closed subset generated by their union. Universal algebra studies the lattice of substructures of a structure in detail.
1195:
8865:, for example, can be regarded as two-sorted structures in the following way. The two-sorted signature of vector spaces consists of two sorts
9409:
2229:
1913:
11210:
10351:
6848:
As seen above, in the standard encoding of graphs as structures the induced substructures are precisely the induced subgraphs. However, a
163:
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723:
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1345:
When a structure (and hence an interpretation function) is given by context, no notational distinction is made between a symbol
182:
5489:{\displaystyle (a_{1},a_{2},\dots ,a_{n})\in R^{\mathcal {A}}\implies (h(a_{1}),h(a_{2}),\dots ,h(a_{n}))\in R^{\mathcal {B}}}
11682:
10300:
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10215:
10193:
10168:
10139:
10117:
10095:
10071:
9636:
3420:
425:
that indicates how the signature is to be interpreted on the domain. To indicate that a structure has a particular signature
288:. Since the 19th century, one main method for proving the consistency of a set of axioms has been to provide a model for it.
10470:
7361:
Thus, for example, a "ring" is a structure for the language of rings that satisfies each of the ring axioms, and a model of
4504:. The rational numbers are the smallest substructure of the real (or complex) numbers that also satisfies the field axioms.
1447:
11537:
10860:
17:
6676:{\displaystyle (a_{1},a_{2},\dots ,a_{n})\in R^{\mathcal {A}}\iff (h(a_{1}),h(a_{2}),\dots ,h(a_{n}))\in R^{\mathcal {B}}}
511:
2782:
2711:
11542:
11532:
11269:
11122:
10475:
3821:
10466:
9386:{\displaystyle \times ^{\mathcal {V}}:|{\mathcal {V}}|_{S}\times |{\mathcal {V}}|_{V}\rightarrow |{\mathcal {V}}|_{V}}
4689:
are connected by an edge. In this encoding, the notion of induced substructure is more restrictive than the notion of
4077:
11678:
10025:
646:
253:
86:
64:
11020:
8737:
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57:
12102:
11775:
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10344:
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9416:
because a logic over a type theory categorically corresponds to one ("total") category, capturing the logic, being
8287:
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4180:
303:
11080:
10773:
6997:
4419:
3900:
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9657:
The fact that such classes constitute a model of the traditional real number system was pointed out by
Dedekind.
7117:
12036:
11501:
11496:
11321:
10742:
2161:
790:
492:
210:", whereas the term "interpretation" generally has a different (although related) meaning in model theory, see
6446:
837:
12031:
11814:
11731:
11444:
11375:
11252:
10494:
4690:
972:
211:
9105:
9058:
8855:
8192:
is more common amongst set theorists, while the opposite convention is more common amongst model theorists.
7844:{\displaystyle (a_{1},\ldots ,a_{n})\in R\Leftrightarrow {\mathcal {M}}\vDash \varphi (a_{1},\ldots ,a_{n})}
11956:
11782:
11468:
11102:
10701:
4611:
4520:
3749:
1091:
12097:
11834:
11829:
11439:
11178:
11107:
10436:
10337:
9502:
7362:
7248:
4793:
605:
but often no notational distinction is made between a structure and its domain (that is, the same symbol
9434:
allows first-order sentences that have the form of universally quantified equations between terms, e.g.
8463:{\displaystyle R=\{(a_{1},\ldots ,a_{n})\in M^{n}:{\mathcal {M}}\vDash \varphi (a_{1},\ldots ,a_{n})\}.}
7695:{\displaystyle R=\{(a_{1},\ldots ,a_{n})\in M^{n}:{\mathcal {M}}\vDash \varphi (a_{1},\ldots ,a_{n})\}.}
1765:
1633:
12112:
11763:
11353:
10747:
10715:
10406:
10185:
10160:
5857:
4051:
3380:
3132:
167:
9701:
8047:
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6691:
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5504:
4143:
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2883:
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2449:
547:
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11358:
10835:
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9739:
8794:
8599:
7337:
4369:
4297:
4133:
3527:
3294:
1420:
581:
10509:
9627:
Hodges, Wilfrid (2009). "Functional
Modelling and Mathematical Models". In Meijers, Anthonie (ed.).
9034:
8713:
8646:
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8263:
8105:
7897:
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608:
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177:
From the model-theoretic point of view, structures are the objects used to define the semantics of
51:
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27:
Mapping of mathematical formulas to a particular meaning, in universal algebra and in model theory
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11703:
11664:
11550:
11491:
11137:
11057:
10901:
10845:
10458:
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Thus an embedding is the same thing as a strong homomorphism which is one-to-one. The category σ-
2972:
2941:
1580:
277:
7498:
7472:
3028:
196:
if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a
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10980:
10946:
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3050:
and therefore structures for such a signature are not algebras, even though they are of course
207:
186:
68:
9461:
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8539:
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10059:
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7147:
7037:
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448:
428:
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9833:
9613:
Some authors refer to structures as "algebras" when generalizing universal algebra to allow
11919:
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11758:
11562:
11402:
11326:
11304:
11132:
11090:
10989:
10956:
10820:
10608:
10519:
9979:
Jeavons, Peter; Cohen, David; Pearson, Justin (1998), "Constraints and universal algebra",
9917:
3060:
1784:
respectively). Thus a structure (algebra) for this signature consists of a set of elements
1368:
960:
7212:
which is interpreted as that element. This relation is defined inductively using Tarski's
3083:
3080:
A structure for this signature consists of a set of elements and an interpretation of the
1316:
1266:
8:
12048:
11939:
11924:
11904:
11861:
11748:
11698:
11624:
11569:
11506:
11299:
11294:
11242:
11010:
10999:
10671:
10571:
10499:
10490:
10486:
10421:
10416:
10250:
7005:
3051:
2683:
2597:{\displaystyle I_{\mathcal {Q}}(\times ):\mathbb {Q} \times \mathbb {Q} \to \mathbb {Q} }
1605:
1025:
assigns functions and relations to the symbols of the signature. To each function symbol
951:
497:
474:
350:
139:
131:
127:
9412:
puts it: "A logic is always a logic over a type theory." This emphasis in turn leads to
8670:
7192:
7041:
4756:
4589:
4546:
4324:
4028:
4001:
3974:
3799:
3554:
2828:
2757:
923:
356:
12077:
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11809:
11794:
11787:
11770:
11574:
11556:
11422:
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11284:
11097:
11006:
10840:
10825:
10785:
10737:
10722:
10710:
10666:
10641:
10411:
10360:
10315:
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10106:
9996:
9897:
9544:
9540:
9534:
8770:
8693:
8579:
8559:
8495:
8243:
8223:
8203:
8129:
8023:
7877:
7857:
7705:
7416:
7396:
7376:
7317:
7297:
7253:
7048:
6849:
6428:
5562:
4736:
4716:
4696:
4672:
4652:
4569:
4405:
4401:
4349:
3729:
3709:
3597:
3577:
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1787:
1400:
1348:
1290:
1175:
1155:
1135:
1069:
1048:
1028:
984:
949:
often contain only function symbols, a signature with no relation symbols is called an
899:
875:
767:
688:
408:
203:
135:
31:
11030:
9563:, it is sometimes useful to consider structures in which the domain of discourse is a
8854:
are part of the signature, and they play the role of names for the different domains.
12072:
12012:
11819:
11629:
11619:
11511:
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11227:
11203:
10984:
10968:
10873:
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10296:
10274:
10237:
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10189:
10164:
10135:
10113:
10091:
10067:
10021:
9632:
9413:
9401:
8821:
7466:
7051:. In connection with first-order logic and model theory, structures are often called
7024:
5846:
4904:
638:
377:
178:
120:
112:
100:
10000:
7365:
is a structure in the language of set theory that satisfies each of the ZFC axioms.
12026:
12021:
11914:
11871:
11693:
11654:
11649:
11634:
11460:
11417:
11314:
11112:
11062:
10636:
10598:
10266:
10045:
9988:
9575:
9498:
8169:
Every element of a structure is definable using the element itself as a parameter.
7016:
5214:{\displaystyle h(f(a_{1},a_{2},\dots ,a_{n}))=f(h(a_{1}),h(a_{2}),\dots ,h(a_{n}))}
4781:
2533:{\displaystyle I_{\mathcal {Q}}(+):\mathbb {Q} \times \mathbb {Q} \to \mathbb {Q} }
713:
707:
281:
226:
10153:
12007:
11997:
11951:
11934:
11889:
11851:
11753:
11673:
11480:
11407:
11380:
11368:
11274:
11188:
11162:
11117:
11085:
10886:
10688:
10631:
10581:
10546:
10504:
10233:
10207:
10087:
10086:, Graduate Texts in Mathematics, vol. 173 (3rd ed.), Berlin, New York:
10051:
9560:
9417:
7012:
4566:
The vertices of the graph form the domain of the structure, and for two vertices
4513:
4493:
4397:
1805:
1309:
218:
155:
11992:
11971:
11929:
11909:
11804:
11659:
11257:
11247:
11237:
11232:
11166:
11040:
10916:
10805:
10800:
10778:
10379:
10177:
10148:
8006:{\displaystyle {\mathcal {M}}\vDash \forall x(x=m\leftrightarrow \varphi (x)).}
7854:
An important special case is the definability of specific elements. An element
4501:
1858:
832:
785:
198:
10270:
9992:
501:. In classical first-order logic, the definition of a structure prohibits the
12091:
11966:
11644:
11151:
10936:
10926:
10896:
10881:
10551:
10258:
7023:
is essentially the same thing as a relational structure. It turns out that a
3000:
10228:
A Course in Model Theory: An
Introduction to Contemporary Mathematical Logic
9505:. Universal algebra solves this problem by adding a unary function symbol .
9400:
In most mathematical endeavours, not much attention is paid to the sorts. A
3551:
that is, if the following condition is satisfied: for every natural number
11866:
11713:
11614:
11606:
11486:
11434:
11343:
11279:
11262:
11193:
11052:
10911:
10613:
10396:
9564:
8862:
6869:
6829:
4777:
502:
159:
143:
104:
6361:{\displaystyle b_{1}=h(a_{1}),\,b_{2}=h(a_{2}),\,\dots ,\,b_{n}=h(a_{n}).}
1563:{\displaystyle f^{\mathcal {A}}:|{\mathcal {A}}|^{2}\to |{\mathcal {A}}|.}
11976:
11856:
11035:
11025:
10972:
10656:
10576:
10561:
10441:
10386:
9733:
9660:
9518:
9405:
6806:
4497:
1833:
1652:
where additional symbols can be derived, such as a unary function symbol
1256:{\displaystyle R^{\mathcal {A}}=I(R)\subseteq A^{\operatorname {ar(R)} }}
6836:. In this case induced substructures also correspond to subobjects in σ-
3057:
The ordinary signature for set theory includes a single binary relation
241:
10906:
10761:
10732:
10538:
10131:
9686:
9584:, structures were also allowed to have a proper class as their domain.
9556:
8850:. A many-sorted structure can have an arbitrary number of domains. The
285:
171:
7000:(CSP) has a translation into the homomorphism problem. Therefore, the
6371:
The strong homomorphisms give rise to a subcategory of the category σ-
4393:
Conversely, the domain of an induced substructure is a closed subset.
12058:
11961:
11014:
10931:
10891:
10855:
10791:
10603:
10593:
10566:
10329:
10292:
6881:
6817:
4137:
479:
The domain of a structure is an arbitrary set; it is also called the
252: with: explicit mention of the term "structure". You can help by
4713:
be a graph consisting of two vertices connected by an edge, and let
4396:
The closed subsets (or induced substructures) of a structure form a
221:, structures with no functions are studied as models for relational
12043:
11841:
11289:
10994:
10588:
9653:
Oxford
English Dictionary, s.v. "model, n., sense I.8.b", July 2023
7213:
7020:
222:
116:
7055:, even when the question "models of what?" has no obvious answer.
3377:
the interpretations of all function and relation symbols agree on
2965:-structure in the same way. In fact, there is no requirement that
11639:
10431:
4509:
2935:
946:
9631:. Handbook of the Philosophy of Science. Vol. 9. Elsevier.
3258:{\displaystyle \sigma ({\mathcal {A}})=\sigma ({\mathcal {B}});}
2653:{\displaystyle I_{\mathcal {Q}}(-):\mathbb {Q} \to \mathbb {Q} }
10104:
Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang (1994),
8824:, every implicitly definable relation is explicitly definable.
9276:{\displaystyle 0_{S}^{\mathcal {V}}=0\in |{\mathcal {V}}|_{S}}
9209:{\displaystyle 0_{V}^{\mathcal {V}}=0\in |{\mathcal {V}}|_{V}}
6230:{\displaystyle (a_{1},a_{2},\dots ,a_{n})\in R^{\mathcal {A}}}
6071:{\displaystyle (b_{1},b_{2},\dots ,b_{n})\in R^{\mathcal {B}}}
4959:{\displaystyle h:|{\mathcal {A}}|\rightarrow |{\mathcal {B}}|}
11183:
10529:
10374:
9685:
A logical system that allows the empty domain is known as an
4472:
be again the standard signature for fields. When regarded as
1066:
917:
273:
202:
when one discusses the notion in the more general setting of
10323:
6150:{\displaystyle a_{1},a_{2},\dots ,a_{n}\in |{\mathcal {A}}|}
5991:{\displaystyle b_{1},b_{2},\dots ,b_{n}\in |{\mathcal {B}}|}
5308:{\displaystyle a_{1},a_{2},\dots ,a_{n}\in |{\mathcal {A}}|}
5050:{\displaystyle a_{1},a_{2},\dots ,a_{n}\in |{\mathcal {A}}|}
4966:
that preserves the functions and relations. More precisely:
192:
For a given theory in model theory, a structure is called a
9420:
over another ("base") category, capturing the type theory.
4733:
be the graph consisting of the same vertices but no edges.
3367:{\displaystyle |{\mathcal {A}}|\subseteq |{\mathcal {B}}|;}
3103:
2151:{\displaystyle \sigma _{f}=(S_{f},\operatorname {ar} _{f})}
2099:
In all three cases we have the standard signature given by
757:{\displaystyle \operatorname {ar} :\ S\to \mathbb {N} _{0}}
6985:{\displaystyle h:{\mathcal {A}}\rightarrow {\mathcal {B}}}
6416:{\displaystyle h:{\mathcal {A}}\rightarrow {\mathcal {B}}}
5896:{\displaystyle h:{\mathcal {A}}\rightarrow {\mathcal {B}}}
5715:{\displaystyle h:{\mathcal {A}}\rightarrow {\mathcal {B}}}
9521:, there are many sorts of variables, each of which has a
5856:
which has σ-structures as objects and σ-homomorphisms as
1342:
can be identified with a constant element of the domain.
