43:
10262:
7921:
6473:, they proved this subtheory consistent, and proved that each of the axioms of extensionality, replacement, and power set is independent of the four remaining axioms of this subtheory. If this subtheory is augmented with the axiom of infinity, each of the axioms of union, choice, and infinity is independent of the five remaining axioms. Because there are non-well-founded models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms.
6626:, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice. (Thus every model of ZF contains a submodel of ZFC, so that Con(ZF) implies Con(ZFC).) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C.
6647:" set theorists argue that usefulness should be the sole ultimate criterion in which axioms to customarily adopt. One school of thought leans on expanding the "iterative" concept of a set to produce a set-theoretic universe with an interesting and complex but reasonably tractable structure by adopting forcing axioms; another school advocates for a tidier, less cluttered universe, perhaps focused on a "core" inner model.
6535:) can be expanded to satisfy the statement in question. A different expansion is then shown to satisfy the negation of the statement. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular
6218:. At stage 0, there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2. The collection of all sets that are obtained in this way, over all the stages, is known as
6629:
Another method of proving independence results, one owing nothing to forcing, is based on Gödel's second incompleteness theorem. This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies
1515:
All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, although he notes that he does so only "for emphasis". Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC
6642:
The project to unify set theorists behind additional axioms to resolve the continuum hypothesis or other meta-mathematical ambiguities is sometimes known as "Gödel's program". Mathematicians currently debate which axioms are the most plausible or "self-evident", which axioms are the most useful in
3671:
The axiom schema of specification must be used to reduce this to a set with exactly these two elements. The axiom of pairing is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement if we are given a set with at least two elements. The existence of a set with at
6443:, which is unprovable in ZFC if ZFC is consistent. Nevertheless, it is deemed unlikely that ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC were inconsistent, that fact would have been uncovered by now. This much is certain — ZFC is immune to the classic paradoxes of
6438:
of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weakly
606:
6369:. This provides a simple notation for classes that can contain sets but need not themselves be sets, while not committing to the ontology of classes (because the notation can be syntactically converted to one that only uses sets). Quine's approach built on the earlier approach of
5195:
having infinitely many members. (It must be established, however, that these members are all different because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.) The minimal set
6344:
Proper classes (collections of mathematical objects defined by a property shared by their members which are too big to be sets) can only be treated indirectly in ZF (and thus ZFC). An alternative to proper classes while staying within ZF and ZFC is the
1558:. Consequently, it is a theorem of every first-order theory that something exists. However, as noted above, because in the intended semantics of ZFC, there are only sets, the interpretation of this logical theorem in the context of ZFC is that some
2267:
6630:
the conditions of Gödel's second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statement allowing such a proof can be proved in ZFC. This method can prove that the existence of
6688:
have both made this point. Some of "mainstream mathematics" (mathematics not directly connected with axiomatic set theory) is beyond Peano arithmetic and second-order arithmetic, but still, all such mathematics can be carried out in ZC
4135:
2892:
865:
There are different ways to formulate the formal language. Some authors may choose a different set of connectives or quantifiers. For example, the logical connective NAND alone can encode the other connectives, a property known as
5409:
4490:
5538:
1679:. ZFC is constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If the variety of first-order logic in which you are constructing set theory does not include equality "
6295:
so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more "narrow" hierarchy, which gives the
3444:
6693:
with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.
3969:
446:
3667:
5988:
192:
defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a
2429:
6764:, a strong supposition incompatible with choice. One attraction of large cardinal axioms is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom. The
5684:
3028:
212:
There are many equivalent formulations of the axioms of
Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, the
4258:
721:. The equality symbol can be treated as either a primitive logical symbol or a high-level abbreviation for having exactly the same elements. The former approach is the most common. The
1812:
2659:
2552:
2362:
328:. Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the
4025:
3998:
3756:
1923:
1671:
2940:
1277:
5576:
2990:
1310:
441:
5187:
3867:
3823:
2188:
1556:
1364:
1337:
657:
5233:
5010:
4878:
3494:
3322:
3135:
2316:
3204:
2480:
1244:
3797:
2181:
6147:
4515:
1520:
must be nonempty. Hence, it is a logical theorem of first-order logic that something exists — usually expressed as the assertion that something is identical to itself,
1473:
1450:
6661:
ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the
4538:
4298:
4183:
3342:
2699:
2601:
2581:
984:
964:
6403:
included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand,
4640:
286:
6112:
3157:
stands in a separate position from which it can't refer to or comprehend itself; therefore, in a certain sense, this axiom schema is saying that in order to build a
3103:
1427:
1387:
1153:
920:
900:
776:
4986:
4956:
4924:
1407:
1197:
1177:
940:
633:
5051:
4843:
4611:
1032:
743:
1723:
1126:
6174:
6082:
6062:
6042:
6022:
5933:
5909:
5885:
5861:
5841:
5818:
5795:
5738:
5718:
5616:
5596:
5471:
5451:
5431:
5330:
5310:
5290:
5266:
4898:
4704:
4684:
4664:
4578:
4558:
4278:
4052:
4045:
3843:
3588:
3568:
3548:
3528:
3470:
3362:
3284:
3264:
3244:
3224:
3175:
3155:
3076:
3052:
2706:
2679:
2500:
2451:
2059:
2039:
2019:
1999:
1965:
1945:
1697:
1217:
1088:
1068:
1008:
859:
839:
817:
797:
326:
306:
254:
234:
663:
operation. Moreover, one of
Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Fraenkel and
5337:
8641:
4305:
5478:
6276:
The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as
2896:
Note that the axiom schema of specification can only construct subsets and does not allow the construction of entities of the more general form:
9316:
8378:
6818:
6714:
6404:
6394:
6277:
601:{\displaystyle \{Z_{0},{\mathcal {P}}(Z_{0}),{\mathcal {P}}({\mathcal {P}}(Z_{0})),{\mathcal {P}}({\mathcal {P}}({\mathcal {P}}(Z_{0}))),...\},}
202:
3369:
3874:
7378:
9399:
8540:
7643:
3595:
1562:
exists. Hence, there is no need for a separate axiom asserting that a set exists. Second, however, even if ZFC is formulated in so-called
6411:
as well as sets; a set is any class that can be a member of another class. NBG and ZFC are equivalent set theories in the sense that any
6423:
352:
2271:
This (along with the axioms of pairing and union) implies, for example, that no set is an element of itself and that every set has an
5940:
4707:
9713:
2369:
6249:
satisfies all the axioms of ZFC if the class of ordinals has appropriate reflection properties. For example, suppose that a set
5623:
7624:
3452:
is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique (does not depend on
6543:. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.
6192:
because it asserts the existence of a choice function but says nothing about how this choice function is to be "constructed".
9871:
7688:
7508:
7417:
7303:
7237:
7158:
6737:
6520:
8659:
667:
independently proposed operationalizing a "definite" property as one that could be formulated as a well-formed formula in a
9726:
9049:
8067:
7887:
6303:, which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whether
10286:
7813:
7798:
7774:
9731:
9721:
9458:
9311:
8664:
8395:
6823:
6773:
6485:
8655:
9867:
7559:
7482:
7452:
7355:
7325:
7190:
7120:
2995:
340:
86:
64:
9209:
57:
9964:
9708:
8533:
7217:
7138:
6600:
7447:(Revised ed.). Cambridge, Massachusetts and London, England: The Belknap Press of Harvard University Press.
7398:
8703:
8253:
4196:
10291:
10225:
9927:
9690:
9685:
9510:
8931:
8615:
6611:
1728:
722:
6430:
can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted in
10220:
10003:
9920:
9633:
9564:
9441:
8683:
8147:
8026:
7767:
2606:
2507:
2287:
714:
676:
6634:
is not provable in ZFC, but cannot prove that assuming such cardinals, given ZFC, is free of contradiction.
1566:, in which it is not provable from logic alone that something exists, the axiom of infinity asserts that an
10145:
9971:
9657:
9291:
8890:
8390:
5700:. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set
1475:
have exactly one. There are countably infinitely many wff, however, each wff has a finite number of nodes.
339:
of
Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the
2325:
10023:
10018:
9628:
9367:
9296:
8625:
8526:
8383:
8021:
7984:
7762:
6813:
6718:
6281:
6226:
can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to
4145:
4003:
3976:
3705:
3290:
1821:
1593:
680:
2901:
1369:
A well-formed formula can be thought as a syntax tree. The leaf nodes are always atomic formulae. Nodes
1250:
9952:
9542:
8936:
8904:
8595:
7751:
7615:
6789:
5548:
3031:
2951:
2262:{\displaystyle \forall x\,(x\neq \varnothing \Rightarrow \exists y(y\in x\land y\cap x=\varnothing )).}
1283:
419:
198:
136:
8038:
6269:
which the axiom of separation can construct is added at (or before) stage α, and that the powerset of
150:
Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a
10242:
10191:
10088:
9586:
9547:
9024:
8669:
8072:
7964:
7952:
7947:
6550:
6328:
5062:
4149:
1489:
There are many equivalent formulations of the ZFC axioms. The following particular axiom set is from
416:
whose existence was taken for granted by most set theorists of the time, notably the cardinal number
8698:
3848:
3804:
1523:
1343:
1316:
638:
412:
in a 1921 letter to
Zermelo, this theory was incapable of proving the existence of certain sets and
10083:
10013:
9552:
9404:
9387:
9110:
8590:
7880:
7265:
6643:
various domains, and about to what degree usefulness should be traded off with plausibility; some "
6469:
of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice. Using
6207:
6189:
5213:
2281:
51:
7369:
6855:, p. 4: "Zermelo-Fraenkel axioms (abbreviated as ZFC where C stands for the axiom of Choice)"
4995:
4848:
3479:
3307:
3114:
2299:
9915:
9892:
9853:
9739:
9680:
9326:
9246:
9090:
9034:
8647:
8492:
8410:
8285:
8237:
8051:
7974:
7707:
7578:
7146:
6828:
6673:
6644:
6589:
The diamond principle implies the continuum hypothesis and the negation of the Suslin hypothesis.
3180:
2456:
1583:
1226:
867:
382:
5692:
define ZF. Alternative forms of these axioms are often encountered, some of which are listed in
3761:
2070:
10205:
9932:
9910:
9877:
9770:
9616:
9601:
9574:
9525:
9409:
9344:
9169:
9135:
9130:
9004:
8835:
8812:
8444:
8325:
8137:
7957:
7851:
7182:
Gödel '96: Logical foundations of mathematics, computer science and physics–Kurt Gödel's legacy
6806:
6749:
6730:
6706:
6698:
6656:
6596:
6540:
6297:
6214:. In this viewpoint, the universe of set theory is built up in stages, with one stage for each
6117:
4497:
4159:
3681:
3473:
3449:
3300:
On the other hand, the axiom schema of specification can be used to prove the existence of the
3294:
1676:
1455:
1432:
206:
68:
4523:
4283:
4168:
3327:
2684:
2586:
2566:
969:
949:
745:, which is a predicate symbol of arity 2 (a binary relation symbol). This symbol symbolizes a
10135:
9988:
9780:
9498:
9234:
9140:
8999:
8984:
8865:
8840:
8360:
8330:
8274:
8194:
8174:
8152:
7745:
7648:
7227:
6777:
6619:
6524:
6477:
6440:
5756:
4616:
1505:
393:, led to the desire for a more rigorous form of set theory that was free of these paradoxes.
364:
259:
165:
7441:
6091:
3082:
1412:
1372:
1132:
905:
885:
755:
675:
were limited to set membership and identity. They also independently proposed replacing the
10108:
10070:
9947:
9751:
9591:
9515:
9493:
9321:
9279:
9178:
9145:
9009:
8797:
8708:
8434:
8424:
8258:
8189:
8142:
8082:
7969:
7683:. Source Books in the History of the Sciences. Harvard University Press. pp. 199–215.
7492:
7462:
6799:
6761:
6757:
6741:
6592:
Martin's axiom plus the negation of the continuum hypothesis implies the Suslin hypothesis.
6556:
6532:
6517:
6452:
6448:
6201:
4965:
4929:
4903:
4581:
2945:
2293:
1392:
1182:
1162:
1043:
925:
611:
401:
390:
344:
329:
132:
124:
31:
5027:
4819:
4587:
4130:{\displaystyle \cup {\mathcal {F}}=\{x\in A:\exists Y(x\in Y\land Y\in {\mathcal {F}})\}.}
2887:{\displaystyle \forall z\forall w_{1}\forall w_{2}\ldots \forall w_{n}\exists y\forall x.}
1017:
728:
8:
10296:
10237:
10128:
10113:
10093:
10050:
9937:
9887:
9813:
9758:
9695:
9488:
9483:
9431:
9199:
9188:
8860:
8760:
8688:
8679:
8675:
8610:
8605:
8429:
8340:
8248:
8243:
8057:
7999:
7937:
7873:
7757:
7676:
7205:
6833:
6677:
6572:
6456:
6427:
6399:
The axiom schemata of replacement and separation each contain infinitely many instances.