126:
Universal algebra studies structures that generalize the
9567:
instead of a set. These structures are sometimes called
3447:{\displaystyle {\mathcal {A}}\subseteq {\mathcal {B}}.}
6952:
of a finite relational signature, find a homomorphism
1484:{\displaystyle f:{\mathcal {A}}^{2}\to {\mathcal {A}}}
955:. A structure with such a signature is also called an
162:
has a different scope that encompasses more arbitrary
9951:
9920:
9900:
9865:
9836:
9814:
9781:
9742:
9704:
9550:
9464:
9440:
9289:
9222:
9155:
9149:, and the obvious functions, such as the vector zero
9108:
9061:
9037:
8797:
8773:
8740:
8716:
8696:
8673:
8649:
8629:
8602:
8582:
8562:
8542:
8518:
8498:
8478:
8345:
8290:
8266:
8246:
8226:
8206:
8152:
8132:
8108:
8088:
8050:
8026:
7953:
7924:
7900:
7880:
7860:
7748:
7728:
7708:
7577:
7522:
7501:
7475:
7439:
7419:
7399:
7379:
7340:
7320:
7300:
7276:
7256:
7225:
7195:
7171:
7150:
7120:
7069:
6958:
6934:
6910:
6780:
6756:
6723:
6694:
6514:
6449:
6389:
6243:
6163:
6084:
6004:
5925:
5869:
5824:
5800:
5766:
5732:
5688:
5664:
5640:
5609:
5585:
5565:
5536:
5507:
5327:
5242:
5069:
4984:
4912:
4885:
4861:
4833:
4809:
4780:
that corresponds to induced substructures is that of
4759:
4739:
4719:
4699:
4675:
4655:
4614:
4592:
4572:
4549:
4529:
4478:
4422:
4372:
4352:
4327:
4300:
4257:
4231:
4183:
4146:
4115:
4080:
4054:
4031:
4004:
3977:
3943:
3903:
3824:
3802:
3752:
3732:
3712:
3644:
3620:
3600:
3580:
3557:
3530:
3502:
3462:
3423:
3383:
3323:
3297:
3273:
3216:
3192:
3168:
3141:
3113:
3086:
3063:
3031:
3009:
2975:
2944:
2918:
2886:
2857:
2831:
2785:
2760:
2714:
2686:
2666:
2612:
2548:
2484:
2452:
2227:
2164:
2105:
1911:
1891:
1866:
1841:
1813:
1790:
1768:
1746:
1724:
1702:
1680:
1658:
1636:
1614:
1583:
1497:
1450:
1423:
1403:
1371:
1351:
1319:
1293:
1269:
1198:
1178:
1158:
1138:
1094:
1072:
1051:
1031:
1007:
987:
926:
902:
878:
840:
793:
770:
726:
691:
649:
611:
584:
550:
514:
451:
431:
411:
385:
359:
306:
10103:
9698:
As a consequence of these conventions, the notation
8643:
in the extended language containing the language of
7189:
together with a constant symbol for each element of
6746:
refers to the interpretation of the relation symbol
959:; this should not be confused with the notion of an
537:{\displaystyle \operatorname {dom} ({\mathcal {A}})}
10043:
9978:
7031:
2818:{\displaystyle I_{\mathcal {Q}}(1)\in \mathbb {Q} }
2747:{\displaystyle I_{\mathcal {Q}}(0)\in \mathbb {Q} }
10225:
10152:
10105:
9962:
9933:
9906:
9886:
9852:
9822:
9800:
9755:
9724:
9470:
9446:
9385:
9275:
9208:
9141:
9094:
9047:
8873:(for scalars) and the following function symbols:
8810:
8779:
8759:
8726:
8702:
8682:
8659:
8635:
8615:
8588:
8568:
8548:
8528:
8504:
8484:
8462:
8331:
8276:
8252:
8232:
8212:
8161:
8138:
8118:
8094:
8070:
8032:
8005:
7939:
7910:
7886:
7866:
7843:
7734:
7714:
7694:
7563:
7507:
7481:
7449:
7425:
7405:
7385:
7353:
7326:
7306:
7286:
7262:
7235:
7204:
7181:
7157:
7136:
7103:
6984:
6944:
6920:
6790:
6766:
6738:
6709:
6675:
6494:
6415:
6360:
6229:
6149:
6070:
5990:
5895:
5834:
5810:
5786:
5752:
5714:
5674:
5650:
5619:
5595:
5571:
5551:
5522:
5488:
5307:
5213:
5049:
4958:
4895:
4871:
4843:
4819:
4768:
4745:
4725:
4705:
4681:
4661:
4641:
4601:
4578:
4558:
4535:
4500:, and the real numbers form a substructure of the
4484:
4464:
4385:
4358:
4338:
4313:
4286:
4243:
4217:
4166:
4124:
4101:
4066:
4040:
4013:
3986:
3963:
3929:
3887:
3811:
3788:
3738:
3718:
3698:
3630:
3606:
3586:
3566:
3543:
3512:
3488:
3446:
3406:
3366:
3310:
3283:
3257:
3202:
3178:
3151:
3123:
3092:
3072:
3042:
3017:
2988:
2957:
2926:
2901:
2872:
2840:
2817:
2769:
2746:
2698:
2672:
2652:
2596:
2532:
2467:
2436:
2213:
2150:
2089:
1897:
1877:
1849:
1824:
1796:
1776:
1754:
1732:
1710:
1688:
1666:
1644:
1622:
1596:
1562:
1483:
1436:
1409:
1389:
1357:
1334:
1299:
1279:
1255:
1184:
1164:
1144:
1124:
1078:
1057:
1037:
1017:
993:
935:
908:
884:
864:
820:
776:
756:
697:
673:
621:
597:
570:
536:
457:
437:
417:
394:
368:
340:
10044:Burris, Stanley N.; Sankappanavar, H. P. (1981),
9981:Annals of Mathematics and Artificial Intelligence
9771:
9769:
9629:Philosophy of technology and engineering sciences
9512:
8832:Structures as defined above are sometimes called
4630:
3888:{\displaystyle f(b_{1},b_{2},\dots ,b_{n})\in B.}
3100:relation as a binary relation on these elements.
12089:
4346:assigns to every symbol of σ the restriction to
4102:{\displaystyle \langle B\rangle _{\mathcal {A}}}
2660:is the function that takes each rational number
272:In the context of mathematical logic, the term "
8015:
7722:is definable if and only if there is a formula
6805:of σ-structures and σ-embeddings is a concrete
4787:
4776:but not an induced substructure. The notion in
674:{\displaystyle \sigma =(S,\operatorname {ar} )}
284:(1831 – 1916), a pioneer in the development of
276:" was first applied in 1940 by the philosopher
9766:
9539:There is more than one possible semantics for
8760:{\displaystyle {\mathcal {M}}\vDash \varphi ,}
8284:is explicitly definable if there is a formula
7165:in the language consisting of the language of
4543:consisting of a single binary relation symbol
3699:{\displaystyle b_{1},b_{2},\dots ,b_{n}\in B,}
629:refers both to the structure and its domain.)
206:. Logicians sometimes refer to structures as "
10345:
8332:{\displaystyle \varphi (x_{1},\ldots ,x_{n})}
7564:{\displaystyle \varphi (x_{1},\ldots ,x_{n})}
5559:is the interpretation of the relation symbol
10255:A Concise Introduction to Mathematical Logic
10057:
8454:
8352:
7686:
7584:
7104:{\displaystyle {\mathcal {M}}=(M,\sigma ,I)}
4794:Universal algebra § Basic constructions
4459:
4429:
4218:{\displaystyle {\mathcal {A}}=(A,\sigma ,I)}
4119:
4116:
4088:
4081:
4061:
4055:
3003:needs an additional binary relation such as
2208:
2178:
341:{\displaystyle {\mathcal {A}}=(A,\sigma ,I)}
4465:{\displaystyle \sigma =\{+,\times ,-,0,1\}}
3930:{\displaystyle B\subseteq |{\mathcal {A}}|}
3489:{\displaystyle B\subseteq |{\mathcal {A}}|}
10537:
10352:
10338:
10249:
9801:{\displaystyle \mathbf {0} ,\mathbf {1} ,}
9763:In practice this never leads to confusion.
8841:to distinguish them from the more general
7137:{\displaystyle {\mathcal {M}}\vDash \phi }
6860:is not induced, the identity map id:
6582:
6578:
5395:
5391:
4404:of two subsets is their intersection. The
966:
10286:
9953:
9914:on the right refer to natural numbers of
9528:
9031:, the corresponding two-sorted structure
8827:
8184:. Broadly speaking, the convention that
7151:
6992:or show that no such homomorphism exists.
6868:is a homomorphism. This map is in fact a
6322:
6315:
6279:
3039:
3032:
3014:
3010:
2920:
2811:
2740:
2646:
2638:
2590:
2582:
2574:
2526:
2518:
2510:
2214:{\displaystyle S_{f}=\{+,\times ,-,0,1\}}
2048:
1991:
1934:
1885:like any other field, can be regarded as
1868:
1843:
1815:
1273:
821:{\displaystyle n=\operatorname {ar} (s).}
744:
87:Learn how and when to remove this message
9423:
8195:
7058:
6750:of the object theory σ in the structure
6495:{\displaystyle a_{1},a_{2},\dots ,a_{n}}
3417:The usual notation for this relation is
3104:Induced substructures and closed subsets
1608:consists of two binary function symbols
865:{\displaystyle n=\operatorname {ar} (s)}
50:This article includes a list of general
10079:
6891:
3524:if it is closed under the functions of
3054:in the usual, loose sense of the word.
487:(especially in universal algebra), its
14:
12090:
10359:
10223:
10201:
10176:
10147:
10125:
10013:
9626:
9596: – Additional mathematical object
9142:{\displaystyle |{\mathcal {V}}|_{S}=F}
9095:{\displaystyle |{\mathcal {V}}|_{V}=V}
7368:
6896:The following problem is known as the
6888:which is not an induced substructure.
5919:of the object theory and any elements
5579:of the object theory in the structure
2604:is multiplication of rational numbers,
10333:
9670:
7038:Model theory § First-order logic
4642:{\displaystyle (a,b)\!\in {\text{E}}}
3937:there is a smallest closed subset of
3789:{\displaystyle b_{1}b_{2}\dots b_{n}}
1125:{\displaystyle f^{\mathcal {A}}=I(f)}
10112:(2nd ed.), New York: Springer,
7004:can be studied using the methods of
6824:. If σ has only function symbols, σ-
6816:Induced substructures correspond to
5722:, although technically the function
4492:-structures in the natural way, the
1132:on the domain. Each relation symbol
236:
36:
10324:Stanford Encyclopedia of Philosophy
9428:
945:Since the signatures that arise in
24:
10128:Fundamentals of Mathematical Logic
9745:
9712:
9551:Structures that are proper classes
9465:
9441:
9366:
9339:
9312:
9296:
9256:
9234:
9189:
9167:
9116:
9069:
9040:
8800:
8743:
8719:
8652:
8605:
8521:
8408:
8269:
8111:
8058:
7964:
7956:
7918:if and only if there is a formula
7903:
7795:
7640:
7502:
7476:
7442:
7343:
7279:
7228:
7174:
7123:
7072:
6977:
6967:
6937:
6913:
6783:
6759:
6730:
6701:
6667:
6572:
6502:, the following equivalence holds:
6408:
6398:
6221:
6137:
6062:
5978:
5888:
5878:
5827:
5803:
5774:
5740:
5707:
5697:
5667:
5643:
5612:
5588:
5543:
5514:
5480:
5385:
5315:, the following implication holds:
5295:
5037:
4946:
4926:
4888:
4864:
4836:
4812:
4375:
4303:
4186:
4154:
4093:
3951:
3917:
3623:
3533:
3505:
3476:
3436:
3426:
3391:
3351:
3331:
3300:
3276:
3244:
3225:
3195:
3171:
3144:
3116:
2938:, which is not a field, is also a
2893:
2864:
2792:
2721:
2619:
2555:
2491:
2459:
2074:
2032:
2017:
1975:
1960:
1918:
1777:{\displaystyle \mathbf {\times } }
1645:{\displaystyle \mathbf {\times } }
1547:
1520:
1504:
1476:
1460:
1426:
1244:
1238:
1235:
1205:
1101:
1010:
614:
587:
558:
526:
309:
280:, in a reference to mathematician
56:it lacks sufficient corresponding
25:
12124:
10309:
10017:Categorical Logic and Type Theory
9732:may also be used to refer to the
9543:, as discussed in the article on
7516:cf. below) if there is a formula
5845:For every signature σ there is a
4519:The most obvious way to define a
4067:{\displaystyle \langle B\rangle }
3407:{\displaystyle |{\mathcal {A}}|.}
491:(especially in model theory, cf.
12071:
9816:
9791:
9783:
9725:{\displaystyle |{\mathcal {A}}|}
8071:{\displaystyle |{\mathcal {M}}|}
7042:Model theory § Definability
7032:Structures and first-order logic
6739:{\displaystyle R^{\mathcal {B}}}
6710:{\displaystyle R^{\mathcal {A}}}
5787:{\displaystyle |{\mathcal {B}}|}
5753:{\displaystyle |{\mathcal {A}}|}
5552:{\displaystyle R^{\mathcal {B}}}
5523:{\displaystyle R^{\mathcal {A}}}
4798:
4523:is a structure with a signature
4167:{\displaystyle |{\mathcal {A}}|}
4125:{\displaystyle \langle \rangle }
3964:{\displaystyle |{\mathcal {A}}|}
2902:{\displaystyle I_{\mathcal {C}}}
2873:{\displaystyle I_{\mathcal {R}}}
2540:is addition of rational numbers,
2468:{\displaystyle I_{\mathcal {Q}}}
1748:
1726:
1704:
1682:
1660:
1616:
571:{\displaystyle |{\mathcal {A}}|}
240:
41:
10263:Springer Science+Business Media
9756:{\displaystyle {\mathcal {A}}.}
9027:is a vector space over a field
8811:{\displaystyle {\mathcal {M}}.}
8616:{\displaystyle {\mathcal {M}}.}
7495:definable with parameters from
7354:{\displaystyle {\mathcal {M}}.}
7294:is the same as the language of
6998:constraint satisfaction problem
5057:, the following equation holds:
4386:{\displaystyle {\mathcal {A}}.}
4314:{\displaystyle {\mathcal {A}},}
3994:It is called the closed subset
3544:{\displaystyle {\mathcal {A}},}
3311:{\displaystyle {\mathcal {B}}:}
1905:-structures in an obvious way:
1696:) and the two constant symbols
1437:{\displaystyle {\mathcal {A}},}
1417:is a binary function symbol of
598:{\displaystyle {\mathcal {A}},}
10007:
9972:
9830:on the left refer to signs of
9718:
9706:
9692:
9679:
9664:
9645:
9620:
9607:
9513:Structures for typed languages
9373:
9360:
9356:
9346:
9333:
9319:
9306:
9263:
9250:
9196:
9183:
9123:
9110:
9076:
9063:
9055:consists of the vector domain
9048:{\displaystyle {\mathcal {V}}}
8727:{\displaystyle {\mathcal {M}}}
8660:{\displaystyle {\mathcal {M}}}
8529:{\displaystyle {\mathcal {M}}}
8512:must be over the signature of
8451:
8419:
8387:
8355:
8326:
8294:
8277:{\displaystyle {\mathcal {M}}}
8119:{\displaystyle {\mathcal {M}}}
8064:
8052:
7997:
7994:
7988:
7982:
7970:
7934:
7928:
7911:{\displaystyle {\mathcal {M}}}
7838:
7806:
7790:
7781:
7749:
7683:
7651:
7619:
7587:
7558:
7526:
7450:{\displaystyle {\mathcal {M}}}
7413:on the universe (i.e. domain)
7287:{\displaystyle {\mathcal {M}}}
7236:{\displaystyle {\mathcal {M}}}
7182:{\displaystyle {\mathcal {M}}}
7098:
7080:
6972:
6945:{\displaystyle {\mathcal {B}}}
6921:{\displaystyle {\mathcal {A}}}
6791:{\displaystyle {\mathcal {B}}}
6767:{\displaystyle {\mathcal {A}}}
6655:
6652:
6639:
6624:
6611:
6602:
6589:
6583:
6579:
6560:
6515:
6403:
6352:
6339:
6309:
6296:
6273:
6260:
6209:
6164:
6143:
6131:
6050:
6005:
5984:
5972:
5883:
5835:{\displaystyle {\mathcal {B}}}
5811:{\displaystyle {\mathcal {A}}}
5780:
5768:
5746:
5734:
5702:
5675:{\displaystyle {\mathcal {B}}}
5651:{\displaystyle {\mathcal {A}}}
5620:{\displaystyle {\mathcal {B}}}
5596:{\displaystyle {\mathcal {A}}}
5468:
5465:
5452:
5437:
5424:
5415:
5402:
5396:
5392:
5373:
5328:
5301:
5289:
5208:
5205:
5192:
5177:
5164:
5155:
5142:
5136:
5127:
5124:
5079:
5073:
5043:
5031:
4952:
4940:
4936:
4932:
4920:
4896:{\displaystyle {\mathcal {B}}}
4872:{\displaystyle {\mathcal {A}}}
4844:{\displaystyle {\mathcal {B}}}
4820:{\displaystyle {\mathcal {A}}}
4627:
4615:
4294:is an induced substructure of
4287:{\displaystyle (B,\sigma ,I')}
4281:
4258:
4212:
4194:
4160:
4148:
3957:
3945:
3923:
3911:
3873:
3828:
3631:{\displaystyle {\mathcal {A}}}
3513:{\displaystyle {\mathcal {A}}}
3482:
3470:
3397:
3385:
3357:
3345:
3337:
3325:
3291:is contained in the domain of
3284:{\displaystyle {\mathcal {A}}}
3249:
3239:
3230:
3220:
3203:{\displaystyle {\mathcal {B}}}
3179:{\displaystyle {\mathcal {A}}}
3152:{\displaystyle {\mathcal {B}}}
3124:{\displaystyle {\mathcal {A}}}
2969:of the field axioms hold in a
2804:
2798:
2733:
2727:
2642:
2631:
2625:
2586:
2567:
2561:
2522:
2503:
2497:
2416:
2410:
2375:
2369:
2334:
2328:
2293:
2287:
2252:
2246:
2145:
2119:
2080:
2044:
2023:
1987:
1966:
1930:
1553:
1541:
1537:
1527:
1514:
1471:
1381:
1375:
1329:
1323:
1247:
1241:
1223:
1217:
1119:
1113:
1018:{\displaystyle {\mathcal {A}}}
859:
853:
812:
806:
739:
668:
656:
622:{\displaystyle {\mathcal {A}}}
564:
552:
531:
521:
335:
317:
13:
1:
12032:History of mathematical logic
10204:Model Theory: An Introduction
10047:A Course in Universal Algebra
10037:
9963:{\displaystyle \mathbb {Q} .}
9404:however naturally leads to a
6378:
4411:
3496:of the domain of a structure
1878:{\displaystyle \mathbb {C} ,}
1825:{\displaystyle \mathbb {Q} ,}
1313:, because its interpretation
764:that ascribes to each symbol
291:
212:interpretation (model theory)
170:structures such as models of
11957:Primitive recursive function
10289:Introduction to Model Theory
9823:{\displaystyle \mathbf {-} }
9671:Quine, Willard V.O. (1940).