5201:
4643:
4186:
4155:
3289:
In some other axiomatizations of ZF, this axiom is redundant in that it follows from the
2319:
1976:
1702:
1574:
set exists, and so, once again, it is superfluous to include an axiom asserting as much.
1517:
1105:
750:
684:
189:
17:
7808:
10266:
10035:
9998:
9983:
9976:
9959:
9763:
9745:
9611:
9537:
9520:
9473:
9286:
9195:
9029:
9014:
8974:
8926:
8911:
8899:
8855:
8830:
8600:
8549:
8352:
8347:
8132:
8087:
7994:
7665:
7471:
7340:
7098:
6690:
6493:
6431:
6408:
6159:
6067:
6047:
6027:
6007:
5918:
5894:
5870:
5846:
5826:
5803:
5780:
5723:
5703:
5697:
5601:
5581:
5456:
5436:
5416:
5315:
5295:
5275:
5251:
5243:
4883:
4689:
4669:
4649:
4563:
4543:
4263:
4030:
3828:
3702:
over the elements of a set exists. For example, the union over the elements of the set
3699:
3573:
3553:
3533:
3513:
3455:
3347:
3269:
3249:
3229:
3209:
3160:
3140:
3061:
3037:
2664:
2485:
2436:
2044:
2024:
2004:
1984:
1950:
1930:
1682:
1202:
1073:
1053:
993:
844:
824:
802:
782:
405:
311:
291:
239:
219:
147:
refers to the axioms of
Zermelo–Fraenkel set theory with the axiom of choice excluded.
9219:
4517:
denotes the existence of exactly one element such that it follows a given statement.)
10261:
10201:
10008:
9818:
9808:
9700:
9581:
9416:
9392:
9173:
9157:
9062:
9039:
8916:
8885:
8850:
8745:
8580:
8209:
8046:
8009:
7979:
7910:
7848:
7827:
7726:
7684:
7669:
7555:
7548:
7504:
7478:
7448:
7413:
7394:
7351:
7321:
7299:
7233:
7186:
7154:
7116:
7058:
6753:
6745:
6681:
6566:
6489:
6435:
4719:
3673:
2275:
1494:
718:
668:
209:
of
Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.
157:
7823:
5404:{\displaystyle (z\subseteq x)\Leftrightarrow (\forall q(q\in z\Rightarrow q\in x)).}
10215:
10210:
10103:
10060:
9882:
9843:
9838:
9823:
9649:
9606:
9503:
9301:
9251:
8825:
8787:
8497:
8487:
8472:
8467:
8335:
7989:
7716:
7657:
7587:
7427:
7275:
7201:
7168:
7082:
6685:
6669:
6444:
6242:
6211:
3505:
3324:, once at least one set is known to exist. One way to do this is to use a property
688:
409:
386:
378:
213:
116:
112:
7180:
7172:
6668:
Many mathematical theorems can be proven in much weaker systems than ZFC, such as
2556:
The axiom schema of specification states that this subset always exists (it is an
2296:. For example, the even integers can be constructed as the subset of the integers
870:. This section attempts to strike a balance between simplicity and intuitiveness.
10196:
10186:
10140:
10123:
10078:
10040:
9942:
9862:
9669:
9596:
9569:
9557:
9463:
9377:
9351:
9306:
9274:
9075:
8877:
8820:
8770:
8735:
8693:
8366:
8304:
8122:
7942:
7793:
7789:
7629:
7538:
7110:
7053:
6726:
6710:
6702:
6631:
6623:
6586:
but not forcing: every model of ZF can be trimmed to become a model of ZFC + V=L.
6497:
6481:
6285:
6150:
6085:
5798:
5766:
5760:
5752:
5208:
3590:
as elements, for example if x = {1,2} and y = {2,3} then z will be {{1,2},{2,3}}
1509:
708:
696:
413:
336:
128:
4485:{\displaystyle \forall A\forall w_{1}\forall w_{2}\ldots \forall w_{n}{\bigl }.}
3286:
can't refer to itself; or, in other words, sets shouldn't refer to themselves).
2282:
Axiom schema of specification (or of separation, or of restricted comprehension)
10181:
10160:
10118:
10098:
9993:
9848:
9446:
9436:
9426:
9421:
9355:
9229:
9105:
8994:
8989:
8967:
8568:
8502:
8299:
8280:
8184:
8169:
8126:
8062:
8004:
7841:
of a version of the separation schema from a version of the replacement schema.
7518:
7252:
7087:
7070:
6215:
3693:
2272:
1095:
746:
672:
664:
351:
of a theory such as ZFC cannot be proved within the theory itself, as shown by
151:
139:. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated
120:
7209:
6415:
not mentioning classes and provable in one theory can be proved in the other.
5720:
whose existence is being asserted are just those sets which the axiom asserts
10280:
10155:
9833:
9340:
9125:
9115:
9085:
9070:
8740:
8507:
8309:
8223:
8218:
7730:
7698:
7639:
7569:
7530:
7365:
7335:
6752:. Some of these conjectures are provable with the addition of axioms such as
6662:
6284:. The cumulative hierarchy is not compatible with other set theories such as
5912:
4190:
2062:
397:
343:
of the axiom of choice from the remaining
Zermelo-Fraenkel axioms and of the
194:
108:
8477:
7592:
7573:
7176:
6261:
is also added at (or before) stage α, because all elements of any subset of
5533:{\displaystyle \forall x\exists y\forall z(z\subseteq x\Rightarrow z\in y).}
1970:
10055:
9902:
9803:
9795:
9675:
9623:
9532:
9468:
9451:
9382:
9241:
9100:
8802:
8585:
8457:
8452:
8270:
8199:
8157:
8016:
7920:
7834:
7785:
7778:
7130:
6780:(encountered in category theory and algebraic geometry) can be formalized.
6765:
6722:
6528:
6470:
6185:
5864:
2557:
1091:
374:
185:
177:
154:
127:. Today, Zermelo–Fraenkel set theory, with the historically controversial
7721:
7702:
10165:
10045:
9224:
9214:
9161:
8845:
8765:
8750:
8630:
8575:
8482:
8117:
7466:
7313:
6794:
6622:
can also be used to demonstrate the consistency and unprovability of the
6607:
6583:
6561:
6536:
3439:{\displaystyle \varnothing =\{u\in w\mid (u\in u)\land \lnot (u\in u)\}.}
348:
119:
that was proposed in the early twentieth century in order to formulate a
6484:
requires. Huge sets of this nature are possible if ZF is augmented with
3964:{\displaystyle \forall {\mathcal {F}}\,\exists A\,\forall Y\,\forall x.}
161:
9095:
8950:
8921:
8727:
8462:
8233:
7896:
7661:
7500:
7295:
6709:
of sets under ZFC is not closed under the elementary operations of the
6177:
5821:
5770:
1563:
370:
100:
6024:
be a set whose members are all nonempty. Then there exists a function
1588:
Two sets are equal (are the same set) if they have the same elements.
1219:
be a metavariable for any variable. These are valid wff constructions:
10247:
10150:
9203:
9120:
9080:
9044:
8980:
8792:
8782:
8755:
8518:
8265:
8228:
8179:
8077:
7856:
6546:
Forcing proves that the following statements are independent of ZFC:
6466:
5543:
3301:
1675:
The converse of this axiom follows from the substitution property of
660:
181:
7846:
7830:
is defined especially to facilitate machine verification of proofs.
7071:"On the Consistency and Independence of Some Set-Theoretical Axioms"
3662:{\displaystyle \forall x\forall y\exists z((x\in z)\land (y\in z)).}
2603:
be any formula in the language of ZFC with all free variables among
10232:
10030:
9478:
9183:
8777:
7838:
7681:
From Frege to Gödel: A Source Book in
Mathematical Logic, 1879–1931
7347:
7270:
Includes annotated
English translations of the classic articles by
7262:
From Frege to Gödel: A Source Book in
Mathematical Logic, 1879–1931
6769:
6382:
6238:
1500:
Axioms 1–8 form ZF, while the axiom 9 turns ZF into ZFC. Following
877:
A countably infinite amount of variables used for representing sets
173:
6603:, the diamond principle, Martin's axiom and the Kurepa hypothesis.
9828:
8620:
7271:
6929:
6705:, ZFC does not admit the existence of a universal set. Hence the
6412:
1816:
In this case, the axiom of extensionality can be reformulated as
6426:
says that a recursively axiomatizable system that can interpret
6407:(NBG) can be finitely axiomatized. The ontology of NBG includes
8290:
8112:
7279:
7229:
Combinatorial Set Theory: With a Gentle Introduction to Forcing
5983:{\displaystyle \forall X\exists R(R\;{\mbox{well-orders}}\;X).}
5888:
5269:
4706:
may be larger than strictly necessary, is sometimes called the
3973:
Although this formula doesn't directly assert the existence of
7430:(1961). "Semantical closure and non-finite axiomatizability".
7318:
Set Theory: The Third Millennium Edition, Revised and Expanded
6265:
were also added before stage α. This means that any subset of
5696:. Some ZF axiomatizations include an axiom asserting that the
184:(elements of sets that are not themselves sets). Furthermore,
9372:
8718:
8563:
8162:
7929:
7865:
7410:
Formalism and Beyond: On the Nature of Mathematical Discourse
5542:
The axiom schema of specification is then used to define the
1725:
may be defined as an abbreviation for the following formula:
1484:
169:
7826:— A concise and nonredundant axiomatization. The background
3845:
containing every element that is a member of some member of
7679:(1967). "Investigations in the foundations of set theory".
7173:"Gödel's program for new axioms: why, where, how and what?"
6257:
was added at a stage earlier than α. Then, every subset of
2424:{\displaystyle \{x\in \mathbb {Z} :x\equiv 0{\pmod {2}}\}.}
1967:
have the same elements, then they belong to the same sets.
6760:
to ZFC. Some others are decided in ZF+AD where AD is the
6195:
5679:{\displaystyle {\mathcal {P}}(x)=\{z\in y:z\subseteq x\}.}
1493:. The axioms in order below are expressed in a mixture of
6882:
3364:
is any existing set, the empty set can be constructed as
1971:
Axiom of regularity (also called the axiom of foundation)
699:(AC) or a statement that is equivalent to it yields ZFC.
6941:
6870:
3030:) and its variants that accompany naive set theory with
6253:
is added at stage α, which means that every element of
7644:"Untersuchungen über die Grundlagen der Mengenlehre I"
7200:
6935:
6900:
6527:, whereby it is shown that every countable transitive
5964:
4047:
in the above using the axiom schema of specification:
691:), to Zermelo set theory yields the theory denoted by
7029:
6965:
6476:
If consistent, ZFC cannot prove the existence of the
6162:
6120:
6094:
6070:
6050:
6030:
6010:
5943:
5921:
5897:
5873:
5849:
5829:
5806:
5783:
5726:
5706:
5626:
5604:
5584:
5551:
5481:
5459:
5439:
5419:
5340:
5318:
5298:
5278:
5254:
5216:
5065:
5030:
4998:
4968:
4932:
4926:
is a valid set by applying the axiom of pairing with
4906:
4886:
4851:
4822:
4692:
4672:
4652:
4619:
4590:
4566:
4546:
4526:
4500:
4308:
4286:
4266:
4199:
4171:
4055:
4033:
4006:
3979:
3877:
3851:
3831:
3807:
3764:
3708:
3598:
3576:
3556:
3536:
3516:
3482:
3458:
3372:
3350:
3330:
3310:
3272:
3252:
3232:
3212:
3183:
3163:
3143:
3117:
3085:
3064:
3040:
2998:
2954:
2904:
2709:
2687:
2667:
2609:
2589:
2569:
2510:
2488:
2459:
2439:
2372:
2328:
2302:
2191:
2073:
2047:
2027:
2007:
1987:
1953:
1933:
1824:
1731:
1705:
1685:
1596:
1526:
1458:
1435:
1415:
1395:
1375:
1346:
1319:
1286:
1253:
1229:
1205:
1185:
1165:
1135:
1108:
1076:
1056:
1020:
996:
972:
952:
928:
908:
888:
847:
827:
805:
785:
758:
731:
641:
614:
449:
422:
314:
294:
262:
242:
222:
7017:
6725:. A further comparative weakness of ZFC is that the
6149:. A third version of the axiom, also equivalent, is
1042:
With this alphabet, the recursive rules for forming
6315:
is more regular and well behaved than that of
6000:, many statements are provably equivalent to axiom
3801:The axiom of union states that for any set of sets
1094:for any variables. These are the two ways to build
7547:
7470:
7440:
7342:Set Theory: An Introduction to Independence Proofs
7339:
7097:
7057:
7005:
6168:
6141:
6106:
6076:
6056:
6036:
6016:
5982:
5927:
5903:
5879:
5855:
5835:
5812:
5789:
5746:
5743:The following axiom is added to turn ZF into ZFC:
5732:
5712:
5678:
5610:
5590:
5570:
5532:
5465:
5445:
5425:
5403:
5324:
5304:
5284:
5260:
5227:
5181:
5045:
5004:
4980:
4950:
4918:
4892:
4872:
4837:
4698:
4678:
4658:
4634:
4605:
4572:
4552:
4532:
4509:
4484:
4292:
4272:
4252:
4177:
4129:
4039:
4019:
3992:
3963:
3861:
3837:
3817:
3791:
3750:
3661:
3582:
3562:
3542:
3522:
3488:
3464:
3438:
3356:
3336:
3316:
3278:
3258:
3238:
3218:
3198:
3169:
3149:
3129:
3097:
3070:
3046:
3022:
2984:
2934:
2886:
2693:
2673:
2653:
2595:
2575:
2546:
2494:
2474:
2445:
2423:
2356:
2310:
2261:
2175:
2053:
2033:
2013:
1993:
1959:
1939:
1917:
1806:
1717:
1691:
1665:
1550:
1467:
1444:
1421:
1401:
1381:
1358:
1331:
1304:
1271:
1238:
1211:
1191:
1171:
1147:
1120:
1082:
1062:
1026:
1002:
978:
958:
934:
914:
894:
853:
833:
811:
791:
770:
737:
651:
627:
600:
435:
320:
300:
280:
248:
228:
7747:Axioms of set Theory - Lec 02 - Frederic Schuller
7069:———; LaMacchia, Samuel (1978).
4154:The axiom schema of replacement asserts that the
3550:are sets, then there exists a set which contains
10278:
7625:"To Settle Infinity Dispute, a New Law of Logic"
6776:, an extension of ZFC, so that proofs involving
6004:. The most common of these goes as follows. Let
7151:Ernst Zermelo: An Approach to His Life and Work
7095:
6370:
5413:The Axiom of power set states that for any set
3023:{\displaystyle y\in y\Leftrightarrow y\notin y}
725:has a single predicate symbol, usually denoted
7259:
7068:
6462:
6156:Since the existence of a choice function when
4139:
8534:
7881:
7545:
7529:
7060:The Theory of Sets and Transfinite Arithmetic
6378:
4474:
4467:
4421:
4359:
172:of Zermelo–Fraenkel set theory refer only to
5670:
5646:
4975:
4969:
4913:
4907:
4864:
4858:
4121:
4069:
3783:
3765:
3745:
3742:
3730:
3724:
3712:
3709:
3672:least two elements is assured by either the
3430:
3379:
2979:
2961:
2926:
2905:
2538:
2511:
2415:
2373:
592:
450:
275:
263:
6291:It is possible to change the definition of
5207:which can also be thought of as the set of
8726:
8541:
8527:
7888:
7874:
7675:
7491:
7461:
7145:
7108:
6988:
6959:
6947:
6876:
6852:
6738:mathematical statements independent of ZFC
6721:(MK), ZFC does not admit the existence of
6610:is equiconsistent with the existence of a
6488:. Assuming that axiom turns the axioms of
5970:
5962:
4253:{\displaystyle x,y,A,w_{1},\dotsc ,w_{n},}
3206:, we need to previously restrict the sets
1577:
1409:have exactly two child nodes, while nodes
332:(also known as the cumulative hierarchy).
30:"ZFC" redirects here. For other uses, see
7720:
7622:
7609:
7591:
7434:. London: Pergamon Press. pp. 45–69.
7225:
7086:
7035:
6888:
6327:should be added to ZFC as an additional "
6273:will be added at the next stage after α.
5218:
5092:
4394:
3902:
3895:
3888:
2383:
2304:
2198:
1807:{\displaystyle \forall z\land \forall w.}
87:Learn how and when to remove this message
7612:Lectures in Logic and Set Theory, Vol. 2
7426:
7167:
7112:Set Theory for the Working Mathematician
7023:
6523:. The independence is usually proved by
6400:
6388:
5769:, is presented here as a property about
5200:satisfying the axiom of infinity is the
5012:, defined axiomatically, is a member of
683:. Appending this schema, as well as the
381:in the 1870s. However, the discovery of
50:This article includes a list of general
7809:"Axioms of Zermelo–Fraenkel Set Theory"
7806:
7784:
7697:
7638:
7246:
6923:
6196:Motivation via the cumulative hierarchy
2944:This restriction is necessary to avoid
2654:{\displaystyle x,z,w_{1},\ldots ,w_{n}}
2547:{\displaystyle \{x\in z:\varphi (x)\}.}
2292:Subsets are commonly constructed using
131:(AC) included, is the standard form of
27:Standard system of axiomatic set theory
14:
10279:
8548:
7568:
7517:
7412:. Walter de Gruyter GmbH & Co KG.
7384:from the original on 7 September 2023.
7289:
7249:The Logical Foundations of Mathematics
7129:
7096:Bernays, Paul; Fraenkel, A.A. (1958).
7011:
6971:
6582:The consistency of V=L is provable by
5765:The last axiom, commonly known as the
5191:More colloquially, there exists a set
201:, thereby avoiding Russell's paradox.
8522:
7869:
7847:
7703:"Über Grenzzahlen und Mengenbereiche"
7600:
7438:
7364:
7334:
7052:
6911:
6864:
6637:
6434:, a small fragment of ZFC. Hence the
6424:Gödel's second incompleteness theorem
6350:
6319:, few mathematicians argue that
6206:One motivation for the ZFC axioms is
5774:
5237:
3079:that don't belong to themselves, and
1501:
1490:
873:The language's alphabet consists of:
353:Gödel's second incompleteness theorem
7574:"On well-ordered subsets of any set"
7550:Introduction to Axiomatic Set Theory
7546:Takeuti, Gaisi; Zaring, W M (1982).
7535:Introduction to Axiomatic Set Theory
7477:. North-Holland Publishing Company.
7407:
7388:
7312:
7222:Fraenkel's final word on ZF and ZFC.
7000:
6936:Fraenkel, Bar-Hillel & Lévy 1973
6901:Fraenkel, Bar-Hillel & Lévy 1973
6819:Von Neumann–Bernays–Gödel set theory
6715:von Neumann–Bernays–Gödel set theory
6701:, ZFC is comparatively weak. Unlike
6405:von Neumann–Bernays–Gödel set theory
6395:Von Neumann–Bernays–Gödel set theory
6374:
6373:. Virtual classes are also used in
6278:Von Neumann–Bernays–Gödel set theory
5693:
4713:
2563:because there is one axiom for each
2357:{\displaystyle x\equiv 0{\pmod {2}}}
203:Von Neumann–Bernays–Gödel set theory
197:(a set containing all sets) nor for
36:
7814:Stanford Encyclopedia of Philosophy
7799:Stanford Encyclopedia of Philosophy
7775:Stanford Encyclopedia of Philosophy
7465:(1977). "Axioms of set theory". In
6729:included in ZFC is weaker than the
6349:notational construct introduced by
4725:First several von Neumann ordinals
4494:(The unique existential quantifier
4020:{\displaystyle \cup {\mathcal {F}}}
3993:{\displaystyle \cup {\mathcal {F}}}
3751:{\displaystyle \{\{1,2\},\{2,3\}\}}
3499:
2407:
2346:
1918:{\displaystyle \forall x\forall y,}
1666:{\displaystyle \forall x\forall y.}
408:. However, as first pointed out by
143:, where C stands for "choice", and
24:
7824:Metamath version of the ZFC axioms
7292:Fundamentals of Mathematical Logic
7185:. Springer-Verlag. pp. 3–22.
7075:Notre Dame Journal of Formal Logic
6339:
6334:
5950:
5944:
5629:
5554:
5494:
5488:
5482:
5362:
5129:
5093:
5086:
5077:
5066:
4501:
4438:
4413:
4404:
4385:
4364:
4344:
4328:
4315:
4309:
4113:
4084:
4061:
4012:
3985:
3935:
3903:
3896:
3889:
3883:
3878:
3854:
3810:
3611:
3605:
3599:
3412:
3344:which no set has. For example, if
2935:{\displaystyle \{x:\varphi (x)\}.}
2764:
2758:
2745:
2729:
2716:
2710:
2214:
2192:
2131:
2128:
2107:
2083:
2074:
1876:
1840:
1831:
1825:
1768:
1732:
1612:
1603:
1597:
1527:
1459:
1436:
1416:
1347:
1320:
1272:{\displaystyle (\phi \land \psi )}
1230:
1199:be metavariables for any wff, and
973:
953:
889:
702:
644:
553:
543:
533:
504:
494:
468:
424:
56:it lacks sufficient corresponding
25:
10308:
7739:
7439:Quine, Willard van Orman (1969).
6531:of ZFC (sometimes augmented with
5571:{\displaystyle {\mathcal {P}}(x)}
4686:. The form stated here, in which
3687:
2985:{\displaystyle y=\{x:x\notin x\}}
2247:
2208:
1305:{\displaystyle (\phi \lor \psi )}
436:{\displaystyle \aleph _{\omega }}
10260:
7919:
6601:generalized continuum hypothesis
6233:It is provable that a set is in
5292:if and only if every element of
4810:{∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}
4540:represents a definable function
4520:In other words, if the relation
2433:In general, the subset of a set
41:
7045:
6994:
6977:
6953:
6676:(as explored by the program of
6553:(which is also not a ZFC axiom)
6551:Axiom of constructibility (V=L)
6511:
6064:to the union of the members of
5747:Axiom of well-ordering (choice)
5182:{\displaystyle \exists X\left.}
3111:been established, even though
2400:
2339:
135:and as such is the most common
7895:
7473:Handbook of Mathematical Logic
7371:The Foundations of Mathematics
7115:. Cambridge University Press.
7109:Ciesielski, Krzysztof (1997).
6917:
6905:
6894:
6858:
6846:
6824:Tarski–Grothendieck set theory
6774:Tarski–Grothendieck set theory
6612:strongly inaccessible cardinal
6516:Many important statements are
6418:
6184:, AC only matters for certain
6130:
6124:
5974:
5956:
5640:
5634:
5565:
5559:
5524:
5512:
5500:
5453:that contains every subset of
5395:
5392:
5380:
5368:
5359:
5356:
5353:
5341:
5168:
5159:
5153:
5147:
5135:
5123:
5108:
5096:
5083:
5040:
5034:
4900:is some set. (We can see that
4832:
4826:
4600:
4594:
4462:
4444:
4435:
4401:
4398:
4382:
4370:
4118:
4090:
3955:
3943:
3940:
3912:
3909:
3862:{\displaystyle {\mathcal {F}}}
3818:{\displaystyle {\mathcal {F}}}
3653:
3650:
3638:
3632:
3620:
3617:
3427:
3415:
3406:
3394:
3193:
3187:
3034:(since under this restriction
3008:
2923:
2917:
2878:
2875:
2872:
2809:
2800:
2788:
2785:
2782:
2770:
2535:
2529:
2469:
2463:
2411:
2401:
2350:
2340:
2253:
2250:
2220:
2211:
2199:
2167:
2164:
2161:
2137:
2113:
2104:
2101:
2089:
2080:
1909:
1906:
1894:
1882:
1873:
1870:
1858:
1846:
1837:
1798:
1786:
1774:
1762:
1750:
1738:
1657:
1645:
1642:
1630:
1618:
1609:
1570:set exists. This implies that
1551:{\displaystyle \exists x(x=x)}
1545:
1533:
1497:and high-level abbreviations.
1359:{\displaystyle \exists x\phi }
1332:{\displaystyle \forall x\phi }
1299:
1287:
1266:
1254:
652:{\displaystyle {\mathcal {P}}}
577:
574:
571:
558:
548:
538:
525:
522:
509:
499:
486:
473:
13:
1:
10221:History of mathematical logic
7852:"Zermelo-Fraenkel Set Theory"
7260:van Heijenoort, Jean (1967).
7226:Halbeisen, Lorenz J. (2011).
6983:For a complete argument that
6750:normal Moore space conjecture
6650:
6618:A variation on the method of
6371:Bernays & Fraenkel (1958)
6353:, where the entire construct
6180:is easily proved from axioms
5228:{\displaystyle \mathbb {N} .}
4189:in the language of ZFC whose
4162:will also fall inside a set.
4158:of a set under any definable
3678:axiom schema of specification
2288:Axiom schema of specification
1516:is typically formalized, the
677:axiom schema of specification
216:says that given any two sets
107:, named after mathematicians
10146:Primitive recursive function
7603:The Philosophy of Set Theory
7232:. Springer. pp. 62–63.