8190:definable without parameters
8178:definable without parameters
8016:Definability with parameters
6904:Given two finite structures
4788:Homomorphisms and embeddings
4244:{\displaystyle B\subseteq A}
2927:{\displaystyle \mathbb {Z} }
2446:The interpretation function
1850:{\displaystyle \mathbb {R} }
1755:{\displaystyle \mathbf {+} }
1733:{\displaystyle \mathbf {1} }
1711:{\displaystyle \mathbf {0} }
1689:{\displaystyle \mathbf {+} }
1667:{\displaystyle \mathbf {-} }
1623:{\displaystyle \mathbf {+} }
681:of a structure consists of:
632:
7:
10287:Rothmaler, Philipp (2000),
10080:Diestel, Reinhard (2005) ,
9941:and to the unary operation
9655:. Oxford University Press.
9587:
9283:, or scalar multiplication
8596:is not in the signature of
8180:, while other authors mean
7940:{\displaystyle \varphi (x)}
7063:Each first-order structure
6850:homomorphism between graphs
4851:of the same signature σ, a
4496:form a substructure of the
2989:{\displaystyle \sigma _{f}}
2958:{\displaystyle \sigma _{f}}
1597:{\displaystyle \sigma _{f}}
1572:
971:Not to be confused with an
300:can be defined as a triple
115:along with a collection of
10:
12129:
11021:Schröder–Bernstein theorem
10748:Monadic predicate calculus
10407:Foundations of mathematics
10186:Cambridge University Press
10161:Cambridge University Press
10020:, Elsevier, pp. 1–4,
9532:
8877:
8492:used to define a relation
8200:Recall from above that an
7508:{\displaystyle \emptyset }
7482:{\displaystyle \emptyset }
7035:
7011:Another application is in
6843:
4791:
3043:{\displaystyle \,\leq ,\,}
1263:on the domain. A nullary (
970:
636:
578:is used for the domain of
472:
232:
150:is used for structures of
29:
12067:
12054:Philosophy of mathematics
12003:Automated theorem proving
11985:
11880:
11712:
11605:
11457:
11174:
11150:
11128:Von Neumann–Bernays–Gödel
11073:
10967:
10871:
10769:
10760:
10687:
10622:
10528:
10450:
10367:
10271:10.1007/978-1-4419-1221-3
8182:definable with parameters
8162:{\displaystyle \varphi .}
8042:definable with parameters
7144:defined for all formulas
4366:of its interpretation in
4251:is a closed subset, then
4134:finitary closure operator
973:interpretation of a model
920:of the interpretation of
468:
445:one can refer to it as a
9600:
9471:{\displaystyle \forall }
9447:{\displaystyle \forall }
8710:is the only relation on
8636:{\displaystyle \varphi }
8549:{\displaystyle \varphi }
8485:{\displaystyle \varphi }
8095:{\displaystyle \varphi }
8082:) if there is a formula
7735:{\displaystyle \varphi }
6375:that was defined above.
5682:is typically denoted as
4512:, rather than that of a
4508:signature, is that of a
3210:have the same signature
3018:{\displaystyle \,<\,}
1740:(uniquely determined by
1674:(uniquely determined by
395:{\displaystyle \sigma ,}
183:Tarski's theory of truth
123:that are defined on it.
30:Not to be confused with
12103:Mathematical structures
11704:Self-verifying theories
11525:Tarski's axiomatization
10476:Tarski's undefinability
10471:incompleteness theorems
9993:10.1023/A:1018941030227
9675:. Vol. vi. Norton.
7158:{\displaystyle \,\phi }
5726:is between the domains
4536:{\displaystyle \sigma }
4485:{\displaystyle \sigma }
3796:is again an element of
3706:the result of applying
2909:are similarly defined.
1898:{\displaystyle \sigma }
1577:The standard signature
1365:and its interpretation
980:interpretation function
967:Interpretation function
508:Sometimes the notation
458:{\displaystyle \sigma }
438:{\displaystyle \sigma }
404:interpretation function
278:Willard Van Orman Quine
71:more precise citations.
12078:Mathematics portal
11689:Proof of impossibility
11337:propositional variable
10647:Propositional calculus
10224:Poizat, Bruno (2000),
10202:Marker, David (2002),
10182:A shorter model theory
9964:
9935:
9908:
9888:
9887:{\displaystyle 0,1,2,}
9854:
9853:{\displaystyle S_{f}.}
9824:
9802:
9757:
9726:
9594:Mathematical structure
9529:Higher-order languages
9472:
9448:
9387:
9277:
9210:
9143:
9096:
9049:
8856:Many-sorted signatures
8828:Many-sorted structures
8812:
8781:
8761:
8728:
8704:
8684:
8661:
8637:
8623:If there is a formula
8617:
8590:
8570:
8550:
8530:
8506:
8486:
8464:
8333:
8278:
8254:
8234:
8214:
8163:
8140:
8120:
8096:
8072:
8034:
8007:
7941:
7912:
7888:
7868:
7845:
7736:
7716:
7696:
7565:
7509:
7483:
7451:
7427:
7407:
7387:
7355:
7328:
7314:and every sentence in
7308:
7288:
7264:
7237:
7206:
7183:
7159:
7138:
7105:
7028:homomorphism problem.
6986:
6946:
6922:
6828:is the subcategory of
6792:
6768:
6740:
6711:
6677:
6496:
6443:of σ and any elements
6417:
6362:
6231:
6151:
6072:
5992:
5897:
5836:
5812:
5794:of the two structures
5788:
5754:
5716:
5676:
5652:
5621:
5597:
5573:
5553:
5524:
5490:
5309:
5236:of σ and any elements
5215:
5051:
4978:of σ and any elements
4960:
4897:
4873:
4845:
4821:
4770:
4747:
4727:
4707:
4683:
4663:
4643:
4603:
4580:
4560:
4537:
4486:
4466:
4387:
4360:
4340:
4315:
4288:
4245:
4219:
4168:
4126:
4103:
4068:
4042:
4015:
3988:
3965:
3931:
3889:
3813:
3790:
3740:
3720:
3700:
3632:
3608:
3588:
3568:
3545:
3514:
3490:
3448:
3408:
3368:
3312:
3285:
3259:
3204:
3180:
3153:
3133:(induced) substructure
3125:
3094:
3074:
3044:
3019:
2990:
2959:
2928:
2903:
2874:
2842:
2819:
2771:
2748:
2700:
2674:
2654:
2598:
2534:
2469:
2438:
2215:
2152:
2091:
1899:
1879:
1851:
1826:
1798:
1778:
1756:
1734:
1712:
1690:
1668:
1646:
1624:
1598:
1564:
1485:
1438:
1411:
1391:
1359:
1336:
1301:
1287:-ary) function symbol
1281:
1257:
1186:
1166:
1146:
1126:
1080:
1059:
1039:
1019:
995:
937:
910:
886:
866:
822:
778:
758:
699:
675:
623:
599:
572:
538:
483:of the structure, its
459:
439:
419:
396:
370:
342:
11947:Kolmogorov complexity
11900:Computably enumerable
11800:Model complete theory
11592:Principia Mathematica
10652:Propositional formula
10481:Banach–Tarski paradox
10014:Jacobs, Bart (1999),
9965:
9936:
9934:{\displaystyle N_{0}}
9909:
9889:
9855:
9825:
9803:
9758:
9727:
9617:as well as functions.
9581:Principia Mathematica
9473:
9449:
9424:Other generalizations
9388:
9278:
9211:
9144:
9097:
9050:
8845:many-sorted structure
8813:
8782:
8762:
8729:
8705:
8685:
8662:
8638:
8618:
8591:
8571:
8551:
8531:
8507:
8487:
8465:
8334:
8279:
8255:
8235:
8215:
8196:Implicit definability
8164:
8141:
8121:
8102:with parameters from
8097:
8073:
8035:
8008:
7942:
7913:
7889:
7869:
7846:
7737:
7717:
7697:
7566:
7510:
7484:
7452:
7428:
7408:
7388:
7356:
7329:
7309:
7289:
7265:
7238:
7207:
7184:
7160:
7139:
7113:satisfaction relation
7106:
7059:Satisfaction relation
6987:
6947:
6923:
6793:
6769:
6741:
6712:
6678:
6497:
6439:-ary relation symbol
6418:
6363:
6232:
6152:
6073:
5993:
5915:-ary relation symbol
5898:
5837:
5813:
5789:
5755:
5717:
5677:
5653:
5622:
5598:
5574:
5554:
5525:
5491:
5310:
5232:-ary relation symbol
5216:
5052:
4974:-ary function symbol
4961:
4898:
4874:
4846:
4822:
4803:Given two structures
4771:
4748:
4728:
4708:
4684:
4664:
4644:
4604:
4581:
4561:
4538:
4487:
4467:
4388:
4361:
4341:
4316:
4289:
4246:
4220:
4169:
4127:
4104:
4069:
4043:
4016:
3989:
3966:
3932:
3890:
3814:
3791:
3741:
3721:
3701:
3633:
3614:(in the signature of
3609:
3594:-ary function symbol
3589:
3569:
3546:
3515:
3491:
3449:
3409:
3369:
3313:
3286:
3260:
3205:
3181:
3154:
3126:
3095:
3075:
3073:{\displaystyle \in .}
3045:
3020:
2991:
2960:
2929:
2904:
2875:
2843:
2820:
2772:
2749:
2701:
2675:
2655:
2599:
2535:
2470:
2439:
2216:
2153:
2092:
1900:
1880:
1852:
1827:
1799:
1779:
1757:
1735:
1713:
1691:
1669:
1647:
1625:
1599:
1565:
1486:
1439:
1412:
1392:
1390:{\displaystyle I(s).}
1360:
1337:
1302:
1282:
1258:
1187:
1167:
1147:
1127:
1081:
1060:
1040:
1020:
996:
938:
911:
887:
867:
823:
779:
759:
700:
676:
624:
600:
573:
539:
460:
440:
420:
397:
371:
343:
11895:Church–Turing thesis
11882:Computability theory
11091:continuum hypothesis
10609:Square of opposition
10467:Gödel's completeness
10251:Rautenberg, Wolfgang
10232:, Berlin, New York:
10206:, Berlin, New York:
10050:, Berlin, New York:
9949:
9918:
9898:
9863:
9834:
9812:
9779:
9740:
9702:
9462:
9438:
9287:
9220:
9153:
9106:
9102:, the scalar domain
9059:
9035:
8836:one-sorted structure
8795:
8789:implicitly definable
8771:
8738:
8714:
8694:
8671:
8647:
8627:
8600:
8580:
8560:
8540:
8516:
8496:
8476:
8343:
8288:
8264:
8244:
8224:
8204:
8150:
8130:
8106:
8086:
8048:
8024:
7951:
7922:
7898:
7878:
7858:
7746:
7726:
7706:
7575:
7520:
7499:
7473:
7463:explicitly definable
7437:
7417:
7397:
7377:
7338:
7318:
7298:
7274:
7254:
7223:
7193:
7169:
7148:
7118:
7067:
6956:
6932:
6908:
6898:homomorphism problem
6892:Homomorphism problem
6778:
6754:
6721:
6692:
6512:
6447:
6387:
6241:
6161:
6082:
6002:
5923:
5903:is sometimes called
5867:
5822:
5798:
5764:
5730:
5686:
5662:
5638:
5607:
5583:
5563:
5534:
5505:
5325:
5240:
5067:
4982:
4910:
4883:
4859:
4831:
4807:
4757:
4737:
4717:
4697:
4673:
4653:
4612:
4590:
4570:
4547:
4527:
4476:
4420:
4370:
4350:
4325:
4298:
4255:
4229:
4181:
4144:
4113:
4078:
4052:
4029:
4002:
3975:
3941:
3901:
3822:
3800:
3750:
3730:
3710:
3642:
3618:
3598:
3578:
3555:
3528:
3500:
3460:
3421:
3381:
3321:
3295:
3271:
3214:
3190:
3166:
3139:
3111:
3093:{\displaystyle \in }
3084:
3061:
3052:algebraic structures
3029:
3007:
2973:
2942:
2916:
2884:
2855:
2829:
2783:
2758:
2712:
2684:
2664:
2610:
2546:
2482:
2450:
2225:
2162:
2103:
1909:
1889:
1864:
1839:
1811:
1788:
1766:
1744:
1722:
1700:
1678:
1656:
1634:
1612:
1581:
1495:
1448:
1421:
1401:
1369:
1349:
1335:{\displaystyle I(c)}
1317:
1291:
1280:{\displaystyle =\,0}
1267:
1196:
1176:
1156:
1136:
1092:
1070:
1049:
1029:
1005:
985:
961:algebra over a field
924:
900:
876:
838:
791:
768:
724:
689:
647:
609:
582:
548:
512:
449:
429:
409:
383:
357:
304:
164:first-order theories
152:first-order theories
128:algebraic structures
18:Relational structure
12049:Mathematical object
11940:P versus NP problem
11905:Computable function
11699:Reverse mathematics
11625:Logical consequence
11502:primitive recursive
11497:elementary function
11270:Free/bound variable
11123:Tarski–Grothendieck
10642:Logical connectives
10572:Logical equivalence
10422:Logical consequence
10126:Hinman, P. (2005),
10058:Chang, Chen Chung;
9239:
9172:
8146:is definable using
7369:Definable relations
7270:if the language of
7006:finite model theory
6383:A (σ-)homomorphism
4693:. For example, let
3638:) and all elements
2699:{\displaystyle -x,}
952:algebraic signature
498:domain of discourse
475:Domain of discourse
204:mathematical models
117:finitary operations
12098:Mathematical logic
11847:Transfer principle
11810:Semantics of logic
11795:Categorical theory
11771:Non-standard model
11285:Logical connective
10412:Information theory
10361:Mathematical logic
10108:Mathematical Logic
10060:Keisler, H. Jerome
9960:
9931:
9904:
9884:
9850:
9820:
9798:
9753:
9722:
9673:Mathematical logic
9545:second-order logic
9541:higher-order logic
9535:Second-order logic
9501:, but it is not a
9468:
9444:
9383:
9273:
9223:
9216:, the scalar zero
9206:
9156:
9139:
9092:
9045:
8869:(for vectors) and
8808:
8777:
8757:
8724:
8700:
8683:{\displaystyle R,}
8680:
8657:
8633:
8613:
8586:
8566:
8546:
8526:
8502:
8482:
8460:
8329:
8274:
8250:
8230:
8210:
8159:
8136:
8116:
8092:
8068:
8030:
8003:
7937:
7908:
7884:
7864:
7841:
7732:
7712:
7692:
7561:
7505:
7479:
7447:
7423:
7403:
7383:
7351:
7324:
7304:
7284:
7260:
7233:
7205:{\displaystyle M,}
7202:
7179:
7155:
7134:
7101:
7049:second-order logic
6982:
6942:
6918:
6872:in the category σ-
6788:
6764:
6736:
6707:
6673:
6492:
6413:
6358:
6227:
6147:
6068:
5988:
5893:
5832:
5808:
5784:
5750:
5712:
5672:
5648:
5617:
5593:
5569:
5549:
5520:
5486:
5305:
5211:
5047:
4956:
4893:
4869:
4841:
4817:
4769:{\displaystyle G,}
4766:
4743:
4723:
4703:
4679:
4659:
4639:
4602:{\displaystyle b,}
4599:
4576:
4559:{\displaystyle E.}
4556:
4533:
4482:
4462:
4383:
4356:
4339:{\displaystyle I'}
4336:
4311:
4284:
4241:
4215:
4164:
4122:
4099:
4064:
4041:{\displaystyle B,}
4038:
4014:{\displaystyle B,}
4011:
3987:{\displaystyle B.}
3984:
3961:
3927:
3885:
3812:{\displaystyle B:}
3809:
3786:
3736:
3716:
3696:
3628:
3604:
3584:
3567:{\displaystyle n,}
3564:
3541:
3510:
3486:
3444:
3404:
3364:
3308:
3281:
3255:
3200:
3176:
3149:
3121:
3090:
3070:
3040:
3015:
2986:
2955:
2924:
2899:
2870:
2841:{\displaystyle 1;}
2838:
2815:
2770:{\displaystyle 0,}
2767:
2744:
2696:
2670:
2650:
2594:
2530:
2465:
2434:
2432:
2211:
2148:
2087:
2085:
1895:
1875:
1847:
1822:
1794:
1774:
1752:
1730:
1708:
1686:
1664:
1642:
1620:
1594:
1560:
1481:
1444:one simply writes
1434:
1407:
1387:
1355:
1332:
1297:
1277:
1253:
1182:
1162:
1142:
1122:
1076:
1055:
1035:
1015:
991:
936:{\displaystyle s.}
933:
916:because it is the
906:
882:
862:
818:
774:
754:
695:
671:
619:
595:
568:
534:
455:
435:
415:
392:
369:{\displaystyle A,}
366:
338:
187:Tarskian semantics
32:Mathematical model
12113:Universal algebra
12085:
12084:
12017:Abstract category
11820:Theories of truth
11630:Rule of inference
11620:Natural deduction
11601:
11600:
11146:
11145:
10851:Cartesian product
10756:
10755:
10662:Many-valued logic
10637:Boolean functions
10520:Russell's paradox
10495:diagonal argument
10392:First-order logic
10302:978-90-5699-313-9
10280:978-1-4419-1220-6
10243:978-0-387-98655-5
10217:978-0-387-98760-6
10195:978-0-521-58713-6
10170:978-0-521-30442-9
10141:978-1-56881-262-5
10119:978-0-387-94258-2
10097:978-3-540-26183-4
10073:978-0-7204-0692-4
9907:{\displaystyle -}
9736:of the domain of
9638:978-0-444-51667-1
9414:categorical logic
9402:many-sorted logic
9021:
9020:
8780:{\displaystyle R}
8703:{\displaystyle R}
8690:and the relation
8667:and a new symbol
8589:{\displaystyle R}
8569:{\displaystyle R}
8505:{\displaystyle R}
8472:Here the formula
8253:{\displaystyle M}
8233:{\displaystyle R}
8213:{\displaystyle n}
8172:Some authors use
8139:{\displaystyle R}
8033:{\displaystyle R}
7887:{\displaystyle M}
7867:{\displaystyle m}
7715:{\displaystyle R}
7467:Beth definability
7433:of the structure
7426:{\displaystyle M}
7406:{\displaystyle R}
7386:{\displaystyle n}
7327:{\displaystyle T}
7307:{\displaystyle T}
7263:{\displaystyle T}
7025:conjunctive query
7002:complexity of CSP
6688:(where as before
5572:{\displaystyle R}
4782:induced subgraphs
4753:is a subgraph of
4746:{\displaystyle H}
4726:{\displaystyle H}
4706:{\displaystyle G}
4682:{\displaystyle b}
4662:{\displaystyle a}
4637:
4579:{\displaystyle a}
4359:{\displaystyle B}
3897:For every subset
3739:{\displaystyle n}
3719:{\displaystyle f}
3607:{\displaystyle f}
3587:{\displaystyle n}
2673:{\displaystyle x}
1797:{\displaystyle A}
1410:{\displaystyle f}
1358:{\displaystyle s}
1300:{\displaystyle c}
1185:{\displaystyle n}
1165:{\displaystyle n}
1145:{\displaystyle R}
1079:{\displaystyle n}
1058:{\displaystyle n}
1038:{\displaystyle f}
994:{\displaystyle I}
975:in another model.