6463:Abian & LaMacchia (1978)
6311:. Although the structure of
5005:{\displaystyle \varnothing }
4873:{\displaystyle w\cup \{w\},}
3489:{\displaystyle \varnothing }
3317:{\displaystyle \varnothing }
3130:{\displaystyle y\subseteq z}
2311:{\displaystyle \mathbb {Z} }
7:
7807:— (31 January 2023).
7763:Encyclopedia of Mathematics
7623:Wolchover, Natalie (2013).
7104:. Amsterdam: North Holland.
6926:, p. 138, def. 1.
6783:
6379:Takeuti & Zaring (1982)
4988:). Then there exists a set
4146:Axiom schema of replacement
4140:Axiom schema of replacement
3291:axiom schema of replacement
3199:{\displaystyle \varphi (x)}
2475:{\displaystyle \varphi (x)}
1239:{\displaystyle \lnot \phi }
749:relation. For example, the
681:axiom schema of replacement
105:Zermelo–Fraenkel set theory
10:
10313:
10287:Foundations of mathematics
9210:Schröder–Bernstein theorem
8937:Monadic predicate calculus
8596:Foundations of mathematics
8379:von Neumann–Bernays–Gödel
7616:Cambridge University Press
7610:Tourlakis, George (2003).
7247:Hatcher, William (1982) .
6790:Foundations of mathematics
6654:
6569:(which is not a ZFC axiom)
6392:
6237:if and only if the set is
6199:
5750:
5598:containing the subsets of
5241:
4717:
4708:axiom schema of collection
4143:
3792:{\displaystyle \{1,2,3\}.}
3691:
3684:applied twice to any set.
3503:
3496:" to the language of ZFC.
3472:). It is common to make a
3177:on the basis of a formula
3032:unrestricted comprehension
2285:
2176:{\displaystyle \forall x.}
1974:
1581:
1482:
1014:The set membership symbol
706:
695:. Adding to ZF either the
362:
358:
199:unrestricted comprehension
123:free of paradoxes such as
29:
10256:
10243:Philosophy of mathematics
10192:Automated theorem proving
10174:
10069:
9901:
9794:
9646:
9363:
9339:
9317:Von Neumann–Bernays–Gödel
9262:
9156:
9060:
8958:
8949:
8876:
8811:
8717:
8639:
8556:
8443:
8406:
8318:
8208:
8180:One-to-one correspondence
8096:
8037:
7928:
7917:
7903:
7214:Foundations of Set Theory
6697:On the other hand, among
6329:axiom of constructibility
6188:. AC is characterized as
6142:{\displaystyle f(Y)\in Y}
5887:such that every nonempty
4797:
4780:
4763:
4746:
4729:
4510:{\displaystyle \exists !}
4150:uniqueness quantification
3226:will regard within a set
1478:
1468:{\displaystyle \exists x}
1445:{\displaystyle \forall x}
799:is an element of the set
205:(NBG) is a commonly used
137:foundation of mathematics
7443:Set Theory and Its Logic
7266:Harvard University Press
7147:Ebbinghaus, Heinz-Dieter
7088:10.1305/ndjfl/1093888220
6839:
6733:included in NBG and MK.
6365:} is simply defined as F
6208:the cumulative hierarchy
5578:as the subset of such a
4992:such that the empty set
4666:is a subset of some set
4533:{\displaystyle \varphi }
4293:{\displaystyle \varphi }
4178:{\displaystyle \varphi }
4027:can be constructed from
3337:{\displaystyle \varphi }
2694:{\displaystyle \varphi }
2596:{\displaystyle \varphi }
2576:{\displaystyle \varphi }
1504:, we use the equivalent
979:{\displaystyle \exists }
959:{\displaystyle \forall }
882:The logical connectives
635:is any infinite set and
168:are such sets. Thus the
9893:Self-verifying theories
9714:Tarski's axiomatization
8665:Tarski's undefinability
8660:incompleteness theorems
7708:Fundamenta Mathematicae
7674:English translation in
7593:10.4064/fm-32-1-176-783
7579:Fundamenta Mathematicae
7408:Link, Godehard (2014).
6829:Constructive set theory
6814:Morse–Kelley set theory
6719:Morse–Kelley set theory
6674:second-order arithmetic
6385:implementation of ZFC.
6282:Morse–Kelley set theory
6280:(often called NBG) and
4635:{\displaystyle x\in A,}
2482:with one free variable
2185:or in modern notation:
1584:Axiom of extensionality
1578:Axiom of extensionality
946:The quantifier symbols
868:functional completeness
281:{\displaystyle \{a,b\}}
71:more precise citations.
10267:Mathematics portal
9878:Proof of impossibility
9526:propositional variable
8836:Propositional calculus
8138:Constructible universe
7965:Constructibility (V=L)
7533:; Zaring, W M (1971).
7393:. Dover Publications.
7290:Hinman, Peter (2005).
6807:axiomatic set theories
6778:Grothendieck universes
6731:axiom of global choice
6699:axiomatic set theories
6657:projective determinacy
6597:constructible universe
6541:constructible universe
6508:above) into theorems.
6478:inaccessible cardinals
6298:constructible universe
6210:of sets introduced by
6170:
6143:
6108:
6107:{\displaystyle Y\in X}
6078:
6058:
6038:
6018:
5984:
5929:
5905:
5881:
5857:
5837:
5814:
5791:
5734:
5714:
5680:
5612:
5592:
5572:
5534:
5467:
5447:
5427:
5405:
5326:
5312:is also an element of
5306:
5286:
5262:
5229:
5183:
5047:
5006:
4982:
4952:
4920:
4894:
4874:
4839:
4700:
4680:
4660:
4636:
4607:
4574:
4554:
4534:
4511:
4486:
4294:
4274:
4260:so that in particular
4254:
4179:
4131:
4041:
4021:
3994:
3965:
3863:
3839:
3819:
3793:
3752:
3682:axiom of the power set
3663:
3584:
3564:
3544:
3524:
3490:
3476:that adds the symbol "
3474:definitional extension
3466:
3450:axiom of the empty set
3440:
3358:
3338:
3318:
3295:axiom of the empty set
3280:
3260:
3240:
3220:
3200:
3171:
3151:
3131:
3099:
3098:{\displaystyle y\in z}
3072:
3048:
3024:
2986:
2936:
2888:
2695:
2675:
2655:
2597:
2577:
2548:
2496:
2476:
2447:
2425:
2358:
2312:
2263:
2177:
2055:
2035:
2015:
1995:
1961:
1941:
1919:
1808:
1719:
1693:
1667:
1552:
1469:
1446:
1423:
1422:{\displaystyle \lnot }
1403:
1383:
1382:{\displaystyle \land }
1360:
1333:
1306:
1273:
1240:
1213:
1193:
1173:
1149:
1148:{\displaystyle x\in y}
1122:
1084:
1064:
1046:(wff) are as follows:
1028:
1004:
980:
960:
936:
916:
915:{\displaystyle \land }
896:
895:{\displaystyle \lnot }
855:
835:
813:
793:
772:
771:{\displaystyle a\in b}
739:
653:
629:
602:
437:
322:
302:
282:
250:
230:
207:conservative extension
10292:Systems of set theory
10136:Kolmogorov complexity
10089:Computably enumerable
9989:Model complete theory
9781:Principia Mathematica
8841:Propositional formula
8670:Banach–Tarski paradox
8361:Principia Mathematica
8195:Transfinite induction
8054:(i.e. set difference)
7722:10.4064/fm-16-1-29-47
7649:Mathematische Annalen
7493:Shoenfield, Joseph R.
7463:Shoenfield, Joseph R.
7389:Levy, Azriel (2002).
6758:large cardinal axioms
6533:large cardinal axioms
6441:inaccessible cardinal
6389:Finite axiomatization
6200:Further information:
6171:
6144:
6109:
6088:", such that for all
6079:
6059:
6039:
6019:
5985:
5930:
5906:
5882:
5858:
5838:
5815:
5792:
5757:Well-ordering theorem
5735:
5715:
5681:
5613:
5593:
5573:
5535:
5468:
5448:
5428:
5406:
5327:
5307:
5287:
5263:
5248:By definition, a set
5230:
5184:
5048:
5007:
4983:
4981:{\displaystyle \{w\}}
4953:
4951:{\displaystyle x=y=w}
4921:
4919:{\displaystyle \{w\}}
4895:
4875:
4840:
4701:
4681:
4661:
4637:
4608:
4575:
4555:
4535:
4512:
4487:
4295:
4275:
4255:
4180:
4132:
4042:
4022:
3995:
3966:
3864:
3840:
3820:
3794:
3753:
3664:
3585:
3565:
3545:
3525:
3491:
3467:
3441:
3359:
3339:
3319:
3281:
3261:
3241:
3221:
3201:
3172:
3152:
3132:
3100:
3073:
3049:
3025:
2987:
2937:
2889:
2696:
2676:
2656:
2598:
2578:
2549:
2497:
2477:
2448:
2426:
2359:
2313:
2264:
2178:
2056:
2036:
2016:
1996:
1962:
1942:
1920:
1809:
1720:
1694:
1668:
1553:
1506:well-ordering theorem
1470:
1447:
1424:
1404:
1402:{\displaystyle \lor }
1384:
1361:
1334:
1307:
1274:
1241:
1214:
1194:
1192:{\displaystyle \psi }
1174:
1172:{\displaystyle \phi }
1150:
1123:
1085:
1065:
1029:
1005:
981:
961:
937:
935:{\displaystyle \lor }
917:
897:
856:
836:
814:
794:
773:
740:
654:
630:
628:{\displaystyle Z_{0}}
603:
438:
365:History of set theory
323:
303:
283:
251:
231:
166:universe of discourse
10084:Church–Turing thesis
10071:Computability theory
9280:continuum hypothesis
8798:Square of opposition
8656:Gödel's completeness
8435:Burali-Forti paradox
8190:Set-builder notation
8143:Continuum hypothesis
8083:Symmetric difference
7811:. In — (ed.).
7677:Heijenoort, Jean van
7601:Tiles, Mary (1989).
7523:Axiomatic Set Theory
7206:Bar-Hillel, Yehoshua
7100:Axiomatic Set Theory
6800:Large cardinal axiom
6762:axiom of determinacy
6742:continuum hypothesis
6740:. These include the
6557:Continuum hypothesis
6453:Burali-Forti paradox
6202:Von Neumann universe
6160:
6118:
6092:
6068:
6048:
6028:
6008:
5941:
5919:
5895:
5871:
5847:
5827:
5804:
5781:
5724:
5704:
5624:
5602:
5582:
5549:
5479:
5457:
5437:
5417:
5338:
5316:
5296:
5276:
5252:
5214:
5063:
5053:is also a member of
5046:{\displaystyle S(y)}
5028:
5016:and, whenever a set
4996:
4966:
4930:
4904:
4884:
4849:
4838:{\displaystyle S(w)}
4820:
4690:
4670:
4650:
4617:
4606:{\displaystyle f(x)}
4588:
4564:
4544:
4524:
4498:
4306:
4284:
4264:
4197:
4169:
4053:
4031:
4004:
3977:
3875:
3849:
3829:
3805:
3762:
3706:
3596:
3574:
3554:
3534:
3514:
3480:
3456:
3370:
3348:
3328:
3308:
3270:
3250:
3230:
3210:
3181:
3161:
3141:
3115:
3083:
3062:
3054:only refers to sets
3038:
2996:
2952:
2902:
2707:
2685:
2665:
2607:
2587:
2567:
2508:
2486:
2457:
2437:
2370:
2326:
2300:
2294:set builder notation
2189:
2071:
2045:
2025:
2005:
1985:
1981:Every non-empty set
1951:
1931:
1822:
1729:
1703:
1683:
1594:
1524:
1456:
1433:
1413:
1393:
1373:
1344:
1317:
1284:
1251:
1227:
1203:
1183:
1163:
1133:
1106:
1098:(the simplest wffs):
1074:
1054:
1044:well-formed formulae
1027:{\displaystyle \in }
1018:
994:
990:The equality symbol
970:
950:
926:
906:
886:
845:
825:
803:
783:
756:
738:{\displaystyle \in }
729:
639:
612:
447:
420:
402:axiomatic set theory
369:The modern study of
345:continuum hypothesis
341:logical independence
330:von Neumann universe
312:
292:
260:
240:
220:
190:mathematical objects
133:axiomatic set theory
32:ZFC (disambiguation)
10238:Mathematical object
10129:P versus NP problem
10094:Computable function
9888:Reverse mathematics
9814:Logical consequence
9691:primitive recursive
9686:elementary function
9459:Free/bound variable
9312:Tarski–Grothendieck
8831:Logical connectives
8761:Logical equivalence
8611:Logical consequence
8396:Tarski–Grothendieck
7788:(31 January 2023).