909:{\displaystyle s}
885:{\displaystyle s}
777:{\displaystyle s}
735:
698:{\displaystyle S}
639:Signature (logic)
418:{\displaystyle I}
270:
269:
227:relational models
225:, in the form of
179:first-order logic
148:universal algebra
101:universal algebra
97:
96:
89:
16:(Redirected from
12120:
12076:
12075:
12027:History of logic
12022:Category of sets
11915:Decision problem
11694:Ordinal analysis
11635:Sequent calculus
11533:Boolean algebras
11473:
11472:
11447:
11418:logical/constant
11172:
11171:
11158:
11081:Zermelo–Fraenkel
10832:Set operations:
10767:
10766:
10704:
10535:
10534:
10515:Löwenheim–Skolem
10402:Formal semantics
10354:
10347:
10340:
10331:
10330:
10305:
10283:
10257:(3rd ed.),
10246:
10231:
10220:
10198:
10173:
10158:
10144:
10122:
10111:
10100:
10076:
10054:
10031:
10030:
10011:
10005:
10004:
9976:
9970:
9969:
9967:
9966:
9961:
9956:
9940:
9938:
9937:
9932:
9930:
9929:
9913:
9911:
9910:
9905:
9893:
9891:
9890:
9885:
9859:
9857:
9856:
9851:
9846:
9845:
9829:
9827:
9826:
9821:
9819:
9807:
9805:
9804:
9799:
9794:
9786:
9773:
9764:
9762:
9760:
9759:
9754:
9749:
9748:
9731:
9729:
9728:
9723:
9721:
9716:
9715:
9709:
9696:
9690:
9683:
9677:
9676:
9668:
9662:
9659:
9649:
9643:
9642:
9624:
9618:
9611:
9576:Bertrand Russell
9555:In the study of
9499:elementary class
9477:
9475:
9474:
9469:
9453:
9451:
9450:
9445:
9429:Partial algebras
9392:
9390:
9389:
9384:
9382:
9381:
9376:
9370:
9369:
9363:
9355:
9354:
9349:
9343:
9342:
9336:
9328:
9327:
9322:
9316:
9315:
9309:
9301:
9300:
9299:
9282:
9280:
9279:
9274:
9272:
9271:
9266:
9260:
9259:
9253:
9238:
9237:
9231:
9215:
9213:
9212:
9207:
9205:
9204:
9199:
9193:
9192:
9186:
9171:
9170:
9164:
9148:
9146:
9145:
9140:
9132:
9131:
9126:
9120:
9119:
9113:
9101:
9099:
9098:
9093:
9085:
9084:
9079:
9073:
9072:
9066:
9054:
9052:
9051:
9046:
9044:
9043:
8876:
8875:
8847:
8846:
8838:
8837:
8817:
8815:
8814:
8809:
8804:
8803:
8786:
8784:
8783:
8778:
8766:
8764:
8763:
8758:
8747:
8746:
8733:
8731:
8730:
8725:
8723:
8722:
8709:
8707:
8706:
8701:
8689:
8687:
8686:
8681:
8666:
8664:
8663:
8658:
8656:
8655:
8642:
8640:
8639:
8634:
8622:
8620:
8619:
8614:
8609:
8608:
8595:
8593:
8592:
8587:
8575:
8573:
8572:
8567:
8556:may not mention
8555:
8553:
8552:
8547:
8535:
8533:
8532:
8527:
8525:
8524:
8511:
8509:
8508:
8503:
8491:
8489:
8488:
8483:
8469:
8467:
8466:
8461:
8450:
8449:
8431:
8430:
8412:
8411:
8402:
8401:
8386:
8385:
8367:
8366:
8338:
8336:
8335:
8330:
8325:
8324:
8306:
8305:
8283:
8281:
8280:
8275:
8273:
8272:
8259:
8257:
8256:
8251:
8240:on the universe
8239:
8237:
8236:
8231:
8219:
8217:
8216:
8211:
8168:
8166:
8165:
8160:
8145:
8143:
8142:
8137:
8125:
8123:
8122:
8117:
8115:
8114:
8101:
8099:
8098:
8093:
8077:
8075:
8074:
8069:
8067:
8062:
8061:
8055:
8039:
8037:
8036:
8031:
8012:
8010:
8009:
8004:
7960:
7959:
7946:
7944:
7943:
7938:
7917:
7915:
7914:
7909:
7907:
7906:
7894:is definable in
7893:
7891:
7890:
7885:
7873:
7871:
7870:
7865:
7850:
7848:
7847:
7842:
7837:
7836:
7818:
7817:
7799:
7798:
7780:
7779:
7761:
7760:
7741:
7739:
7738:
7733:
7721:
7719:
7718:
7713:
7702:In other words,
7701:
7699:
7698:
7693:
7682:
7681:
7663:
7662:
7644:
7643:
7634:
7633:
7618:
7617:
7599:
7598:
7570:
7568:
7567:
7562:
7557:
7556:
7538:
7537:
7514:
7512:
7511:
7506:
7488:
7486:
7485:
7480:
7456:
7454:
7453:
7448:
7446:
7445:
7432:
7430:
7429:
7424:
7412:
7410:
7409:
7404:
7392:
7390:
7389:
7384:
7360:
7358:
7357:
7352:
7347:
7346:
7334:is satisfied by
7333:
7331:
7330:
7325:
7313:
7311:
7310:
7305:
7293:
7291:
7290:
7285:
7283:
7282:
7269:
7267:
7266:
7261:
7243:is said to be a
7242:
7240:
7239:
7234:
7232:
7231:
7211:
7209:
7208:
7203:
7188:
7186:
7185:
7180:
7178:
7177:
7164:
7162:
7161:
7156:
7143:
7141:
7140:
7135:
7127:
7126:
7110:
7108:
7107:
7102:
7076:
7075:
7017:relational model
6991:
6989:
6988:
6983:
6981:
6980:
6971:
6970:
6951:
6949:
6948:
6943:
6941:
6940:
6927:
6925:
6924:
6919:
6917:
6916:
6876:, and therefore
6797:
6795:
6794:
6789:
6787:
6786:
6773:
6771:
6770:
6765:
6763:
6762:
6745:
6743:
6742:
6737:
6735:
6734:
6733:
6716:
6714:
6713:
6708:
6706:
6705:
6704:
6682:
6680:
6679:
6674:
6672:
6671:
6670:
6651:
6650:
6623:
6622:
6601:
6600:
6577:
6576:
6575:
6559:
6558:
6540:
6539:
6527:
6526:
6501:
6499:
6498:
6493:
6491:
6490:
6472:
6471:
6459:
6458:
6423:is called a (σ-)
6422:
6420:
6419:
6414:
6412:
6411:
6402:
6401:
6367:
6365:
6364:
6359:
6351:
6350:
6332:
6331:
6308:
6307:
6289:
6288:
6272:
6271:
6253:
6252:
6236:
6234:
6233:
6228:
6226:
6225:
6224:
6208:
6207:
6189:
6188:
6176:
6175:
6156:
6154:
6153:
6148:
6146:
6141:
6140:
6134:
6126:
6125:
6107:
6106:
6094:
6093:
6077:
6075:
6074:
6069:
6067:
6066:
6065:
6049:
6048:
6030:
6029:
6017:
6016:
5997:
5995:
5994:
5989:
5987:
5982:
5981:
5975:
5967:
5966:
5948:
5947:
5935:
5934:
5902:
5900:
5899:
5894:
5892:
5891:
5882:
5881:
5841:
5839:
5838:
5833:
5831:
5830:
5817:
5815:
5814:
5809:
5807:
5806:
5793:
5791:
5790:
5785:
5783:
5778:
5777:
5771:
5759:
5757:
5756:
5751:
5749:
5744:
5743:
5737:
5721:
5719:
5718:
5713:
5711:
5710:
5701:
5700:
5681:
5679:
5678:
5673:
5671:
5670:
5657:
5655:
5654:
5649:
5647:
5646:
5626:
5624:
5623:
5618:
5616:
5615:
5602:
5600:
5599:
5594:
5592:
5591:
5578:
5576:
5575:
5570:
5558:
5556:
5555:
5550:
5548:
5547:
5546:
5529:
5527:
5526:
5521:
5519:
5518:
5517:
5495:
5493:
5492:
5487:
5485:
5484:
5483:
5464:
5463:
5436:
5435:
5414:
5413:
5390:
5389:
5388:
5372:
5371:
5353:
5352:
5340:
5339:
5314:
5312:
5311:
5306:
5304:
5299:
5298:
5292:
5284:
5283:
5265:
5264:
5252:
5251:
5220:
5218:
5217:
5212:
5204:
5203:
5176:
5175:
5154:
5153:
5123:
5122:
5104:
5103:
5091:
5090:
5056:
5054:
5053:
5048:
5046:
5041:
5040:
5034:
5026:
5025:
5007:
5006:
4994:
4993:
4965:
4963:
4962:
4957:
4955:
4950:
4949:
4943:
4935:
4930:
4929:
4923:
4902:
4900:
4899:
4894:
4892:
4891:
4878:
4876:
4875:
4870:
4868:
4867:
4853:(σ-)homomorphism
4850:
4848:
4847:
4842:
4840:
4839:
4826:
4824:
4823:
4818:
4816:
4815:
4775:
4773:
4772:
4767:
4752:
4750:
4749:
4744:
4732:
4730:
4729:
4724:
4712:
4710:
4709:
4704:
4688:
4686:
4685:
4680:
4668:
4666:
4665:
4660:
4648:
4646:
4645:
4640:
4638:
4635:
4608:
4606:
4605:
4600:
4585:
4583:
4582:
4577:
4565:
4563:
4562:
4557:
4542:
4540:
4539:
4534:
4494:rational numbers
4491:
4489:
4488:
4483:
4471:
4469:
4468:
4463:
4392:
4390:
4389:
4384:
4379:
4378:
4365:
4363:
4362:
4357:
4345:
4343:
4342:
4337:
4335:
4320:
4318:
4317:
4312:
4307:
4306:
4293:
4291:
4290:
4285:
4280:
4250:
4248:
4247:
4242:
4224:
4222:
4221:
4216:
4190:
4189:
4173:
4171:
4170:
4165:
4163:
4158:
4157:
4151:
4131:
4129:
4128:
4123:
4108:
4106:
4105:
4100:
4098:
4097:
4096:
4073:
4071:
4070:
4065:
4047:
4045:
4044:
4039:
4020:
4018:
4017:
4012:
3993:
3991:
3990:
3985:
3970:
3968:
3967:
3962:
3960:
3955:
3954:
3948:
3936:
3934:
3933:
3928:
3926:
3921:
3920:
3914:
3894:
3892:
3891:
3886:
3872:
3871:
3853:
3852:
3840:
3839:
3818:
3816:
3815:
3810:
3795:
3793:
3792:
3787:
3785:
3784:
3772:
3771:
3762:
3761:
3745:
3743:
3742:
3737:
3725:
3723:
3722:
3717:
3705:
3703:
3702:
3697:
3686:
3685:
3667:
3666:
3654:
3653:
3637:
3635:
3634:
3629:
3627:
3626:
3613:
3611:
3610:
3605:
3593:
3591:
3590:
3585:
3573:
3571:
3570:
3565:
3550:
3548:
3547:
3542:
3537:
3536:
3519:
3517:
3516:
3511:
3509:
3508:
3495:
3493:
3492:
3487:
3485:
3480:
3479:
3473:
3453:
3451:
3450:
3445:
3440:
3439:
3430:
3429:
3413:
3411:
3410:
3405:
3400:
3395:
3394:
3388:
3373:
3371:
3370:
3365:
3360:
3355:
3354:
3348:
3340:
3335:
3334:
3328:
3317:
3315:
3314:
3309:
3304:
3303:
3290:
3288:
3287:
3282:
3280:
3279:
3264:
3262:
3261:
3256:
3248:
3247:
3229:
3228:
3209:
3207:
3206:
3201:
3199:
3198:
3185:
3183:
3182:
3177:
3175:
3174:
3158:
3156:
3155:
3150:
3148:
3147:
3130:
3128:
3127:
3122:
3120:
3119:
3099:
3097:
3096:
3091:
3079:
3077:
3076:
3071:
3049:
3047:
3046:
3041:
3024:
3022:
3021:
3016:
2999:A signature for
2995:
2993:
2992:
2987:
2985:
2984:
2964:
2962:
2961:
2956:
2954:
2953:
2933:
2931:
2930:
2925:
2923:
2908:
2906:
2905:
2900:
2898:
2897:
2896:
2879:
2877:
2876:
2871:
2869:
2868:
2867:
2847:
2845:
2844:
2839:
2824:
2822:
2821:
2816:
2814:
2797:
2796:
2795:
2776:
2774:
2773:
2768:
2753:
2751:
2750:
2745:
2743:
2726:
2725:
2724:
2705:
2703:
2702:
2697:
2679:
2677:
2676:
2671:
2659:
2657:
2656:
2651:
2649:
2641:
2624:
2623:
2622:
2603:
2601:
2600:
2595:
2593:
2585:
2577:
2560:
2559:
2558:
2539:
2537:
2536:
2531:
2529:
2521:
2513:
2496:
2495:
2494:
2474:
2472:
2471:
2466:
2464:
2463:
2462:
2443:
2441:
2440:
2435:
2433:
2420:
2405:
2404:
2379:
2364:
2363:
2338:
2323:
2322:
2297:
2282:
2281:
2256:
2241:
2240:
2220:
2218:
2217:
2212:
2174:
2173:
2157:
2155:
2154:
2149:
2144:
2143:
2131:
2130:
2115:
2114:
2096:
2094:
2093:
2088:
2086:
2079:
2078:
2077:
2064:
2063:
2051:
2036:
2035:
2022:
2021:
2020:
2007:
2006:
1994:
1979:
1978:
1965:
1964:
1963:
1950:
1949:
1937:
1922:
1921:
1904:
1902:
1901:
1896:
1884:
1882:
1881:
1876:
1871:
1856:
1854:
1853:
1848:
1846:
1831:
1829:
1828:
1823:
1818:
1806:rational numbers
1803:
1801:
1800:
1795:
1783:
1781:
1780:
1775:
1773:
1761:
1759:
1758:
1753:
1751:
1739:
1737:
1736:
1731:
1729:
1717:
1715:
1714:
1709:
1707:
1695:
1693:
1692:
1687:
1685:
1673:
1671:
1670:
1665:
1663:
1651:
1649:
1648:
1643:
1641:
1629:
1627:
1626:
1621:
1619:
1603:
1601:
1600:
1595:
1593:
1592:
1569:
1567:
1566:
1561:
1556:
1551:
1550:
1544:
1536:
1535:
1530:
1524:
1523:
1517:
1509:
1508:
1507:
1490:
1488:
1487:
1482:
1480:
1479:
1470:
1469:
1464:
1463:
1443:
1441:
1440:
1435:
1430:
1429:
1416:
1414:
1413:
1408:
1397:For example, if
1396:
1394:
1393:
1388:
1364:
1362:
1361:
1356:
1341:
1339:
1338:
1333:
1306:
1304:
1303:
1298:
1286:
1284:
1283:
1278:
1262:
1260:
1259:
1254:
1252:
1251:
1250:
1210:
1209:
1208:
1191:
1189:
1188:
1183:
1171:
1169:
1168:
1163:
1151:
1149:
1148:
1143:
1131:
1129:
1128:
1123:
1106:
1105:
1104:
1085:
1083:
1082:
1077:
1064:
1062:
1061:
1056:
1044:
1042:
1041:
1036:
1024:
1022:
1021:
1016:
1014:
1013:
1000:
998:
997:
992:
942:
940:
939:
934:
915:
913:
912:
907:
891:
889:
888:
883:
871:
869:
868:
863:
827:
825:
824:
819:
783:
781:
780:
775:
763:
761:
760:
755:
753:
752:
747:
733:
714:relation symbols
708:function symbols
704:
702:
701:
696:
680:
678:
677:
672:
628:
626:
625:
620:
618:
617:
604:
602:
601:
596:
591:
590:
577:
575:
574:
569:
567:
562:
561:
555:
543:
541:
540:
535:
530:
529:
464:
462:
461:
456:
444:
442:
441:
436:
424:
422:
421:
416:
401:
399:
398:
393:
375:
373:
372:
367:
348:consisting of a
347:
345:
344:
339:
313:
312:
282:Richard Dedekind
265:
262:
244:
237:
156:relation symbols
92:
85:
81:
78:
72:
67:this article by
58:inline citations
45:
44:
37:
21:
12128:
12127:
12123:
12122:
12121:
12119:
12118:
12117:
12088:
12087:
12086:
12081:
12070:
12063:
12008:Category theory
11998:Algebraic logic
11981:
11952:Lambda calculus
11890:Church encoding
11876:
11852:Truth predicate
11708:
11674:Complete theory
11597:
11466:
11462:
11458:
11453:
11445:
11165: and
11161:
11156:
11142:
11118:New Foundations
11086:axiom of choice
11069:
11031:Gödel numbering
10971: and
10963:
10867:
10752:
10702:
10683:
10632:Boolean algebra
10618:
10582:Equiconsistency
10547:Classical logic
10524:
10505:Halting problem
10493: and
10469: and
10457: and
10456:
10451:Theorems (
10446:
10363:
10358:
10320:Classical Logic
10312:
10303:
10281:
10244:
10234:Springer-Verlag
10218:
10208:Springer-Verlag
10196:
10178:Hodges, Wilfrid
10171:
10149:Hodges, Wilfrid
10142:
10120:
10098:
10088:Springer-Verlag
10074:
10052:Springer-Verlag
10040:
10035:
10034:
10028:
10012:
10008:
9977:
9973:
9952:
9950:
9947:
9946:
9925:
9921:
9919:
9916:
9915:
9899:
9896:
9895:
9864:
9861:
9860:
9841:
9837:
9835:
9832:
9831:
9815:
9813:
9810:
9809:
9790:
9782:
9780:
9777:
9776:
9774:
9767:
9744:
9743:
9741:
9738:
9737:
9717:
9711:
9710:
9705:
9703:
9700:
9699:
9697:
9693:
9687:inclusive logic
9684:
9680:
9669:
9665:
9651:
9650:
9646:
9639:
9625:
9621:
9612:
9608:
9603:
9590:
9561:category theory
9553:
9537:
9531:
9515:
9463:
9460:
9459:
9439:
9436:
9435:
9431:
9426:
9377:
9372:
9371:
9365:
9364:
9359:
9350:
9345:
9344:
9338:
9337:
9332:
9323:
9318:
9317:
9311:
9310:
9305:
9295:
9294:
9290:
9288:
9285:
9284:
9267:
9262:
9261:
9255:
9254:
9249:
9233:
9232:
9227:
9221:
9218:
9217:
9200:
9195:
9194:
9188:
9187:
9182:
9166:
9165:
9160:
9154:
9151:
9150:
9127:
9122:
9121:
9115:
9114:
9109:
9107:
9104:
9103:
9080:
9075:
9074:
9068:
9067:
9062:
9060:
9057:
9056:
9039:
9038:
9036:
9033:
9032:
8991:
8974:
8953:
8936:
8930:
8913:
8892:
8886:
8844:
8843:
8835:
8834:
8830:
8799:
8798:
8796:
8793:
8792:
8772:
8769:
8768:
8742:
8741:
8739:
8736:
8735:
8718:
8717:
8715:
8712:
8711:
8695:
8692:
8691:
8672:
8669:
8668:
8651:
8650:
8648:
8645:
8644:
8628:
8625:
8624:
8604:
8603:
8601:
8598:
8597:
8581:
8578:
8577:
8561:
8558:
8557:
8541:
8538:
8537:
8520:
8519:
8517:
8514:
8513:
8497:
8494:
8493:
8477:
8474:
8473:
8445:
8441:
8426:
8422:
8407:
8406:
8397:
8393:
8381:
8377:
8362:
8358:
8344:
8341:
8340:
8320:
8316:
8301:
8297:
8289:
8286:
8285:
8268:
8267:
8265:
8262:
8261:
8245:
8242:
8241:
8225:
8222:
8221:
8205:
8202:
8201:
8198:
8151:
8148:
8147:
8131:
8128:
8127:
8110:
8109:
8107:
8104:
8103:
8087:
8084:
8083:
8063:
8057:
8056:
8051:
8049:
8046:
8045:
8025:
8022:
8021:
8018:
7955:
7954:
7952:
7949:
7948:
7923:
7920:
7919:
7902:
7901:
7899:
7896:
7895:
7879:
7876:
7875:
7859:
7856:
7855:
7832:
7828:
7813:
7809:
7794:
7793:
7775:
7771:
7756:
7752:
7747:
7744:
7743:
7727:
7724:
7723:
7707:
7704:
7703:
7677:
7673:
7658:
7654:
7639:
7638:
7629:
7625:
7613:
7609:
7594:
7590:
7576:
7573:
7572:
7552:
7548:
7533:
7529:
7521:
7518:
7517:
7500:
7497:
7496:
7474:
7471:
7470:
7441:
7440:
7438:
7435:
7434:
7418:
7415:
7414:
7398:
7395:
7394:
7378:
7375:
7374:
7371:
7342:
7341:
7339:
7336:
7335:
7319:
7316:
7315:
7299:
7296:
7295:
7278:
7277:
7275:
7272:
7271:
7255:
7252:
7251:
7227:
7226:
7224:
7221:
7220:
7194:
7191:
7190:
7173:
7172:
7170:
7167:
7166:
7149:
7146:
7145:
7122:
7121:
7119:
7116:
7115:
7071:
7070:
7068:
7065:
7064:
7061:
7044:
7034:
7013:database theory
6976:
6975:
6966:
6965:
6957:
6954:
6953:
6936:
6935:
6933:
6930:
6929:
6912:
6911:
6909:
6906:
6905:
6894:
6846:
6798:respectively).
6782:
6781:
6779:
6776:
6775:
6758:
6757:
6755:
6752:
6751:
6729:
6728:
6724:
6722:
6719:
6718:
6700:
6699:
6695:
6693:
6690:
6689:
6666:
6665:
6661:
6646:
6642:
6618:
6614:
6596:
6592:
6571:
6570:
6566:
6554:
6550:
6535:
6531:
6522:
6518:
6513:
6510:
6509:
6486:
6482:
6467:
6463:
6454:
6450:
6448:
6445:
6444:
6407:
6406:
6397:
6396:
6388:
6385:
6384:
6381:
6346:
6342:
6327:
6323:
6303:
6299:
6284:
6280:
6267:
6263:
6248:
6244:
6242:
6239:
6238:
6220:
6219:
6215:
6203:
6199:
6184:
6180:
6171:
6167:
6162:
6159:
6158:
6142:
6136:
6135:
6130:
6121:
6117:
6102:
6098:
6089:
6085:
6083:
6080:
6079:
6061:
6060:
6056:
6044:
6040:
6025:
6021:
6012:
6008:
6003:
6000:
5999:
5983:
5977:
5976:
5971:
5962:
5958:
5943:
5939:
5930:
5926:
5924:
5921:
5920:
5887:
5886:
5877:
5876:
5868:
5865:
5864:
5863:A homomorphism
5826:
5825:
5823:
5820:
5819:
5802:
5801:
5799:
5796:
5795:
5779:
5773:
5772:
5767:
5765:
5762:
5761:
5745:
5739:
5738:
5733:
5731:
5728:
5727:
5706:
5705:
5696:
5695:
5687:
5684:
5683:
5666:
5665:
5663:
5660:
5659:
5642:
5641:
5639:
5636:
5635:
5630:A homomorphism
5611:
5610:
5608:
5605:
5604:
5587:
5586:
5584:
5581:
5580:
5564:
5561:
5560:
5542:
5541:
5537:
5535:
5532:
5531:
5513:
5512:
5508:
5506:
5503:
5502:
5479:
5478:
5474:
5459:
5455:
5431:
5427:
5409:
5405:
5384:
5383:
5379:
5367:
5363:
5348:
5344:
5335:
5331:
5326:
5323:
5322:
5300:
5294:
5293:
5288:
5279:
5275:
5260:
5256:
5247:
5243:
5241:
5238:
5237:
5199:
5195:
5171:
5167:
5149:
5145:
5118:
5114:
5099:
5095:
5086:
5082:
5068:
5065:
5064:
5042:
5036:
5035:
5030:
5021:
5017:
5002:
4998:
4989:
4985:
4983:
4980:
4979:
4951:
4945:
4944:
4939:
4931:
4925:
4924:
4919:
4911:
4908:
4907:
4887:
4886:
4884:
4881:
4880:
4863:
4862:
4860:
4857:
4856:
4835:
4834:
4832:
4829:
4828:
4811:
4810:
4808:
4805:
4804:
4801:
4796:
4790:
4758:
4755:
4754:
4738:
4735:
4734:
4718:
4715:
4714:
4698:
4695:
4694:
4674:
4671:
4670:
4654:
4651:
4650:
4634:
4613:
4610:
4609:
4591:
4588:
4587:
4571:
4568:
4567:
4548:
4545:
4544:
4528:
4525:
4524:
4502:complex numbers
4477:
4474:
4473:
4421:
4418:
4417:
4414:
4374:
4373:
4371:
4368:
4367:
4351:
4348:
4347:
4328:
4326:
4323:
4322:
4302:
4301:
4299:
4296:
4295:
4273:
4256:
4253:
4252:
4230:
4227:
4226:
4185:
4184:
4182:
4179:
4178:
4159:
4153:
4152:
4147:
4145:
4142:
4141:
4114:
4111:
4110:
4109:. The operator
4092:
4091:
4087:
4079:
4076:
4075:
4053:
4050:
4049:
4048:and denoted by
4030:
4027:
4026:
4003:
4000:
3999:
3976:
3973:
3972:
3956:
3950:
3949:
3944:
3942:
3939:
3938:
3922:
3916:
3915:
3910:
3902:
3899:
3898:
3867:
3863:
3848:
3844:
3835:
3831:
3823:
3820:
3819:
3801:
3798:
3797:
3780:
3776:
3767:
3763:
3757:
3753:
3751:
3748:
3747:
3731:
3728:
3727:
3711:
3708:
3707:
3681:
3677:
3662:
3658:
3649:
3645:
3643:
3640:
3639:
3622:
3621:
3619:
3616:
3615:
3599:
3596:
3595:
3579:
3576:
3575:
3556:
3553:
3552:
3532:
3531:
3529:
3526:
3525:
3504:
3503:
3501:
3498:
3497:
3481:
3475:
3474:
3469:
3461:
3458:
3457:
3435:
3434:
3425:
3424:
3422:
3419:
3418:
3396:
3390:
3389:
3384:
3382:
3379:
3378:
3356:
3350:
3349:
3344:
3336:
3330:
3329:
3324:
3322:
3319:
3318:
3299:
3298:
3296:
3293:
3292:
3275:
3274:
3272:
3269:
3268:
3243:
3242:
3224:
3223:
3215:
3212:
3211:
3194:
3193:
3191:
3188:
3187:
3170:
3169:
3167:
3164:
3163:
3143:
3142:
3140:
3137:
3136:
3115:
3114:
3112:
3109:
3108:
3106:
3085:
3082:
3081:
3062:
3059:
3058:
3030:
3027:
3026:
3008:
3005:
3004:
2980:
2976:
2974:
2971:
2970:
2949:
2945:
2943:
2940:
2939:
2919:
2917:
2914:
2913:
2892:
2891:
2887:
2885:
2882:
2881:
2863:
2862:
2858:
2856:
2853:
2852:
2830:
2827:
2826:
2810:
2791:
2790:
2786:
2784:
2781:
2780:
2759:
2756:
2755:
2739:
2720:
2719:
2715:
2713:
2710:
2709:
2685:
2682:
2681:
2665:
2662:
2661:
2645:
2637:
2618:
2617:
2613:
2611:
2608:
2607:
2589:
2581:
2573:
2554:
2553:
2549:
2547:
2544:
2543:
2525:
2517:
2509:
2490:
2489:
2485:
2483:
2480:
2479:
2458:
2457:
2453:
2451:
2448:
2447:
2431:
2430:
2419:
2406:
2400:
2396:
2393:
2392:
2378:
2365:
2359:
2355:
2352:
2351:
2337:
2324:
2318:
2314:
2311:
2310:
2296:
2283:
2277:
2273:
2270:
2269:
2255:
2242:
2236:
2232:
2228:
2226:
2223:
2222:
2169:
2165:
2163:
2160:
2159:
2139:
2135:
2126:
2122:
2110:
2106:
2104:
2101:
2100:
2084:
2083:
2073:
2072:
2068:
2059:
2055:
2047:
2037:
2031:
2030:
2027:
2026:
2016:
2015:
2011:
2002:
1998:
1990:
1980:
1974:
1973:
1970:
1969:
1959:
1958:
1954:
1945:
1941:
1933:
1923:
1917:
1916:
1912:
1910:
1907:
1906:
1890:
1887:
1886:
1867:
1865:
1862:
1861:
1859:complex numbers
1842:
1840:
1837:
1836:
1814:
1812:
1809:
1808:
1789:
1786:
1785:
1769:
1767:
1764:
1763:
1747:
1745:
1742:
1741:
1725:
1723:
1720:
1719:
1703:
1701:
1698:
1697:
1681:
1679:
1676:
1675:
1659:
1657:
1654:
1653:
1637:
1635:
1632:
1631:
1615:
1613:
1610:
1609:
1588:
1584:
1582:
1579:
1578:
1575:
1552:
1546:
1545:
1540:
1531:
1526:
1525:
1519:
1518:
1513:
1503:
1502:
1498:
1496:
1493:
1492:
1475:
1474:
1465:
1459:
1458:
1457:
1449:
1446:
1445:
1425:
1424:
1422:
1419:
1418:
1402:
1399:
1398:
1370:
1367:
1366:
1350:
1347:
1346:
1318:
1315:
1314:
1310:constant symbol
1292:
1289:
1288:
1268:
1265:
1264:
1234:
1233:
1229:
1204:
1203:
1199:
1197:
1194:
1193:
1177:
1174:
1173:
1172:is assigned an
1157:
1154:
1153:
1137:
1134:
1133:
1100:
1099:
1095:
1093:
1090:
1089:
1071:
1068:
1067:
1065:is assigned an
1050:
1047:
1046:
1030:
1027:
1026:
1009:
1008:
1006:
1003:
1002:
986:
983:
982:
976:
969:
925:
922:
921:
901:
898:
897:
877:
874:
873:
839:
836:
835:
792:
789:
788:
769:
766:
765:
748:
743:
742:
725:
722:
721:
690:
687:
686:
648:
645:
644:
641:
635:
613:
612:
610:
607:
606:
586:
585:
583:
580:
579:
563:
557:
556:
551:
549:
546:
545:
525:
524:
513:
510:
509:
477:
471:
450:
447:
446:
430:
427:
426:
410:
407:
406:
384:
381:
380:
358:
355:
354:
308:
307:
305:
302:
301:
294:
266:
260:
257:
250:needs expansion
235:
219:database theory
208:interpretations
93:
82:
76:
73:
63:Please help to
62:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
12126:
12116:
12115:
12110:
12105:
12100:
12083:
12082:
12068:
12065:
12064:
12062:
12061:
12056:
12051:
12046:
12041:
12040:
12039:
12029:
12024:
12019:
12010:
12005:
12000:
11995:
11993:Abstract logic
11989:
11987:
11983:
11982:
11980:
11979:
11974:
11972:Turing machine
11969:
11964:
11959:
11954:
11949:
11944:
11943:
11942:
11937:
11932:
11927:
11922:
11912:
11910:Computable set
11907:
11902:
11897:
11892:
11886:
11884:
11878:
11877:
11875:
11874:
11869:
11864:
11859:
11854:
11849:
11844:
11839:
11838:
11837:
11832:
11827:
11817:
11812:
11807:
11805:Satisfiability
11802:
11797:
11792:
11791:
11790:
11780:
11779:
11778:
11768:
11767:
11766:
11761:
11756:
11751:
11746:
11736:
11735:
11734:
11729:
11722:Interpretation
11718:
11716:
11710:
11709:
11707:
11706:
11701:
11696:
11691:
11686:
11676:
11671:
11670:
11669:
11668:
11667:
11657:
11652:
11642:
11637:
11632:
11627:
11622:
11617:
11611:
11609:
11603:
11602:
11599:
11598:
11596:
11595:
11587:
11586:
11585:
11584:
11579:
11578:
11577:
11572:
11567:
11547:
11546:
11545:
11543:minimal axioms
11540:
11529:
11528:
11527:
11516:
11515:
11514:
11509:
11504:
11499:
11494:
11489:
11476:
11474:
11455:
11454:
11452:
11451:
11450:
11449:
11437:
11432:
11431:
11430:
11425:
11420:
11415:
11405:
11400:
11395:
11390:
11389:
11388:
11383:
11373:
11372:
11371:
11366:
11361:
11356:
11346:
11341:
11340:
11339:
11334:
11329:
11319:
11318:
11317:
11312:
11307:
11302:
11297:
11292:
11282:
11277:
11272:
11267:
11266:
11265:
11260:
11255:
11250:
11240:
11235:
11233:Formation rule
11230:
11225:
11224:
11223:
11218:
11208:
11207:
11206:
11196:
11191:
11186:
11181:
11175:
11169:
11152:Formal systems
11148:
11147:
11144:
11143:
11141:
11140:
11135:
11130:
11125:
11120:
11115:
11110:
11105:
11100:
11095:
11094:
11093:
11088:
11077:
11075:
11071:
11070:
11068:
11067:
11066:
11065:
11055:
11050:
11049:
11048:
11041:Large cardinal
11038:
11033:
11028:
11023:
11018:
11004:
11003:
11002:
10997:
10992:
10977:
10975:
10965:
10964:
10962:
10961:
10960:
10959:
10954:
10949:
10939:
10934:
10929:
10924:
10919:
10914:
10909:
10904:
10899:
10894:
10889:
10884:
10878:
10876:
10869:
10868:
10866:
10865:
10864:
10863:
10858:
10853:
10848:
10843:
10838:
10830:
10829:
10828:
10823:
10813:
10808:
10806:Extensionality
10803:
10801:Ordinal number
10798:
10788:
10783:
10782:
10781:
10770:
10764:
10758:
10757:
10754:
10753:
10751:
10750:
10745:
10740:
10735:
10730:
10725:
10720:
10719:
10718:
10708:
10707:
10706:
10693:
10691:
10685:
10684:
10682:
10681:
10680:
10679:
10674:
10669:
10659:
10654:
10649:
10644:
10639:
10634:
10628:
10626:
10620:
10619:
10617:
10616:
10611:
10606:
10601:
10596:
10591:
10586:
10585:
10584:
10574:
10569:
10564:
10559:
10554:
10549:
10543:
10541:
10532:
10526:
10525:
10523:
10522:
10517:
10512:
10507:
10502:
10497:
10485:Cantor's
10483:
10478:
10473:
10463:
10461:
10448:
10447:
10445:
10444:
10439:
10434:
10429:
10424:
10419:
10414:
10409:
10404:
10399:
10394:
10389:
10384:
10383:
10382:
10371:
10369:
10365:
10364:
10357:
10356:
10349:
10342:
10334:
10328:
10327:
10311:
10310:External links
10308:
10307:
10306:
10301:
10284:
10279:
10247:
10242:
10221:
10216:
10199:
10194:
10174:
10169:
10145:
10140:
10123:
10118:
10101:
10096:
10077:
10072:
10055:
10039:
10036:
10033:
10032:
10026:
10006:
9971:
9959:
9955:
9928:
9924:
9903:
9883:
9880:
9877:
9874:
9871:
9868:
9849:
9844:
9840:
9818:
9797:
9793:
9789:
9785:
9765:
9752:
9747:
9720:
9714:
9708:
9691:
9678:
9663:
9644:
9637:
9619:
9605:
9604:
9602:
9599:
9598:
9597:
9589:
9586:
9552:
9549:
9533:Main article:
9530:
9527:
9514:
9511:
9467:
9443:
9430:
9427:
9425:
9422:
9380:
9375:
9368:
9362:
9358:
9353:
9348:
9341:
9335:
9331:
9326:
9321:
9314:
9308:
9304:
9298:
9293:
9270:
9265:
9258:
9252:
9248:
9245:
9242:
9236:
9230:
9226:
9203:
9198:
9191:
9185:
9181:
9178:
9175:
9169:
9163:
9159:
9138:
9135:
9130:
9125:
9118:
9112:
9091:
9088:
9083:
9078:
9071:
9065:
9042:
9019:
9018:
9017:
9016:
8999:
8998:
8997:
8987:
8984:
8970:
8967:
8949:
8944:
8943:
8942:
8932:
8926:
8923:
8909:
8906:
8888:
8882:
8829:
8826:
8822:Beth's theorem
8807:
8802:
8787:is said to be
8776:
8756:
8753:
8750:
8745:
8721:
8699:
8679:
8676:
8654:
8632:
8612:
8607:
8585:
8576:itself, since
8565:
8545:
8523:
8501:
8481:
8459:
8456:
8453:
8448:
8444:
8440:
8437:
8434:
8429:
8425:
8421:
8418:
8415:
8410:
8405:
8400:
8396:
8392:
8389:
8384:
8380:
8376:
8373:
8370:
8365:
8361:
8357:
8354:
8351:
8348:
8328:
8323:
8319:
8315:
8312:
8309:
8304:
8300:
8296:
8293:
8271:
8249:
8229:
8220:-ary relation
8209:
8197:
8194:
8158:
8155:
8135:
8113:
8091:
8066:
8060:
8054:
8040:is said to be
8029:
8017:
8014:
8002:
7999:
7996:
7993:
7990:
7987:
7984:
7981:
7978:
7975:
7972:
7969:
7966:
7963:
7958:
7936:
7933:
7930:
7927:
7905:
7883:
7863:
7840:
7835:
7831:
7827:
7824:
7821:
7816:
7812:
7808:
7805:
7802:
7797:
7792:
7789:
7786:
7783:
7778:
7774:
7770:
7767:
7764:
7759:
7755:
7751:
7731:
7711:
7691:
7688:
7685:
7680:
7676:
7672:
7669:
7666:
7661:
7657:
7653:
7650:
7647:
7642:
7637:
7632:
7628:
7624:
7621:
7616:
7612:
7608:
7605:
7602:
7597:
7593:
7589:
7586:
7583:
7580:
7560:
7555:
7551:
7547:
7544:
7541:
7536:
7532:
7528:
7525:
7504:
7478:
7457:is said to be
7444:
7422:
7402:
7393:-ary relation
7382:
7370:
7367:
7363:ZFC set theory
7350:
7345:
7323:
7303:
7281:
7259:
7230:
7201:
7198:
7176:
7154:
7133:
7130:
7125:
7100:
7097:
7094:
7091:
7088:
7085:
7082:
7079:
7074:
7060:
7057:
7033:
7030:
6994:
6993:
6979:
6974:
6969:
6964:
6961:
6939:
6915:
6893:
6890:
6845:
6842:
6785:
6761:
6732:
6727:
6703:
6698:
6686:
6685:
6684:
6683:
6669:
6664:
6660:
6657:
6654:
6649:
6645:
6641:
6638:
6635:
6632:
6629:
6626:
6621:
6617:
6613:
6610:
6607:
6604:
6599:
6595:
6591:
6588:
6585:
6581:
6574:
6569:
6565:
6562:
6557:
6553:
6549:
6546:
6543:
6538:
6534:
6530:
6525:
6521:
6517:
6504:
6503:
6489:
6485:
6481:
6478:
6475:
6470:
6466:
6462:
6457:
6453:
6410:
6405:
6400:
6395:
6392:
6380:
6377:
6369:
6368:
6357:
6354:
6349:
6345:
6341:
6338:
6335:
6330:
6326:
6321:
6318:
6314:
6311:
6306:
6302:
6298:
6295:
6292:
6287:
6283:
6278:
6275:
6270:
6266:
6262:
6259:
6256:
6251:
6247:
6223:
6218:
6214:
6211:
6206:
6202:
6198:
6195:
6192:
6187:
6183:
6179:
6174:
6170:
6166:
6145:
6139:
6133:
6129:
6124:
6120:
6116:
6113:
6110:
6105:
6101:
6097:
6092:
6088:
6064:
6059:
6055:
6052:
6047:
6043:
6039:
6036:
6033:
6028:
6024:
6020:
6015:
6011:
6007:
5986:
5980:
5974:
5970:
5965:
5961:
5957:
5954:
5951:
5946:
5942:
5938:
5933:
5929:
5890:
5885:
5880:
5875:
5872:
5829:
5805:
5782:
5776:
5770:
5748:
5742:
5736:
5709:
5704:
5699:
5694:
5691:
5669:
5645:
5627:respectively.
5614:
5590:
5568:
5545:
5540:
5516:
5511:
5499:
5498:
5497:
5496:
5482:
5477:
5473:
5470:
5467:
5462:
5458:
5454:
5451:
5448:
5445:
5442:
5439:
5434:
5430:
5426:
5423:
5420:
5417:
5412:
5408:
5404:
5401:
5398:
5394:
5387:
5382:
5378:
5375:
5370:
5366:
5362:
5359:
5356:
5351:
5347:
5343:
5338:
5334:
5330:
5317:
5316:
5303:
5297:
5291:
5287:
5282:
5278:
5274:
5271:
5268:
5263:
5259:
5255:
5250:
5246:
5225:
5224:
5223:
5222:
5210:
5207:
5202:
5198:
5194:
5191:
5188:
5185:
5182:
5179:
5174:
5170:
5166:
5163:
5160:
5157:
5152:
5148:
5144:
5141:
5138:
5135:
5132:
5129:
5126:
5121:
5117:
5113:
5110:
5107:
5102:
5098:
5094:
5089:
5085:
5081:
5078:
5075:
5072:
5059:
5058:
5045:
5039:
5033:
5029:
5024:
5020:
5016:
5013:
5010:
5005:
5001:
4997:
4992:
4988:
4954:
4948:
4942:
4938:
4934:
4928:
4922:
4918:
4915:
4890:
4866:
4838:
4814:
4800:
4797:
4789:
4786:
4765:
4762:
4742:
4722:
4702:
4678:
4658:
4633:
4629:
4626:
4623:
4620:
4617:
4598:
4595:
4575:
4555:
4552:
4532:
4481:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4413:
4410:
4382:
4377:
4355:
4334:
4331:
4310:
4305:
4283:
4279:
4276:
4272:
4269:
4266:
4263:
4260:
4240:
4237:
4234:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4188:
4162:
4156:
4150:
4138:set of subsets
4121:
4118:
4095:
4090:
4086:
4083:
4063:
4060:
4057:
4037:
4034:
4010:
4007:
3983:
3980:
3971:that contains
3959:
3953:
3947:
3925:
3919:
3913:
3909:
3906:
3884:
3881:
3878:
3875:
3870:
3866:
3862:
3859:
3856:
3851:
3847:
3843:
3838:
3834:
3830:
3827:
3808:
3805:
3783:
3779:
3775:
3770:
3766:
3760:
3756:
3735:
3715:
3695:
3692:
3689:
3684:
3680:
3676:
3673:
3670:
3665:
3661:
3657:
3652:
3648:
3625:
3603:
3583:
3563:
3560:
3540:
3535:
3507:
3484:
3478:
3472:
3468:
3465:
3443:
3438:
3433:
3428:
3415:
3414:
3403:
3399:
3393:
3387:
3375:
3363:
3359:
3353:
3347:
3343:
3339:
3333:
3327:
3307:
3302:
3278:
3267:the domain of
3265:
3254:
3251:
3246:
3241:
3238:
3235:
3232:
3227:
3222:
3219:
3197:
3173:
3146:
3118:
3105:
3102:
3089:
3069:
3066:
3038:
3035:
3013:
3001:ordered fields
2983:
2979:
2968:
2952:
2948:
2922:
2895:
2890:
2866:
2861:
2849:
2848:
2837:
2834:
2825:is the number
2813:
2809:
2806:
2803:
2800:
2794:
2789:
2778:
2766:
2763:
2754:is the number
2742:
2738:
2735:
2732:
2729:
2723:
2718:
2707:
2695:
2692:
2689:
2669:
2648:
2644:
2640:
2636:
2633:
2630:
2627:
2621:
2616:
2605:
2592:
2588:
2584:
2580:
2576:
2572:
2569:
2566:
2563:
2557:
2552:
2541:
2528:
2524:
2520:
2516:
2512:
2508:
2505:
2502:
2499:
2493:
2488:
2461:
2456:
2429:
2426:
2423:
2421:
2418:
2415:
2412:
2409:
2407:
2403:
2399:
2395:
2394:
2391:
2388:
2385:
2382:
2380:
2377:
2374:
2371:
2368:
2366:
2362:
2358:
2354:
2353:
2350:
2347:
2344:
2341:
2339:
2336:
2333:
2330:
2327:
2325:
2321:
2317:
2313:
2312:
2309:
2306:
2303:
2300:
2298:
2295:
2292:
2289:
2286:
2284:
2280:
2276:
2272:
2271:
2268:
2265:
2262:
2259:
2257:
2254:
2251:
2248:
2245:
2243:
2239:
2235:
2231:
2230:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2172:
2168:
2147:
2142:
2138:
2134:
2129:
2125:
2121:
2118:
2113:
2109:
2082:
2076:
2071:
2067:
2062:
2058:
2054:
2050:
2046:
2043:
2040:
2038:
2034:
2029:
2028:
2025:
2019:
2014:
2010:
2005:
2001:
1997:
1993:
1989:
1986:
1983:
1981:
1977:
1972:
1971:
1968:
1962:
1957:
1953:
1948:
1944:
1940:
1936:
1932:
1929:
1926:
1924:
1920:
1915:
1914:
1894:
1874:
1870:
1845:
1821:
1817:
1793:
1772:
1750:
1728:
1706:
1684:
1662:
1640:
1618:
1591:
1587:
1574:
1571:
1559:
1555:
1549:
1543:
1539:
1534:
1529:
1522:
1516:
1512:
1506:
1501:
1478:
1473:
1468:
1462:
1456:
1453:
1433:
1428:
1406:
1386:
1383:
1380:
1377:
1374:
1354:
1331:
1328:
1325:
1322:
1296:
1276:
1272:
1249:
1246:
1243:
1240:
1237:
1232:
1228:
1225:
1222:
1219:
1216:
1213:
1207:
1202:
1192:-ary relation
1181:
1161:
1141:
1121:
1118:
1115:
1112:
1109:
1103:
1098:
1075:
1054:
1034:
1012:
990:
968:
965:
932:
929:
905:
892:is called the
881:
861:
858:
855:
852:
849:
846:
843:
833:natural number
829:
828:
817:
814:
811:
808:
805:
802:
799:
796:
786:natural number
773:
751:
746:
741:
738:
732:
729:
718:
694:
670:
667:
664:
661:
658:
655:
652:
643:The signature
637:Main article:
634:
631:
616:
594:
589:
566:
560:
554:
533:
528:
523:
520:
517:
500:
490:
486:
482:
481:underlying set
473:Main article:
470:
467:
454:
434:
414:
391:
388:
365:
362:
337:
334:
331:
328:
325:
322:
319:
316:
311:
293:
290:
268:
267:
247:
245:
234:
231:
199:semantic model
111:consists of a
95:
94:
49:
47:
40:
26:
9:
6:
4:
3:
2:
12125:
12114:
12111:
12109:
12106:
12104:
12101:
12099:
12096:
12095:
12093:
12080:
12079:
12074:
12066:
12060:
12057:
12055:
12052:
12050:
12047:
12045:
12042:
12038:
12035:
12034:
12033:
12030:
12028:
12025:
12023:
12020:
12018:
12014:
12011:
12009:
12006:
12004:
12001:
11999:
11996:
11994:
11991:
11990:
11988:
11984:
11978:
11975:
11973:
11970:
11968:
11967:Recursive set
11965:
11963:
11960:
11958:
11955:
11953:
11950:
11948:
11945:
11941:
11938:
11936:
11933:
11931:
11928:
11926:
11923:
11921:
11918:
11917:
11916:
11913:
11911:
11908:
11906:
11903:
11901:
11898:
11896:
11893:
11891:
11888:
11887:
11885:
11883:
11879:
11873:
11870:
11868:
11865:
11863:
11860:
11858:
11855:
11853:
11850:
11848:
11845:
11843:
11840:
11836:
11833:
11831:
11828:
11826:
11823:
11822:
11821:
11818:
11816:
11813:
11811:
11808:
11806:
11803:
11801:
11798:
11796:
11793:
11789:
11786:
11785:
11784:
11781:
11777:
11776:of arithmetic
11774:
11773:
11772:
11769:
11765:
11762:
11760:
11757:
11755:
11752:
11750:
11747:
11745:
11742:
11741:
11740:
11737:
11733:
11730:
11728:
11725:
11724:
11723:
11720:
11719:
11717:
11715:
11711:
11705:
11702:
11700:
11697:
11695:
11692:
11690:
11687:
11684:
11683:from ZFC
11680:
11677:
11675:
11672:
11666:
11663:
11662:
11661:
11658:
11656:
11653:
11651:
11648:
11647:
11646:
11643:
11641:
11638:
11636:
11633:
11631:
11628:
11626:
11623:
11621:
11618:
11616:
11613:
11612:
11610:
11608:
11604:
11594:
11593:
11589:
11588:
11583:
11582:non-Euclidean
11580:
11576:
11573:
11571:
11568:
11566:
11565:
11561:
11560:
11558:
11555:
11554:
11552:
11548:
11544:
11541:
11539:
11536:
11535:
11534:
11530:
11526:
11523:
11522:
11521:
11517:
11513:
11510:
11508:
11505:
11503:
11500:
11498:
11495:
11493:
11490:
11488:
11485:
11484:
11482:
11478:
11477:
11475:
11470:
11464:
11459:Example
11456:
11448:
11443:
11442:
11441:
11438:
11436:
11433:
11429:
11426:
11424:
11421:
11419:
11416:
11414:
11411:
11410:
11409:
11406:
11404:
11401:
11399:
11396:
11394:
11391:
11387:
11384:
11382:
11379:
11378:
11377:
11374:
11370:
11367:
11365:
11362:
11360:
11357:
11355:
11352:
11351:
11350:
11347:
11345:
11342:
11338:
11335:
11333:
11330:
11328:
11325:
11324:
11323:
11320:
11316:
11313:
11311:
11308:
11306:
11303:
11301:
11298:
11296:
11293:
11291:
11288:
11287:
11286:
11283:
11281:
11278:
11276:
11273:
11271:
11268:
11264:
11261:
11259:
11256:
11254:
11251:
11249:
11246:
11245:
11244:
11241:
11239:
11236:
11234:
11231:
11229:
11226:
11222:
11219:
11217:
11216:by definition
11214:
11213:
11212:
11209:
11205:
11202:
11201:
11200:
11197:
11195:
11192:
11190:
11187:
11185:
11182:
11180:
11177:
11176:
11173:
11170:
11168:
11164:
11159:
11153:
11149:
11139:
11136:
11134:
11131:
11129:
11126:
11124:
11121:
11119:
11116:
11114:
11111:
11109:
11106:
11104:
11103:Kripke–Platek
11101:
11099:
11096:
11092:
11089:
11087:
11084:
11083:
11082:
11079:
11078:
11076:
11072:
11064:
11061:
11060:
11059:
11056:
11054:
11051:
11047:
11044:
11043:
11042:
11039:
11037:
11034:
11032:
11029:
11027:
11024:
11022:
11019:
11016:
11012:
11008:
11005:
11001:
10998:
10996:
10993:
10991:
10988:
10987:
10986:
10982:
10979:
10978:
10976:
10974:
10970:
10966:
10958:
10955:
10953:
10950:
10948:
10947:constructible
10945:
10944:
10943:
10940:
10938:
10935:
10933:
10930:
10928:
10925:
10923:
10920:
10918:
10915:
10913:
10910:
10908:
10905:
10903:
10900:
10898:
10895:
10893:
10890:
10888:
10885:
10883:
10880:
10879:
10877:
10875:
10870:
10862:
10859:
10857:
10854:
10852:
10849:
10847:
10844:
10842:
10839:
10837:
10834:
10833:
10831:
10827:
10824:
10822:
10819:
10818:
10817:
10814:
10812:
10809:
10807:
10804:
10802:
10799:
10797:
10793:
10789:
10787:
10784:
10780:
10777:
10776:
10775:
10772:
10771:
10768:
10765:
10763:
10759:
10749:
10746:
10744:
10741:
10739:
10736:
10734:
10731:
10729:
10726:
10724:
10721:
10717:
10714:
10713:
10712:
10709:
10705:
10700:
10699:
10698:
10695:
10694:
10692:
10690:
10686:
10678:
10675:
10673:
10670:
10668:
10665:
10664:
10663:
10660:
10658:
10655:
10653:
10650:
10648:
10645:
10643:
10640:
10638:
10635:
10633:
10630:
10629:
10627:
10625:
10624:Propositional
10621:
10615:
10612:
10610:
10607:
10605:
10602:
10600:
10597:
10595:
10592:
10590:
10587:
10583:
10580:
10579:
10578:
10575:
10573:
10570:
10568:
10565:
10563:
10560:
10558:
10555:
10553:
10552:Logical truth
10550:
10548:
10545:
10544:
10542:
10540:
10536:
10533:
10531:
10527:
10521:
10518:
10516:
10513:
10511:
10508:
10506:
10503:
10501:
10498:
10496:
10492:
10488:
10484:
10482:
10479:
10477:
10474:
10472:
10468:
10465:
10464:
10462:
10460:
10454:
10449:
10443:
10440:
10438:
10435:
10433:
10430:
10428:
10425:
10423:
10420:
10418:
10415:
10413:
10410:
10408:
10405:
10403:
10400:
10398:
10395:
10393:
10390:
10388:
10385:
10381:
10378:
10377:
10376:
10373:
10372:
10370:
10366:
10362:
10355:
10350:
10348:
10343:
10341:
10336:
10335:
10332:
10325:
10322:(an entry of
10321:
10317:
10314:
10313:
10304:
10298:
10294:
10290:
10285:
10282:
10276:
10272:
10268:
10264:
10260:
10256:
10252:
10248:
10245:
10239:
10235:
10230:
10229:
10222:
10219:
10213:
10209:
10205:
10200:
10197:
10191:
10187:
10184:, Cambridge:
10183:
10179:
10175:
10172:
10166:
10162:
10159:, Cambridge:
10157:
10156:
10150:
10146:
10143:
10137:
10133:
10129:
10124:
10121:
10115:
10110:
10109:
10102:
10099:
10093:
10089:
10085:
10084:
10078:
10075:
10069:
10065:
10061:
10056:
10053:
10049:
10048:
10042:
10041:
10029:
10027:9780080528700
10023:
10019:
10018:
10010:
10002:
9998:
9994:
9990:
9986:
9982:
9975:
9957:
9944:
9926:
9922:
9901:
9881:
9878:
9875:
9872:
9869:
9866:
9847:
9842:
9838:
9795:
9787:
9772:
9770:
9750:
9735:
9695:
9688:
9682:
9674:
9667:
9661:
9658:
9654:
9648:
9640:
9634:
9630:
9623:
9616:
9610:
9606:
9595:
9592:
9591:
9585:
9583:
9582:
9577:
9572:
9570:
9566:
9562:
9558:
9548:
9546:
9542:
9536:
9526:
9524:
9520:
9510:
9506:
9504:
9500:
9496:
9493: +
9492:
9489: =
9488:
9485: +
9484:
9480:
9457:
9421:
9419:
9415:
9411:
9407:
9403:
9398:
9394:
9378:
9351:
9329:
9324:
9302:
9291:
9268:
9246:
9243:
9240:
9228:
9224:
9201:
9179:
9176:
9173:
9161:
9157:
9136:
9133:
9128:
9089:
9086:
9081:
9030:
9026:
9014:
9010:
9006:
9002:
9001:
9000:
8995:
8990:
8985:
8982:
8978:
8973:
8968:
8965:
8961:
8957:
8952:
8947:
8946:
8945:
8940:
8935:
8929:
8924:
8921:
8917:
8912:
8907:
8904:
8900:
8896:
8891:
8885:
8880:
8879:
8878:
8874:
8872:
8868:
8864:
8863:Vector spaces
8860:
8857:
8853:
8849:
8840:
8825:
8823:
8818:
8805:
8790:
8774:
8754:
8751:
8748:
8697:
8677:
8674:
8630:
8610:
8583:
8563:
8543:
8499:
8479:
8470:
8457:
8446:
8442:
8438:
8435:
8432:
8427:
8423:
8416:
8413:
8403:
8398:
8394:
8390:
8382:
8378:
8374:
8371:
8368:
8363:
8359:
8349:
8346:
8321:
8317:
8313:
8310:
8307:
8302:
8298:
8291:
8247:
8227:
8207:
8193:
8191:
8187:
8183:
8179:
8175:
8170:
8156:
8153:
8133:
8089:
8081:
8043:
8027:
8013:
8000:
7991:
7985:
7979:
7976:
7973:
7967:
7961:
7931:
7925:
7881:
7861:
7852:
7833:
7829:
7825:
7822:
7819:
7814:
7810:
7803:
7800:
7787:
7784:
7776:
7772:
7768:
7765:
7762:
7757:
7753:
7729:
7709:
7689:
7678:
7674:
7670:
7667:
7664:
7659:
7655:
7648:
7645:
7635:
7630:
7626:
7622:
7614:
7610:
7606:
7603:
7600:
7595:
7591:
7581:
7578:
7553:
7549:
7545:
7542:
7539:
7534:
7530:
7523:
7515:
7492:
7468:
7464:
7460:
7420:
7400:
7380:
7366:
7364:
7348:
7321:
7301:
7257:
7250:
7246:
7217:
7215:
7199:
7196:
7152:
7131:
7128:
7114:
7095:
7092:
7089:
7086:
7083:
7077:
7056:
7054:
7050:
7043:
7039:
7029:
7026:
7022:
7018:
7014:
7009:
7007:
7003:
6999:
6962:
6959:
6903:
6902:
6901:
6899:
6889:
6887:
6883:
6879:
6875:
6871:
6867:
6864: →
6863:
6859:
6855:
6851:
6841:
6839:
6835:
6831:
6830:monomorphisms
6827:
6823:
6819:
6814:
6812:
6808:
6804:
6799:
6749:
6725:
6696:
6662:
6658:
6647:
6643:
6636:
6633:
6630:
6627:
6619:
6615:
6608:
6605:
6597:
6593:
6586:
6567:
6563:
6555:
6551:
6547:
6544:
6541:
6536:
6532:
6528:
6523:
6519:
6508:
6507:
6506:
6505:
6487:
6483:
6479:
6476:
6473:
6468:
6464:
6460:
6455:
6451:
6442:
6438:
6434:
6433:
6432:
6430:
6426:
6393:
6390:
6376:
6374:
6355:
6347:
6343:
6336:
6333:
6328:
6324:
6319:
6316:
6312:
6304:
6300:
6293:
6290:
6285:
6281:
6276:
6268:
6264:
6257:
6254:
6249:
6245:
6216:
6212:
6204:
6200:
6196:
6193:
6190:
6185:
6181:
6177:
6172:
6168:
6127:
6122:
6118:
6114:
6111:
6108:
6103:
6099:
6095:
6090:
6086:
6057:
6053:
6045:
6041:
6037:
6034:
6031:
6026:
6022:
6018:
6013:
6009:
5968:
5963:
5959:
5955:
5952:
5949:
5944:
5940:
5936:
5931:
5927:
5918:
5914:
5910:
5909:
5908:
5906:
5873:
5870:
5861:
5859:
5855:
5851:
5848:
5843:
5725:
5692:
5689:
5633:
5628:
5566:
5538:
5509:
5475:
5471:
5460:
5456:
5449:
5446:
5443:
5440:
5432:
5428:
5421:
5418:
5410:
5406:
5399:
5380:
5376:
5368:
5364:
5360:
5357:
5354:
5349:
5345:
5341:
5336:
5332:
5321:
5320:
5319:
5318:
5285:
5280:
5276:
5272:
5269:
5266:
5261:
5257:
5253:
5248:
5244:
5235:
5231:
5227:
5226:
5200:
5196:
5189:
5186:
5183:
5180:
5172:
5168:
5161:
5158:
5150:
5146:
5139:
5133:
5130:
5119:
5115:
5111:
5108:
5105:
5100:
5096:
5092:
5087:
5083:
5076:
5070:
5063:
5062:
5061:
5060:
5027:
5022:
5018:
5014:
5011:
5008:
5003:
4999:
4995:
4990:
4986:
4977:
4973:
4969:
4968:
4967:
4916:
4913:
4906:
4854:
4799:Homomorphisms
4795:
4785:
4783:
4779:
4763:
4760:
4740:
4720:
4700:
4692:
4676:
4656:
4631:
4624:
4621:
4618:
4596:
4593:
4573:
4553:
4550:
4530:
4522:
4517:
4515:
4511:
4505:
4503:
4499:
4495:
4479:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4426:
4423:
4409:
4407:
4403:
4399:
4394:
4380:
4353:
4332:
4329:
4308:
4277:
4274:
4270:
4267:
4264:
4261:
4238:
4235:
4232:
4209:
4206:
4203:
4200:
4197:
4191:
4175:
4139:
4135:
4084:
4058:
4035:
4032:
4024:
4008:
4005:
3997:
3981:
3978:
3907:
3904:
3895:
3882:
3879:
3876:
3868:
3864:
3860:
3857:
3854:
3849:
3845:
3841:
3836:
3832:
3825:
3806:
3803:
3781:
3777:
3773:
3768:
3764:
3758:
3754:
3733:
3713:
3693:
3690:
3687:
3682:
3678:
3674:
3671:
3668:
3663:
3659:
3655:
3650:
3646:
3601:
3581:
3561:
3558:
3538:
3523:
3466:
3463:
3454:
3441:
3431:
3401:
3376:
3361:
3341:
3305:
3266:
3252:
3236:
3233:
3217:
3162:
3161:
3160:
3134:
3131:is called an
3101:
3087:
3067:
3064:
3055:
3053:
3036:
3033:
3011:
3002:
2997:
2981:
2977:
2966:
2950:
2946:
2937:
2912:But the ring
2910:
2888:
2859:
2835:
2832:
2807:
2801:
2787:
2779:
2764:
2761:
2736:
2730:
2716:
2708:
2693:
2690:
2687:
2667:
2634:
2628:
2614:
2606:
2578:
2570:
2564:
2550:
2542:
2514:
2506:
2500:
2486:
2478:
2477:
2476:
2454:
2444:
2427:
2424:
2422:
2413:
2408:
2401:
2397:
2389:
2386:
2383:
2381:
2372:
2367:
2360:
2356:
2348:
2345:
2342:
2340:
2331:
2326:
2319:
2315:
2307:
2304:
2301:
2299:
2290:
2285:
2278:
2274:
2266:
2263:
2260:
2258:
2249:
2244:
2237:
2233:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2175:
2170:
2166:
2140:
2136:
2132:
2127:
2123:
2116:
2111:
2107:
2097:
2069:
2065:
2060:
2056:
2052:
2041:
2039:
2012:
2008:
2003:
1999:
1995:
1984:
1982:
1955:
1951:
1946:
1942:
1938:
1927:
1925:
1892:
1872:
1860:
1835:
1819:
1807:
1791:
1770:
1638:
1607:
1589:
1585:
1570:
1557:
1532:
1510:
1499:
1466:
1454:
1451:
1431:
1404:
1384:
1378:
1372:
1352:
1343:
1326:
1320:
1312:
1311:
1294:
1274:
1270:
1230:
1226:
1220:
1214:
1211:
1200:
1179:
1159:
1139:
1116:
1110:
1107:
1096:
1087:
1073:
1052:
1032:
988:
981:
974:
964:
962:
958:
954:
953:
948:
943:
930:
927:
919:
903:
895:
879:
856:
850:
847:
844:
841:
834:
815:
809:
803:
800:
797:
794:
787:
771:
749:
736:
730:
727:
719:
716:
715:
710:
709:
692:
684:
683:
682:
665:
662:
659:
653:
650:
640:
630:
592:
518:
515:
506:
504:
499:
496:
494:
488:
484:
480:
476:
466:
452:
432:
412:
405:
389:
386:
379:
363:
360:
353:
352:
332:
329:
326:
323:
320:
314:
299:
289:
287:
283:
279:
275:
264:
261:November 2023
255:
251:
248:This section
246:
243:
239:
238:
230:
228:
224:
220:
215:
213:
209:
205:
201:
200:
195:
190:
188:
184:
180:
175:
173:
169:
165:
161:
157:
153:
149:
145:
144:vector spaces
141:
137:
133:
129:
124:
122:
118:
114:
110:
106:
102:
91:
88:
80:
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
12108:Model theory
12069:
11867:Ultraproduct
11738:
11714:Model theory
11679:Independence
11615:Formal proof
11607:Proof theory
11590:
11563:
11520:real numbers
11492:second-order
11403:Substitution
11280:Metalanguage
11221:conservative
11194:Axiom schema
11138:Constructive
11108:Morse–Kelley
11074:Set theories
11053:Aleph number
11046:inaccessible
10952:Grothendieck
10836:intersection
10723:Higher-order
10711:Second-order
10657:Truth tables
10614:Venn diagram
10426:
10397:Formal proof
10288:
10254:
10227:
10203:
10181:
10155:Model theory
10154:
10127:
10107:
10083:Graph Theory
10082:
10066:, Elsevier,
10064:Model Theory
10063:
10046:
10016:
10009:
9984:
9980:
9974:
9942:
9694:
9681:
9672:
9666:
9656:
9652:
9647:
9628:
9622:
9609:
9579:
9573:
9569:class models
9568:
9565:proper class
9554:
9538:
9522:
9516:
9507:
9494:
9490:
9486:
9482:
9478:
9455:
9432:
9399:
9395:
9028:
9024:
9022:
9012:
9008:
9004:
9003:× of arity (
8993:
8988:
8980:
8976:
8971:
8963:
8959:
8955:
8950:
8938:
8933:
8927:
8919:
8915:
8910:
8902:
8898:
8894:
8889:
8883:
8870:
8866:
8861:
8851:
8842:
8833:
8831:
8819:
8788:
8471:
8199:
8189:
8185:
8181:
8177:
8173:
8171:
8079:
8041:
8019:
7853:
7851:is correct.
7494:
7490:
7462:
7458:
7372:
7244:
7219:A structure
7218:
7112:
7062:
7052:
7045:
7010:
6995:
6897:
6895:
6885:
6877:
6873:
6870:monomorphism
6865:
6861:
6857:
6853:
6847:
6837:
6833:
6825:
6821:
6815:
6810:
6802:
6800:
6747:
6687:
6440:
6436:
6424:
6382:
6372:
6370:
6078:, there are
5916:
5912:
5904:
5862:
5853:
5844:
5723:
5631:
5629:
5500:
5233:
5229:
4975:
4971:
4852:
4802:
4778:graph theory
4518:
4506:
4498:real numbers
4415:
4395:
4176:
4022:
3995:
3896:
3521:
3455:
3416:
3107:
3056:
2998:
2996:-structure.
2911:
2850:
2445:
2098:
1834:real numbers
1576:
1491:rather than
1344:
1308:
1307:is called a
979:
977:
956:
950:
944:
893:
872:of a symbol
830:
717:, along with
712:
706:
642:
507:
503:empty domain
478:
465:-structure.
403:
349:
297:
296:Formally, a
295:
271:
258:
254:adding to it
249:
216:
197:
193:
191:
176:
168:foundational
166:, including
160:Model theory
147:
125:
108:
105:model theory
98:
83:
74:
55:
11977:Type theory
11925:undecidable
11857:Truth value
11744:equivalence
11423:non-logical
11036:Enumeration
11026:Isomorphism
10973:cardinality
10957:Von Neumann
10922:Ultrafilter
10887:Uncountable
10821:equivalence
10738:Quantifiers
10728:Fixed-point
10697:First-order
10577:Consistency
10562:Proposition
10539:Traditional
10510:Lindström's
10500:Compactness
10442:Type theory
10387:Cardinality
10318:section in
9734:cardinality
9519:type theory
9410:Bart Jacobs
9406:type theory
8020:A relation
6807:subcategory
4649:means that
720:a function
181:, cf. also
146:. The term
69:introducing
12092:Categories
11788:elementary
11481:arithmetic
11349:Quantifier
11327:functional
11199:Expression
10917:Transitive
10861:identities
10846:complement
10779:hereditary
10762:Set theory
10291:, London:
10132:A K Peters
10038:References
9557:set theory
8992:of arity (
8975:of arity (
8954:of arity (
8937:of arity (
8914:of arity (
8893:of arity (
8734:such that
8339:such that
8126:such that
7947:such that
7742:such that
7571:such that
7036:See also:
7015:, where a
6818:subobjects
6435:for every
6429:one-to-one
6379:Embeddings
6157:such that
5998:such that
5911:For every
5228:For every
4970:For every
4792:See also:
3520:is called
495:), or its
292:Definition
286:set theory
172:set theory
77:April 2010
52:references
12059:Supertask
11962:Recursion
11920:decidable
11754:saturated
11732:of models
11655:deductive
11650:axiomatic
11570:Hilbert's
11557:Euclidean
11538:canonical
11461:axiomatic
11393:Signature
11322:Predicate
11211:Extension
11133:Ackermann
11058:Operation
10937:Universal
10927:Recursive
10902:Singleton
10897:Inhabited
10882:Countable
10872:Types of
10856:power set
10826:partition
10743:Predicate
10689:Predicate
10604:Syllogism
10594:Soundness
10567:Inference
10557:Tautology
10459:paradoxes
10316:Semantics
10293:CRC Press
10062:(1989) ,
9987:: 51–67,
9902:−
9817:−
9615:relations
9466:∀
9442:∀
9357:→
9330:×
9292:×
9247:∈
9180:∈
8752:φ
8749:⊨
8631:φ
8544:φ
8480:φ
8436:…
8417:φ
8414:⊨
8391:∈
8372:…
8311:…
8292:φ
8186:definable
8174:definable
8154:φ
8090:φ
8080:definable
7986:φ
7983:↔
7965:∀
7962:⊨
7926:φ
7823:…
7804:φ
7801:⊨
7791:⇔
7785:∈
7766:…
7730:φ
7668:…
7649:φ
7646:⊨
7623:∈
7604:…
7543:…
7524:φ
7503:∅
7491:definable
7477:∅
7459:definable
7153:ϕ
7132:ϕ
7129:⊨
7090:σ
6973:→
6882:subobject
6659:∈
6631:…
6580:⟺
6564:∈
6545:…
6477:…
6427:if it is
6425:embedding
6404:→
6317:…
6213:∈
6194:…
6128:∈
6112:…
6054:∈
6035:…
5969:∈
5953:…
5884:→
5858:morphisms
5703:→
5472:∈
5444:…
5393:⟹
5377:∈
5358:…
5286:∈
5270:…
5184:…
5109:…
5028:∈
5012:…
4937:→
4632:∈
4531:σ
4480:σ
4445:−
4439:×
4424:σ
4268:σ
4236:⊆
4204:σ
4120:⟩
4117:⟨
4089:⟩
4082:⟨
4062:⟩
4056:⟨
3996:generated
3908:⊆
3877:∈
3858:…
3774:…
3688:∈
3672:…
3467:⊆
3456:A subset
3432:⊆
3342:⊆
3237:σ
3218:σ
3088:∈
3065:∈
3034:≤
2978:σ
2947:σ
2808:∈
2737:∈
2688:−
2643:→
2629:−
2587:→
2579:×
2565:×
2523:→
2515:×
2332:−
2291:×
2194:−
2188:×
2108:σ
2057:σ
2000:σ
1943:σ
1893:σ
1771:×
1661:−
1639:×
1586:σ
1538:→
1472:→
1227:⊆
1152:of arity
1088:function
1045:of arity
851:
804:
740:→
651:σ
633:Signature
519:
453:σ
433:σ
387:σ
378:signature
327:σ
298:structure
223:databases
121:relations
109:structure
12044:Logicism
12037:timeline
12013:Concrete
11872:Validity
11842:T-schema
11835:Kripke's
11830:Tarski's
11825:semantic
11815:Strength
11764:submodel
11759:spectrum
11727:function
11575:Tarski's
11564:Elements
11551:geometry
11507:Robinson
11428:variable
11413:function
11386:spectrum
11376:Sentence
11332:variable
11275:Language
11228:Relation
11189:Automata
11179:Alphabet
11163:language
11017:-jection
10995:codomain
10981:Function
10942:Universe
10912:Infinite
10816:Relation
10599:Validity
10589:Argument
10487:theorem,
10259:New York
10253:(2010),
10180:(1997),
10151:(1993),
10001:15244028
9588:See also
8176:to mean
7214:T-schema
7021:database
5850:category
5847:concrete
4691:subgraph
4514:subfield
4412:Examples
4333:′
4278:′
2936:integers
1857:and the
1573:Examples
493:universe
489:universe
154:with no
130:such as
11986:Related
11783:Diagram
11681: (
11660:Hilbert
11645:Systems
11640:Theorem
11518:of the
11463:systems
11243:Formula
11238:Grammar
11154: (
11098:General
10811:Forcing
10796:Element
10716:Monadic
10491:paradox
10432:Theorem
10368:General
9503:variety
9481: (
9011:;
9007:,
8979:;
8962:;
8958:,
8918:;
8901:;
8897:,
8536:and so
6844:Example
4510:subring
4398:lattice
4136:on the
4021:or the
3746:-tuple
3726:to the
957:algebra
947:algebra
485:carrier
402:and an
233:History
103:and in
65:improve
11749:finite
11512:Skolem
11465:
11440:Theory
11408:Symbol
11398:String
11381:atomic
11258:ground
11253:closed
11248:atomic
11204:ground
11167:syntax
11063:binary
10990:domain
10907:Finite
10672:finite
10530:Logics
10489:
10437:Theory
10299:
10277:
10240:
10214:
10192:
10167:
10138:
10116:
10094:
10070:
10024:
9999:
9775:Note:
9635:
9458:
9454:
9418:fibred
8188:means
7249:theory
7111:has a
7053:models
7040:, and
6996:Every
5905:strong
5501:where
4400:. The
4321:where
3574:every
3522:closed
1606:fields
734:
685:a set
469:Domain
351:domain
140:fields
132:groups
54:, but
11739:Model
11487:Peano
11344:Proof
11184:Arity
11113:Naive
11000:image
10932:Fuzzy
10892:Empty
10841:union
10786:Class
10427:Model
10417:Lemma
10375:Axiom
9997:S2CID
9943:minus
9601:Notes
9408:. As
8931:and 1
8887:and ×
8852:sorts
8791:over
8767:then
7493:, or
7469:, or
7247:of a
7245:model
7019:of a
6880:is a
6832:of σ-
6820:in σ-
6809:of σ-
5634:from
4903:is a
4855:from
4521:graph
4132:is a
2158:with
918:arity
894:arity
274:model
194:model
136:rings
11862:Type
11665:list
11469:list
11446:list
11435:Term
11369:rank
11263:open
11157:list
10969:Maps
10874:sets
10733:Free
10703:list
10453:list
10380:list
10297:ISBN
10275:ISBN
10238:ISBN
10212:ISBN
10190:ISBN
10165:ISBN
10136:ISBN
10114:ISBN
10092:ISBN
10068:ISBN
10022:ISBN
9894:and
9808:and
9633:ISBN
9559:and
9523:type
8044:(or
7465:cf.
7461:(or
7047:for
6928:and
6431:and
6237:and
5907:if:
4827:and
4669:and
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4416:Let
4406:join
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4225:and
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3186:and
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2880:and
2851:and
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2221:and
1832:the
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1630:and
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978:The
831:The
711:and
142:and
119:and
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11549:of
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8801:M
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5690:h
5668:B
5644:A
5632:h
5613:B
5589:A
5567:R
5544:B
5539:R
5515:A
5510:R
5481:B
5476:R
5469:)
5466:)
5461:n
5457:a
5453:(
5450:h
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5441:,
5438:)
5433:2
5429:a
5425:(
5422:h
5419:,
5416:)
5411:1
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5400:h
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5386:A
5381:R
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5369:n
5365:a
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5350:2
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5329:(
5302:|
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5281:n
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5004:2
5000:a
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4991:1
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4972:n
4953:|
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4941:|
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4927:A
4921:|
4917::
4914:h
4889:B
4865:A
4837:B
4813:A
4764:,
4761:G
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4721:H
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4239:A
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4195:(
4192:=
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4059:B
4036:,
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3924:|
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3807::
3804:B
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3602:f
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2018:R
2013:I
2009:,
2004:f
1996:,
1992:R
1988:(
1985:=
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1967:)
1961:Q
1956:I
1952:,
1947:f
1939:,
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1928:=
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