7368:(29 October 2007).
6834:Internal set theory
6736:There are numerous
6678:reverse mathematics
6606:The failure of the
6428:Robinson arithmetic
5202:von Neumann ordinal
4726:
4613:is a set for every
2502:may be written as:
1977:Axiom of regularity
1927:which says that if
1718:{\displaystyle x=y}
1518:domain of discourse
1121:{\displaystyle x=y}
713:Formally, ZFC is a
687:(first proposed by
685:axiom of regularity
400:proposed the first
288:containing exactly
256:there is a new set
10036:Transfer principle
9999:Semantics of logic
9984:Categorical theory
9960:Non-standard model
9474:Logical connective
8601:Information theory
8550:Mathematical logic
7985:Limitation of size
7849:Weisstein, Eric W.
7662:10.1007/BF01449999
7497:Mathematical Logic
7432:Infinistic Methods
6987:satisfies ZFC see
6691:Zermelo set theory
6638:Proposed additions
6432:general set theory
6166:
6139:
6104:
6074:
6054:
6034:
6014:
5980:
5968:
5925:
5901:
5877:
5853:
5833:
5810:
5787:
5730:
5710:
5676:
5608:
5588:
5568:
5530:
5463:
5443:
5423:
5401:
5322:
5302:
5282:
5258:
5244:Axiom of power set
5238:Axiom of power set
5225:
5179:
5043:
5002:
4978:
4948:
4916:
4890:
4870:
4835:
4724:
4696:
4676:
4656:
4632:
4603:
4570:
4550:
4530:
4507:
4482:
4290:
4270:
4250:
4175:
4127:
4037:
4017:
3990:
3961:
3859:
3835:
3815:
3789:
3748:
3659:
3580:
3560:
3540:
3520:
3486:
3462:
3436:
3354:
3334:
3314:
3276:
3256:
3236:
3216:
3196:
3167:
3147:
3127:
3095:
3068:
3044:
3020:
2982:
2932:
2884:
2691:
2671:
2651:
2593:
2573:
2544:
2492:
2472:
2453:obeying a formula
2443:
2421:
2354:
2308:
2259:
2173:
2051:
2031:
2011:
2001:contains a member
1991:
1957:
1937:
1915:
1804:
1715:
1689:
1663:
1548:
1465:
1442:
1419:
1399:
1379:
1356:
1329:
1302:
1269:
1236:
1209:
1189:
1169:
1145:
1118:
1080:
1060:
1024:
1000:
976:
956:
932:
912:
892:
851:
831:
809:
789:
768:
735:
649:
625:
598:
433:
406:Zermelo set theory
318:
298:
278:
246:
226:
10274:
10273:
10206:Abstract category
10009:Theories of truth
9819:Rule of inference
9809:Natural deduction
9790:
9789:
9335:
9334:
9040:Cartesian product
8945:
8944:
8851:Many-valued logic
8826:Boolean functions
8709:Russell's paradox
8684:diagonal argument
8581:First-order logic
8516:
8515:
8425:Russell's paradox
8374:Zermelo–Fraenkel
8275:Dedekind-infinite
8148:Diagonal argument
8047:Cartesian product
7911:Set (mathematics)
7828:first order logic
7690:978-0-674-32449-7
7510:978-1-56881-135-2
7428:Montague, Richard
7419:978-1-61451-829-7
7305:978-1-56881-262-5
7239:978-1-4471-2172-5
7202:Fraenkel, Abraham
7169:Feferman, Solomon
7160:978-3-540-49551-2
6989:Shoenfield (1977)
6962:, section 2.
6891:, pp. 62–63.
6746:Whitehead problem
6682:Saunders Mac Lane
6608:Kurepa hypothesis
6573:Suslin hypothesis
6562:Diamond principle
6539:, such as in the
6449:Russell's paradox
6169:{\displaystyle X}
6077:{\displaystyle X}
6057:{\displaystyle X}
6037:{\displaystyle f}
6017:{\displaystyle X}
5967:
5928:{\displaystyle R}
5904:{\displaystyle X}
5880:{\displaystyle X}
5856:{\displaystyle R}
5836:{\displaystyle X}
5813:{\displaystyle R}
5797:, there exists a
5790:{\displaystyle X}
5733:{\displaystyle x}
5713:{\displaystyle x}
5611:{\displaystyle x}
5591:{\displaystyle y}
5466:{\displaystyle x}
5446:{\displaystyle y}
5433:, there is a set
5426:{\displaystyle x}
5325:{\displaystyle x}
5305:{\displaystyle z}
5285:{\displaystyle x}
5261:{\displaystyle z}
4893:{\displaystyle w}
4814:
4813:
4720:Axiom of infinity
4714:Axiom of infinity
4699:{\displaystyle B}
4679:{\displaystyle B}
4659:{\displaystyle f}
4573:{\displaystyle A}
4553:{\displaystyle f}
4412:
4273:{\displaystyle B}
4040:{\displaystyle A}
3838:{\displaystyle A}
3825:, there is a set
3674:axiom of infinity
3583:{\displaystyle y}
3563:{\displaystyle x}
3543:{\displaystyle y}
3523:{\displaystyle x}
3465:{\displaystyle w}
3357:{\displaystyle w}
3279:{\displaystyle y}
3259:{\displaystyle y}
3239:{\displaystyle z}
3219:{\displaystyle y}
3170:{\displaystyle y}
3150:{\displaystyle y}
3071:{\displaystyle z}
3047:{\displaystyle y}
2946:Russell's paradox
2674:{\displaystyle y}
2583:). Formally, let
2495:{\displaystyle x}
2446:{\displaystyle z}
2320:congruence modulo
2054:{\displaystyle y}
2034:{\displaystyle x}
2014:{\displaystyle y}
1994:{\displaystyle x}
1960:{\displaystyle y}
1940:{\displaystyle x}
1692:{\displaystyle =}
1495:first order logic
1212:{\displaystyle x}
1083:{\displaystyle y}
1063:{\displaystyle x}
1003:{\displaystyle =}
854:{\displaystyle b}
834:{\displaystyle a}
812:{\displaystyle b}
792:{\displaystyle a}
719:first-order logic
715:one-sorted theory
669:first-order logic
391:Russell's paradox
373:was initiated by
321:{\displaystyle b}
301:{\displaystyle a}
249:{\displaystyle b}
229:{\displaystyle a}
125:Russell's paradox
97:
96:
89:
16:(Redirected from
10304:
10265:
10264:
10216:History of logic
10211:Category of sets
10104:Decision problem
9883:Ordinal analysis
9824:Sequent calculus
9722:Boolean algebras
9662:
9661:
9636:
9607:logical/constant
9361:
9360:
9347:
9270:Zermelo–Fraenkel
9021:Set operations:
8956:
8955:
8893:
8724:
8723:
8704:Löwenheim–Skolem
8591:Formal semantics
8543:
8536:
8529:
8520:
8519:
8498:Bertrand Russell
8488:John von Neumann
8473:Abraham Fraenkel
8468:Richard Dedekind
8430:Suslin's problem
8341:Cantor's theorem
8058:De Morgan's laws
7923:
7890:
7883:
7876:
7867:
7866:
7862:
7861:
7818:
7803:
7794:Zalta, Edward N.
7771:
7748:
7734:
7724:
7694:
7673:
7634:
7619:
7606:
7605:. Dover reprint.
7597:
7595:
7565:
7553:
7542:
7526:
7525:. Dover reprint.
7514:
7499:(2nd ed.).
7488:
7476:
7458:
7446:
7435:
7423:
7404:
7391:Basic Set Theory
7385:
7383:
7376:
7361:
7345:
7331:
7309:
7269:
7256:
7243:
7221:
7196:
7164:
7142:
7126:
7105:
7103:
7092:
7090:
7065:
7063:
7054:Abian, Alexander
7039:
7033:
7027:
7021:
7015:
7009:
7003:
6998:
6992:
6981:
6975:
6969:
6963:
6957:
6951:
6945:
6939:
6933:
6927:
6921:
6915:
6909:
6903:
6898:
6892:
6886:
6880:
6874:
6868:
6862:
6856:
6850:
6686:Solomon Feferman
6670:Peano arithmetic
6457:Cantor's paradox
6445:naive set theory
6212:John von Neumann
6175:
6173:
6172:
6167:
6148:
6146:
6145:
6140:
6113:
6111:
6110:
6105:
6083:
6081:
6080:
6075:
6063:
6061:
6060:
6055:
6043:
6041:
6040:
6035:
6023:
6021:
6020:
6015:
5989:
5987:
5986:
5981:
5969:
5965:
5934:
5932:
5931:
5926:
5915:under the order
5910:
5908:
5907:
5902:
5886:
5884:
5883:
5878:
5862:
5860:
5859:
5854:
5842:
5840:
5839:
5834:
5819:
5817:
5816:
5811:
5796:
5794:
5793:
5788:
5739:
5737:
5736:
5731:
5719:
5717:
5716:
5711:
5698:empty set exists
5685:
5683:
5682:
5677:
5633:
5632:
5617:
5615:
5614:
5609:
5597:
5595:
5594:
5589:
5577:
5575:
5574:
5569:
5558:
5557:
5539:
5537:
5536:
5531:
5472:
5470:
5469:
5464:
5452:
5450:
5449:
5444:
5432:
5430:
5429:
5424:
5410:
5408:
5407:
5402:
5331:
5329:
5328:
5323:
5311:
5309:
5308:
5303:
5291:
5289:
5288:
5283:
5267:
5265:
5264:
5259:
5234:
5232:
5231:
5226:
5221:
5206:
5199:
5194:
5188:
5186:
5185:
5180:
5175:
5171:
5056:
5052:
5050:
5049:
5044:
5023:
5019:
5015:
5011:
5009:
5008:
5003:
4991:
4987:
4985:
4984:
4979:
4961:
4958:so that the set
4957:
4955:
4954:
4949:
4925:
4923:
4922:
4917:
4899:
4897:
4896:
4891:
4879:
4877:
4876:
4871:
4844:
4842:
4841:
4836:
4793:{∅,{∅},{∅,{∅}}}
4727:
4723:
4705:
4703:
4702:
4697:
4685:
4683:
4682:
4677:
4665:
4663:
4662:
4657:
4641:
4639:
4638:
4633:
4612:
4610:
4609:
4604:
4579:
4577:
4576:
4571:
4559:
4557:
4556:
4551:
4539:
4537:
4536:
4531:
4516:
4514:
4513:
4508:
4491:
4489:
4488:
4483:
4478:
4477:
4471:
4470:
4425:
4424:
4410:
4363:
4362:
4356:
4355:
4340:
4339:
4327:
4326:
4299:
4297:
4296:
4291:
4279:
4277:
4276:
4271:
4259:
4257:
4256:
4251:
4246:
4245:
4227:
4226:
4184:
4182:
4181:
4176:
4136:
4134:
4133:
4128:
4117:
4116:
4065:
4064:
4046:
4044:
4043:
4038:
4026:
4024:
4023:
4018:
4016:
4015:
3999:
3997:
3996:
3991:
3989:
3988:
3970:
3968:
3967:
3962:
3939:
3938:
3887:
3886:
3868:
3866:
3865:
3860:
3858:
3857:
3844:
3842:
3841:
3836:
3824:
3822:
3821:
3816:
3814:
3813:
3798:
3796:
3795:
3790:
3757:
3755:
3754:
3749:
3679:
3668:
3666:
3665:
3660:
3589:
3587:
3586:
3581:
3569:
3567:
3566:
3561:
3549:
3547:
3546:
3541:
3529:
3527:
3526:
3521:
3506:Axiom of pairing
3500:Axiom of pairing
3495:
3493:
3492:
3487:
3471:
3469:
3468:
3463:
3445:
3443:
3442:
3437:
3363:
3361:
3360:
3355:
3343:
3341:
3340:
3335:
3323:
3321:
3320:
3315:
3285:
3283:
3282:
3277:
3265:
3263:
3262:
3257:
3245:
3243:
3242:
3237:
3225:
3223:
3222:
3217:
3205:
3203:
3202:
3197:
3176:
3174:
3173:
3168:
3156:
3154:
3153:
3148:
3137:is the case, so
3136:
3134:
3133:
3128:
3104:
3102:
3101:
3096:
3077:
3075:
3074:
3069:
3053:
3051:
3050:
3045:
3029:
3027:
3026:
3021:
2991:
2989:
2988:
2983:
2941:
2939:
2938:
2933:
2893:
2891:
2890:
2885:
2865:
2864:
2840:
2839:
2827:
2826:
2757:
2756:
2741:
2740:
2728:
2727:
2700:
2698:
2697:
2692:
2680:
2678:
2677:
2672:
2660:
2658:
2657:
2652:
2650:
2649:
2631:
2630:
2602:
2600:
2599:
2594:
2582:
2580:
2579:
2574:
2553:
2551:
2550:
2545:
2501:
2499:
2498:
2493:
2481:
2479:
2478:
2473:
2452:
2450:
2449:
2444:
2430:
2428:
2427:
2422:
2414:
2386:
2363:
2361:
2360:
2355:
2353:
2317:
2315:
2314:
2309:
2307:
2268:
2266:
2265:
2260:
2182:
2180:
2179:
2174:
2060:
2058:
2057:
2052:
2040:
2038:
2037:
2032:
2020:
2018:
2017:
2012:
2000:
1998:
1997:
1992:
1966:
1964:
1963:
1958:
1946:
1944:
1943:
1938:
1924:
1922:
1921:
1916:
1813:
1811:
1810:
1805:
1724:
1722:
1721:
1716:
1698:
1696:
1695:
1690:
1672:
1670:
1669:
1664:
1557:
1555:
1554:
1549:
1508:in place of the
1474:
1472:
1471:
1466:
1451:
1449:
1448:
1443:
1428:
1426:
1425:
1420:
1408:
1406:
1405:
1400:
1388:
1386:
1385:
1380:
1365:
1363:
1362:
1357:
1338:
1336:
1335:
1330:
1311:
1309:
1308:
1303:
1278:
1276:
1275:
1270:
1245:
1243:
1242:
1237:
1218:
1216:
1215:
1210:
1198:
1196:
1195:
1190:
1178:
1176:
1175:
1170:
1154:
1152:
1151:
1146:
1127:
1125:
1124:
1119:
1089:
1087:
1086:
1081:
1069:
1067:
1066:
1061:
1033:
1031:
1030:
1025:
1009:
1007:
1006:
1001:
985:
983:
982:
977:
965:
963:
962:
957:
941:
939:
938:
933:
921:
919:
918:
913:
901:
899:
898:
893:
860:
858:
857:
852:
840:
838:
837:
832:
818:
816:
815:
810:
798:
796:
795:
790:
777:
775:
774:
769:
744:
742:
741:
736:
689:John von Neumann
658:
656:
655:
650:
648:
647:
634:
632:
631:
626:
624:
623:
607:
605:
604:
599:
570:
569:
557:
556:
547:
546:
537:
536:
521:
520:
508:
507:
498:
497:
485:
484:
472:
471:
462:
461:
442:
440:
439:
434:
432:
431:
414:cardinal numbers
410:Abraham Fraenkel
387:naive set theory
379:Richard Dedekind
327:
325:
324:
319:
307:
305:
304:
299:
287:
285:
284:
279:
255:
253:
252:
247:
235:
233:
232:
227:
214:axiom of pairing
188:(collections of
180:from containing
176:and prevent its
117:axiomatic system
113:Abraham Fraenkel
92:
85:
81:
78:
72:
67:this article by
58:inline citations
45:
44:
37:
21:
10312:
10311:
10307:
10306:
10305:
10303:
10302:
10301:
10277:
10276:
10275:
10270:
10259:
10252:
10197:Category theory
10187:Algebraic logic
10170:
10141:Lambda calculus
10079:Church encoding
10065:
10041:Truth predicate
9897:
9863:Complete theory
9786:
9655:
9651:
9647:
9642:
9634:
9354: and
9350:
9345:
9331:
9307:New Foundations
9275:axiom of choice
9258:
9220:Gödel numbering
9160: and
9152:
9056:
8941:
8891:
8872:
8821:Boolean algebra
8807:
8771:Equiconsistency
8736:Classical logic
8713:
8694:Halting problem
8682: and
8658: and
8646: and
8645:
8640:Theorems (
8635:
8552:
8547:
8517:
8512:
8439:
8418:
8402:
8367:New Foundations
8314:
8204:
8123:Cardinal number
8106:
8092:
8033:
7924:
7915:
7899:
7894:
7756:
7746:
7742:
7737:
7691:
7630:Quanta Magazine
7562:
7539:Springer-Verlag
7519:Suppes, Patrick
7511:
7485:
7455:
7420:
7401:
7381:
7374:
7358:
7328:
7306:
7240:
7193:
7161:
7135:The Joy of Sets
7123:
7064:. W B Saunders.
7048:
7043:
7042:
7034:
7030:
7022:
7018:
7010:
7006:
6999:
6995:
6982:
6978:
6970:
6966:
6960:Shoenfield 1977
6958:
6954:
6948:Shoenfield 2001
6946:
6942:
6934:
6930:
6922:
6918:
6910:
6906:
6899:
6895:
6887:
6883:
6877:Ebbinghaus 2007
6875:
6871:
6863:
6859:
6853:Ciesielski 1997
6851:
6847:
6842:
6786:
6727:axiom of choice
6711:algebra of sets
6703:New Foundations
6659:
6653:
6640:
6632:large cardinals
6624:axiom of choice
6514:
6504: –
6482:category theory
6421:
6401:Montague (1961)
6397:
6391:
6342:
6340:Virtual classes
6337:
6335:Metamathematics
6286:New Foundations
6204:
6198:
6190:nonconstructive
6161:
6158:
6157:
6119:
6116:
6115:
6093:
6090:
6089:
6086:choice function
6069:
6066:
6065:
6049:
6046:
6045:
6029:
6026:
6025:
6009:
6006:
6005:
5996: –
5990:
5963:
5942:
5939:
5938:
5920:
5917:
5916:
5896:
5893:
5892:
5872:
5869:
5868:
5848:
5845:
5844:
5828:
5825:
5824:
5805:
5802:
5801:
5799:binary relation
5782:
5779:
5778:
5767:axiom of choice
5763:
5753:Axiom of choice
5751:Main articles:
5749:
5725:
5722:
5721:
5705:
5702:
5701:
5686:
5628:
5627:
5625:
5622:
5621:
5603:
5600:
5599:
5583:
5580:
5579:
5553:
5552:
5550:
5547:
5546:
5540:
5480:
5477:
5476:
5458:
5455:
5454:
5438:
5435:
5434:
5418:
5415:
5414:
5411:
5339:
5336:
5335:
5317:
5314:
5313:
5297:
5294:
5293:
5277:
5274:
5273:
5253:
5250:
5249:
5246:
5240:
5217:
5215:
5212:
5211:
5209:natural numbers
5204:
5197:
5192:
5189:
5076:
5072:
5064:
5061:
5060:
5054:
5029:
5026:
5025:
5021:
5020:is a member of
5017:
5013:
4997:
4994:
4993:
4989:
4967:
4964:
4963:
4959:
4931:
4928:
4927:
4905:
4902:
4901:
4885:
4882:
4881:
4850:
4847:
4846:
4821:
4818:
4817:
4722:
4716:
4691:
4688:
4687:
4671:
4668:
4667:
4651:
4648:
4647:
4618:
4615:
4614:
4589:
4586:
4585:
4580:represents its
4565:
4562:
4561:
4545:
4542:
4541:
4525:
4522:
4521:
4499:
4496:
4495:
4492:
4473:
4472:
4466:
4465:
4420:
4419:
4358:
4357:
4351:
4347:
4335:
4331:
4322:
4318:
4307:
4304:
4303:
4285:
4282:
4281:
4280:is not free in
4265:
4262:
4261:
4241:
4237:
4222:
4218:
4198:
4195:
4194:
4170:
4167:
4166:
4152:
4144:Main articles:
4142:
4137:
4112:
4111:
4060:
4059:
4054:
4051:
4050:
4032:
4029:
4028:
4011:
4010:
4005:
4002:
4001:
3984:
3983:
3978:
3975:
3974:
3971:
3934:
3933:
3882:
3881:
3876:
3873:
3872:
3853:
3852:
3850:
3847:
3846:
3830:
3827:
3826:
3809:
3808:
3806:
3803:
3802:
3763:
3760:
3759:
3707:
3704:
3703:
3696:
3690:
3677:
3669:
3597:
3594:
3593:
3575:
3572:
3571:
3555:
3552:
3551:
3535:
3532:
3531:
3515:
3512:
3511:
3508:
3502:
3481:
3478:
3477:
3457:
3454:
3453:
3446:
3371:
3368:
3367:
3349:
3346:
3345:
3329:
3326:
3325:
3309:
3306:
3305:
3271:
3268:
3267:
3251:
3248:
3247:
3231:
3228:
3227:
3211:
3208:
3207:
3182:
3179:
3178:
3162:
3159:
3158:
3142:
3139:
3138:
3116:
3113:
3112:
3084:
3081:
3080:
3063:
3060:
3059:
3039:
3036:
3035:
2997:
2994:
2993:
2953:
2950:
2949:
2942:
2903:
2900:
2899:
2894:
2860:
2856:
2835:
2831:
2822:
2818:
2752:
2748:
2736:
2732:
2723:
2719:
2708:
2705:
2704:
2686:
2683:
2682:
2681:is not free in
2666:
2663:
2662:
2645:
2641:
2626:
2622:
2608:
2605:
2604:
2588:
2585:
2584:
2568:
2565:
2564:
2554:
2509:
2506:
2505:
2487:
2484:
2483:
2458:
2455:
2454:
2438:
2435:
2434:
2431:
2399:
2382:
2371:
2368:
2367:
2338:
2327:
2324:
2323:
2318:satisfying the
2303:
2301:
2298:
2297:
2290:
2284:
2190:
2187:
2186:
2183:
2072:
2069:
2068:
2046:
2043:
2042:
2026:
2023:
2022:
2006:
2003:
2002:
1986:
1983:
1982:
1979:
1973:
1952:
1949:
1948:
1932:
1929:
1928:
1925:
1823:
1820:
1819:
1730:
1727:
1726:
1704:
1701:
1700:
1684:
1681:
1680:
1673:
1595:
1592:
1591:
1586:
1580:
1525:
1522:
1521:
1510:axiom of choice
1487:
1481:
1457:
1454:
1453:
1434:
1431:
1430:
1414:
1411:
1410:
1394:
1391:
1390:
1374:
1371:
1370:
1345:
1342:
1341:
1318:
1315:
1314:
1285:
1282:
1281:
1252:
1249:
1248:
1228:
1225:
1224:
1204:
1201:
1200:
1184:
1181:
1180:
1164:
1161:
1160:
1134:
1131:
1130:
1107:
1104:
1103:
1096:atomic formulae
1075:
1072:
1071:
1055:
1052:
1051:
1019:
1016:
1015:
995:
992:
991:
971:
968:
967:
951:
948:
947:
927:
924:
923:
907:
904:
903:
887:
884:
883:
846:
843:
842:
841:is a member of
826:
823:
822:
804:
801:
800:
784:
781:
780:
757:
754:
753:
730:
727:
726:
711:
709:Formal language
705:
703:Formal language
697:axiom of choice
673:atomic formulas
643:
642:
640:
637:
636:
619:
615:
613:
610:
609:
565:
561:
552:
551:
542:
541:
532:
531:
516:
512:
503:
502:
493:
492:
480:
476:
467:
466:
457:
453:
448:
445:
444:
427:
423:
421:
418:
417:
367:
361:
337:metamathematics
313:
310:
309:
293:
290:
289:
261:
258:
257:
241:
238:
237:
221:
218:
217:
129:axiom of choice
93:
82:
76:
73:
63:Please help to
62:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
10310:
10300:
10299:
10294:
10289:
10272:
10271:
10257:
10254:
10253:
10251:
10250:
10245:
10240:
10235:
10230:
10229:
10228:
10218:
10213:
10208:
10199:
10194:
10189:
10184:
10182:Abstract logic
10178:
10176:
10172:
10171:
10169:
10168:
10163:
10161:Turing machine
10158:
10153:
10148:
10143:
10138:
10133:
10132:
10131:
10126:
10121:
10116:
10111:
10101:
10099:Computable set
10096:
10091:
10086:
10081:
10075:
10073:
10067:
10066:
10064:
10063:
10058:
10053:
10048:
10043:
10038:
10033:
10028:
10027:
10026:
10021:
10016:
10006:
10001:
9996:
9994:Satisfiability
9991:
9986:
9981:
9980:
9979:
9969:
9968:
9967:
9957:
9956:
9955:
9950:
9945:
9940:
9935:
9925:
9924:
9923:
9918:
9911:Interpretation
9907:
9905:
9899:
9898:
9896:
9895:
9890:
9885:
9880:
9875:
9865:
9860:
9859:
9858:
9857:
9856:
9846:
9841:
9831:
9826:
9821:
9816:
9811:
9806:
9800:
9798:
9792:
9791:
9788:
9787:
9785:
9784:
9776:
9775:
9774:
9773:
9768:
9767:
9766:
9761:
9756:
9736:
9735:
9734:
9732:minimal axioms
9729:
9718:
9717:
9716:
9705:
9704:
9703:
9698:
9693:
9688:
9683:
9678:
9665:
9663:
9644:
9643:
9641:
9640:
9639:
9638:
9626:
9621:
9620:
9619:
9614:
9609:
9604:
9594:
9589:
9584:
9579:
9578:
9577:
9572:
9562:
9561:
9560:
9555:
9550:
9545:
9535:
9530:
9529:
9528:
9523:
9518:
9508:
9507:
9506:
9501:
9496:
9491:
9486:
9481:
9471:
9466:
9461:
9456:
9455:
9454:
9449:
9444:
9439:
9429:
9424:
9422:Formation rule
9419:
9414:
9413:
9412:
9407:
9397:
9396:
9395:
9385:
9380:
9375:
9370:
9364:
9358:
9341:Formal systems
9337:
9336:
9333:
9332:
9330:
9329:
9324:
9319:
9314:
9309:
9304:
9299:
9294:
9289:
9284:
9283:
9282:
9277:
9266:
9264:
9260:
9259:
9257:
9256:
9255:
9254:
9244:
9239:
9238:
9237:
9230:Large cardinal
9227:
9222:
9217:
9212:
9207:
9193:
9192:
9191:
9186:
9181:
9166:
9164:
9154:
9153:
9151:
9150:
9149:
9148:
9143:
9138:
9128:
9123:
9118:
9113:
9108:
9103:
9098:
9093:
9088:
9083:
9078:
9073:
9067:
9065:
9058:
9057:
9055:
9054:
9053:
9052:
9047:
9042:
9037:
9032:
9027:
9019:
9018:
9017:
9012:
9002:
8997:
8995:Extensionality
8992:
8990:Ordinal number
8987:
8977:
8972:
8971:
8970:
8959:
8953:
8947:
8946:
8943:
8942:
8940:
8939:
8934:
8929:
8924:
8919:
8914:
8909:
8908:
8907:
8897:
8896:
8895:
8882:
8880:
8874:
8873:
8871:
8870:
8869:
8868:
8863:
8858:
8848:
8843:
8838:
8833:
8828:
8823:
8817:
8815:
8809:
8808:
8806:
8805:
8800:
8795:
8790:
8785:
8780:
8775:
8774:
8773:
8763:
8758:
8753:
8748:
8743:
8738:
8732:
8730:
8721:
8715:
8714:
8712:
8711:
8706:
8701:
8696:
8691:
8686:
8674:Cantor's
8672:
8667:
8662:
8652:
8650:
8637:
8636:
8634:
8633:
8628:
8623:
8618:
8613:
8608:
8603:
8598:
8593:
8588:
8583:
8578:
8573:
8572:
8571:
8560:
8558:
8554:
8553:
8546:
8545:
8538:
8531:
8523:
8514:
8513:
8511:
8510:
8505:
8503:Thoralf Skolem
8500:
8495:
8490:
8485:
8480:
8475:
8470:
8465:
8460:
8455:
8449:
8447:
8441:
8440:
8438:
8437:
8432:
8427:
8421:
8419:
8417:
8416:
8413:
8407:
8404:
8403:
8401:
8400:
8399:
8398:
8393:
8388:
8387:
8386:
8371:
8370:
8369:
8357:
8356:
8355:
8344:
8343:
8338:
8333:
8328:
8322:
8320:
8316:
8315:
8313:
8312:
8307:
8302:
8297:
8288:
8283:
8278:
8268:
8263:
8262:
8261:
8256:
8251:
8241:
8231:
8226:
8221:
8215:
8213:
8206:
8205:
8203:
8202:
8197:
8192:
8187:
8185:Ordinal number
8182:
8177:
8172:
8167:
8166:
8165:
8160:
8150:
8145:
8140:
8135:
8130:
8120:
8115:
8109:
8107:
8105:
8104:
8101:
8097:
8094:
8093:
8091:
8090:
8085:
8080:
8075:
8070:
8065:
8063:Disjoint union
8060:
8055:
8049:
8043:
8041:
8035:
8034:
8032:
8031:
8030:
8029:
8024:
8013:
8012:
8010:Martin's axiom
8007:
8002:
7997:
7992:
7987:
7982:
7977:
7975:Extensionality
7972:
7967:
7962:
7961:
7960:
7955:
7950:
7940:
7934:
7932:
7926:
7925:
7918:
7916:
7914:
7913:
7907:
7905:
7901:
7900:
7893:
7892:
7885:
7878:
7870:
7864:
7863:
7844:
7843:
7842:
7821:
7820:
7819:
7804:
7772:
7754:
7741:
7740:External links
7738:
7736:
7735:
7699:Zermelo, Ernst
7695:
7689:
7656:(2): 261–281.
7640:Zermelo, Ernst
7636:
7620:
7607:
7598:
7570:Tarski, Alfred
7566:
7560:
7543:
7531:Takeuti, Gaisi
7527:
7515:
7509:
7489:
7483:
7467:Barwise, K. J.
7459:
7453:
7436:
7424:
7418:
7405:
7399:
7386:
7366:Kunen, Kenneth
7362:
7356:
7336:Kunen, Kenneth
7332:
7326:
7310:
7304:
7287:
7257:
7253:Pergamon Press
7244:
7238:
7223:
7198:
7191:
7165:
7159:
7143:
7127:
7121:
7106:
7093:
7066:
7049:
7047:
7044:
7041:
7040:
7036:Wolchover 2013
7028:
7016:
7004:
6993:
6976:
6974:, p. 467.
6964:
6952:
6950:, p. 239.
6940:
6928:
6916:
6904:
6893:
6889:Halbeisen 2011
6881:
6879:, p. 136.
6869:
6857:
6844:
6843:
6841:
6838:
6837:
6836:
6831:
6826:
6821:
6816:
6803:
6802:
6797:
6792:
6785:
6782:
6754:Martin's axiom
6723:proper classes
6652:
6649:
6639:
6636:
6616:
6615:
6604:
6599:satisfies the
6593:
6590:
6587:
6576:
6575:
6570:
6567:Martin's axiom
6564:
6559:
6554:
6513:
6510:
6486:Tarski's axiom
6420:
6417:
6409:proper classes
6393:Main article:
6390:
6387:
6341:
6338:
6336:
6333:
6222:. The sets in
6216:ordinal number
6197:
6194:
6165:
6138:
6135:
6132:
6129:
6126:
6123:
6103:
6100:
6097:
6073:
6053:
6033:
6013:
5979:
5976:
5973:
5961:
5958:
5955:
5952:
5949:
5946:
5937:
5924:
5900:
5876:
5852:
5843:. This means
5832:
5809:
5786:
5777:. For any set
5748:
5745:
5740:must contain.
5729:
5709:
5675:
5672:
5669:
5666:
5663:
5660:
5657:
5654:
5651:
5648:
5645:
5642:
5639:
5636:
5631:
5620:
5607:
5587:
5567:
5564:
5561:
5556:
5529:
5526:
5523:
5520:
5517:
5514:
5511:
5508:
5505:
5502:
5499:
5496:
5493:
5490:
5487:
5484:
5475:
5462:
5442:
5422:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5334:
5321:
5301:
5281:
5257:
5242:Main article:
5239:
5236:
5224:
5220:
5178:
5174:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5091:
5088:
5085:
5082:
5079:
5075:
5071:
5068:
5059:
5042:
5039:
5036:
5033:
5001:
4977:
4974:
4971:
4947:
4944:
4941:
4938:
4935:
4915:
4912:
4909:
4889:
4869:
4866:
4863:
4860:
4857:
4854:
4834:
4831:
4828:
4825:
4812:
4811:
4808:
4805:
4802:
4799:
4795:
4794:
4791:
4788:
4785:
4782:
4778:
4777:
4774:
4771:
4768:
4765:
4761:
4760:
4757:
4754:
4751:
4748:
4744:
4743:
4740:
4737:
4734:
4731:
4718:Main article:
4715:
4712:
4695:
4675:
4655:
4631:
4628:
4625:
4622:
4602:
4599:
4596:
4593:
4569:
4549:
4529:
4506:
4503:
4481:
4476:
4469:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4423:
4418:
4415:
4409:
4406:
4403:
4400:
4397:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4361:
4354:
4350:
4346:
4343:
4338:
4334:
4330:
4325:
4321:
4317:
4314:
4311:
4302:
4289:
4269:
4249:
4244:
4240:
4236:
4233:
4230:
4225:
4221:
4217:
4214:
4211:
4208:
4205:
4202:
4191:free variables
4174:
4165:Formally, let
4141:
4138:
4126:
4123:
4120:
4115:
4110:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4063:
4058:
4049:
4036:
4014:
4009:
3987:
3982:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3937:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3901:
3898:
3894:
3891:
3885:
3880:
3871:
3856:
3834:
3812:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3694:Axiom of union
3692:Main article:
3689:
3688:Axiom of union
3686:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3592:
3579:
3559:
3539:
3519:
3504:Main article:
3501:
3498:
3485:
3461:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3366:
3353:
3333:
3313:
3275:
3255:
3235:
3215:
3195:
3192:
3189:
3186:
3166:
3146:
3126:
3123:
3120:
3094:
3091:
3088:
3067:
3043:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2898:
2883:
2880:
2877:
2874:
2871:
2868:
2863:
2859:
2855:
2852:
2849:
2846:
2843:
2838:
2834:
2830:
2825:
2821:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2755:
2751:
2747:
2744:
2739:
2735:
2731:
2726:
2722:
2718:
2715:
2712:
2703:
2690:
2670:
2648:
2644:
2640:
2637:
2634:
2629:
2625:
2621:
2618:
2615:
2612:
2592:
2572:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2504:
2491:
2471:
2468:
2465:
2462:
2442:
2420:
2417:
2413:
2410:
2406:
2403:
2398:
2395:
2392:
2389:
2385:
2381:
2378:
2375:
2366:
2352:
2349:
2345:
2342:
2337:
2334:
2331:
2306:
2286:Main article:
2283:
2280:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2197:
2194:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2067:
2050:
2030:
2010:
1990:
1975:Main article:
1972:
1969:
1956:
1936:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1818:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1714:
1711:
1708:
1688:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1590:
1582:Main article:
1579:
1576:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1480:
1477:
1464:
1461:
1441:
1438:
1418:
1398:
1378:
1367:
1366:
1355:
1352:
1349:
1339:
1328:
1325:
1322:
1312:
1301:
1298:
1295:
1292:
1289:
1279:
1268:
1265:
1262:
1259:
1256:
1246:
1235:
1232:
1221:
1220:
1208:
1188:
1168:
1156:
1155:
1144:
1141:
1138:
1128:
1117:
1114:
1111:
1100:
1099:
1079:
1059:
1040:
1039:
1035:
1034:
1023:
1011:
1010:
999:
987:
986:
975:
955:
943:
942:
931:
911:
891:
879:
878:
861:
850:
830:
820:(also read as
819:
808:
788:
767:
764:
761:
747:set membership
734:
704:
701:
665:Thoralf Skolem
646:
622:
618:
597:
594:
591:
588:
585:
582:
579:
576:
573:
568:
564:
560:
555:
550:
545:
540:
535:
530:
527:
524:
519:
515:
511:
506:
501:
496:
491:
488:
483:
479:
475:
470:
465:
460:
456:
452:
430:
426:
363:Main article:
360:
357:
347:from ZFC. The
317:
297:
277:
274:
271:
268:
265:
245:
225:
186:proper classes
160:, so that all
121:theory of sets
95:
94:
49:
47:
40:
26:
9:
6:
4:
3:
2:
10309:
10298:
10295:
10293:
10290:
10288:
10285:
10284:
10282:
10269:
10268:
10263:
10255:
10249:
10246:
10244:
10241:
10239:
10236:
10234:
10231:
10227:
10224:
10223:
10222:
10219:
10217:
10214:
10212:
10209:
10207:
10203:
10200:
10198:
10195:
10193:
10190:
10188:
10185:
10183:
10180:
10179:
10177:
10173:
10167:
10164:
10162:
10159:
10157:
10156:Recursive set
10154:
10152:
10149:
10147:
10144:
10142:
10139:
10137:
10134:
10130:
10127:
10125:
10122:
10120:
10117:
10115:
10112:
10110:
10107:
10106:
10105:
10102:
10100:
10097:
10095:
10092:
10090:
10087:
10085:
10082:
10080:
10077:
10076:
10074:
10072:
10068:
10062:
10059:
10057:
10054:
10052:
10049:
10047:
10044:
10042:
10039:
10037:
10034:
10032:
10029:
10025:
10022:
10020:
10017:
10015:
10012:
10011:
10010:
10007:
10005:
10002:
10000:
9997:
9995:
9992:
9990:
9987:
9985:
9982:
9978:
9975:
9974:
9973:
9970:
9966:
9965:of arithmetic
9963:
9962:
9961:
9958:
9954:
9951:
9949:
9946:
9944:
9941:
9939:
9936:
9934:
9931:
9930:
9929:
9926:
9922:
9919:
9917:
9914:
9913:
9912:
9909:
9908:
9906:
9904:
9900:
9894:
9891:
9889:
9886:
9884:
9881:
9879:
9876:
9873:
9872:from ZFC
9869:
9866:
9864:
9861:
9855:
9852:
9851:
9850:
9847:
9845:
9842:
9840:
9837:
9836:
9835:
9832:
9830:
9827:
9825:
9822:
9820:
9817:
9815:
9812:
9810:
9807:
9805:
9802:
9801:
9799:
9797:
9793:
9783:
9782:
9778:
9777:
9772:
9771:non-Euclidean
9769:
9765:
9762:
9760:
9757:
9755:
9754:
9750:
9749:
9747:
9744:
9743:
9741:
9737:
9733:
9730:
9728:
9725:
9724:
9723:
9719:
9715:
9712:
9711:
9710:
9706:
9702:
9699:
9697:
9694:
9692:
9689:
9687:
9684:
9682:
9679:
9677:
9674:
9673:
9671:
9667:
9666:
9664:
9659:
9653:
9648:Example
9645:
9637:
9632:
9631:
9630:
9627:
9625:
9622:
9618:
9615:
9613:
9610:
9608:
9605:
9603:
9600:
9599:
9598:
9595:
9593:
9590:
9588:
9585:
9583:
9580:
9576:
9573:
9571:
9568:
9567:
9566:
9563:
9559:
9556:
9554:
9551:
9549:
9546:
9544:
9541:
9540:
9539:
9536:
9534:
9531:
9527:
9524:
9522:
9519:
9517:
9514:
9513:
9512:
9509:
9505:
9502:
9500:
9497:
9495:
9492:
9490:
9487:
9485:
9482:
9480:
9477:
9476:
9475:
9472:
9470:
9467:
9465:
9462:
9460:
9457:
9453:
9450:
9448:
9445:
9443:
9440:
9438:
9435:
9434:
9433:
9430:
9428:
9425:
9423:
9420:
9418:
9415:
9411:
9408:
9406:
9405:by definition
9403:
9402:
9401:
9398:
9394:
9391:
9390:
9389:
9386:
9384:
9381:
9379:
9376:
9374:
9371:
9369:
9366:
9365:
9362:
9359:
9357:
9353:
9348:
9342:
9338:
9328:
9325:
9323:
9320:
9318:
9315:
9313:
9310:
9308:
9305:
9303:
9300:
9298:
9295:
9293:
9292:Kripke–Platek
9290:
9288:
9285:
9281:
9278:
9276:
9273:
9272:
9271:
9268:
9267:
9265:
9261:
9253:
9250:
9249:
9248:
9245:
9243:
9240:
9236:
9233:
9232:
9231:
9228:
9226:
9223:
9221:
9218:
9216:
9213:
9211:
9208:
9205:
9201:
9197:
9194:
9190:
9187:
9185:
9182:
9180:
9177:
9176:
9175:
9171:
9168:
9167:
9165:
9163:
9159:
9155:
9147:
9144:
9142:
9139:
9137:
9136:constructible
9134:
9133:
9132:
9129:
9127:
9124:
9122:
9119:
9117:
9114:
9112:
9109:
9107:
9104:
9102:
9099:
9097:
9094:
9092:
9089:
9087:
9084:
9082:
9079:
9077:
9074:
9072:
9069:
9068:
9066:
9064:
9059:
9051:
9048:
9046:
9043:
9041:
9038:
9036:
9033:
9031:
9028:
9026:
9023:
9022:
9020:
9016:
9013:
9011:
9008:
9007:
9006:
9003:
9001:
8998:
8996:
8993:
8991:
8988:
8986:
8982:
8978:
8976:
8973:
8969:
8966:
8965:
8964:
8961:
8960:
8957:
8954:
8952:
8948:
8938:
8935:
8933:
8930:
8928:
8925:
8923:
8920:
8918:
8915:
8913:
8910:
8906:
8903:
8902:
8901:
8898:
8894:
8889:
8888:
8887:
8884:
8883:
8881:
8879:
8875:
8867:
8864:
8862:
8859:
8857:
8854:
8853:
8852:
8849:
8847:
8844:
8842:
8839:
8837:
8834:
8832:
8829:
8827:
8824:
8822:
8819:
8818:
8816:
8814:
8813:Propositional
8810:
8804:
8801:
8799:
8796:
8794:
8791:
8789:
8786:
8784:
8781:
8779:
8776:
8772:
8769:
8768:
8767:
8764:
8762:
8759:
8757:
8754:
8752:
8749:
8747:
8744:
8742:
8741:Logical truth
8739:
8737:
8734:
8733:
8731:
8729:
8725:
8722:
8720:
8716:
8710:
8707:
8705:
8702:
8700:
8697:
8695:
8692:
8690:
8687:
8685:
8681:
8677:
8673:
8671:
8668:
8666:
8663:
8661:
8657:
8654:
8653:
8651:
8649:
8643:
8638:
8632:
8629:
8627:
8624:
8622:
8619:
8617:
8614:
8612:
8609:
8607:
8604:
8602:
8599:
8597:
8594:
8592:
8589:
8587:
8584:
8582:
8579:
8577:
8574:
8570:
8567:
8566:
8565:
8562:
8561:
8559:
8555:
8551:
8544:
8539:
8537:
8532:
8530:
8525:
8524:
8521:
8509:
8508:Ernst Zermelo
8506:
8504:
8501:
8499:
8496:
8494:
8493:Willard Quine
8491:
8489:
8486:
8484:
8481:
8479:
8476:
8474:
8471:
8469:
8466:
8464:
8461:
8459:
8456:
8454:
8451:
8450:
8448:
8446:
8445:Set theorists
8442:
8436:
8433:
8431:
8428:
8426:
8423:
8422:
8420:
8414:
8412:
8409:
8408:
8405:
8397:
8394:
8392:
8391:Kripke–Platek
8389:
8385:
8382:
8381:
8380:
8377:
8376:
8375:
8372:
8368:
8365:
8364:
8363:
8362:
8358:
8354:
8351:
8350:
8349:
8346:
8345:
8342:
8339:
8337:
8334:
8332:
8329:
8327:
8324:
8323:
8321:
8317:
8311:
8308:
8306:
8303:
8301:
8298:
8296:
8294:
8289:
8287:
8284:
8282:
8279:
8276:
8272:
8269:
8267:
8264:
8260:
8257:
8255:
8252:
8250:
8247:
8246:
8245:
8242:
8239:
8235:
8232:
8230:
8227:
8225:
8222:
8220:
8217:
8216:
8214:
8211:
8207:
8201:
8198:
8196:
8193:
8191:
8188:
8186:
8183:
8181:
8178:
8176:
8173:
8171:
8168:
8164:
8161:
8159:
8156:
8155:
8154:
8151:
8149:
8146:
8144:
8141:
8139:
8136:
8134:
8131:
8128:
8124:
8121:
8119:
8116:
8114:
8111:
8110:
8108:
8102:
8099:
8098:
8095:
8089:
8086:
8084:
8081:
8079:
8076:
8074:
8071:
8069:
8066:
8064:
8061:
8059:
8056:
8053:
8050:
8048:
8045:
8044:
8042:
8040:
8036:
8028:
8027:specification
8025:
8023:
8020:
8019:
8018:
8015:
8014:
8011:
8008:
8006:
8003:
8001:
7998:
7996:
7993:
7991:
7988:
7986:
7983:
7981:
7978:
7976:
7973:
7971:
7968:
7966:
7963:
7959:
7956:
7954:
7951:
7949:
7946:
7945:
7944:
7941:
7939:
7936:
7935:
7933:
7931:
7927:
7922:
7912:
7909:
7908:
7906:
7902:
7898:
7891:
7886:
7884:
7879:
7877:
7872:
7871:
7868:
7859:
7858:
7853:
7850:
7845:
7840:
7836:
7832:
7831:
7829:
7825:
7822:
7816:
7815:
7810:
7805:
7801:
7800:
7795:
7791:
7787:
7786:Bagaria, Joan
7783:
7782:
7780:
7776:
7773:
7769:
7765:
7764:
7759:
7755:
7753:
7749:
7744:
7743:
7732:
7728:
7723:
7718:
7714:
7710:
7709:
7704:
7700:
7696:
7692:
7686:
7682:
7678:
7671:
7667:
7663:
7659:
7655:
7651:
7650:
7645:
7641:
7637:
7632:
7631:
7626:
7621:
7617:
7613:
7608:
7604:
7599:
7594:
7589:
7585:
7581:
7580:
7575:
7571:
7567:
7563:
7561:9780387906836
7557:
7552:
7551:
7544:
7540:
7536:
7532:
7528:
7524:
7520:
7516:
7512:
7506:
7502:
7498:
7494:
7490:
7486:
7484:0-7204-2285-X
7480:
7475:
7474:
7468:
7464:
7460:
7456:
7454:0-674-80207-1
7450:
7445:
7444:
7437:
7433:
7429:
7425:
7421:
7415:
7411:
7406:
7402:
7396:
7392:
7387:
7380:
7373:
7372:
7367:
7363:
7359:
7357:0-444-86839-9
7353:
7349:
7344:
7343:
7337:
7333:
7329:
7327:3-540-44085-2
7323:
7319:
7315:
7311:
7307:
7301:
7297:
7293:
7288:
7285:
7281:
7277:
7273:
7267:
7263:
7258:
7254:
7250:
7245:
7241:
7235:
7231:
7230:
7224:
7219:
7218:North-Holland
7215:
7211:
7207:
7203:
7199:
7194:
7192:3-540-61434-6
7188:
7184:
7183:
7178:
7174:
7170:
7166:
7162:
7156:
7152:
7148:
7144:
7140:
7136:
7132:
7131:Devlin, Keith
7128:
7124:
7122:0-521-59441-3
7118:
7114:
7113:
7107:
7102:
7101:
7094:
7089:
7084:
7080:
7076:
7072:
7067:
7062:
7061:
7055:
7051:
7050:
7037:
7032:
7025:
7024:Feferman 1996
7020:
7013:
7008:
7002:
6997:
6990:
6986:
6980:
6973:
6968:
6961:
6956:
6949:
6944:
6937:
6932:
6925:
6920:
6914:, p. 10.
6913:
6908:
6902:
6897:
6890:
6885:
6878:
6873:
6866:
6861:
6854:
6849:
6845:
6835:
6832:
6830:
6827:
6825:
6822:
6820:
6817:
6815:
6812:
6811:
6810:
6808:
6801:
6798:
6796:
6793:
6791:
6788:
6787:
6781:
6779:
6775:
6772:have adopted
6771:
6767:
6763:
6759:
6755:
6751:
6747:
6743:
6739:
6734:
6732:
6728:
6724:
6720:
6716:
6712:
6708:
6704:
6700:
6695:
6692:
6687:
6683:
6679:
6675:
6671:
6666:
6664:
6663:universal set
6658:
6648:
6646:
6635:
6633:
6627:
6625:
6621:
6613:
6609:
6605:
6602:
6598:
6594:
6591:
6588:
6585:
6581:
6580:
6579:
6574:
6571:
6568:
6565:
6563:
6560:
6558:
6555:
6552:
6549:
6548:
6547:
6544:
6542:
6538:
6534:
6530:
6526:
6522:
6519:
6509:
6507:
6503:
6499:
6495:
6491:
6487:
6483:
6479:
6474:
6472:
6468:
6464:
6460:
6458:
6454:
6450:
6446:
6442:
6437:
6433:
6429:
6425:
6416:
6414:
6410:
6406:
6402:
6396:
6386:
6384:
6381:, and in the
6380:
6376:
6372:
6368:
6364:
6360:
6356:
6352:
6348:
6347:virtual class
6332:
6330:
6326:
6322:
6318:
6314:
6310:
6307: =
6306:
6302:
6299:
6294:
6289:
6287:
6283:
6279:
6274:
6272:
6268:
6264:
6260:
6256:
6252:
6248:
6244:
6240:
6236:
6231:
6229:
6225:
6221:
6217:
6213:
6209:
6203:
6193:
6191:
6187:
6186:infinite sets
6183:
6179:
6163:
6154:
6152:
6136:
6133:
6127:
6121:
6101:
6098:
6095:
6087:
6071:
6051:
6031:
6011:
6003:
5999:
5995:
5992:Given axioms
5977:
5971:
5959:
5953:
5947:
5936:
5922:
5914:
5913:least element
5898:
5890:
5874:
5866:
5850:
5830:
5823:
5807:
5800:
5784:
5776:
5772:
5768:
5762:
5758:
5754:
5744:
5741:
5727:
5707:
5699:
5695:
5691:
5673:
5667:
5664:
5661:
5658:
5655:
5652:
5649:
5643:
5637:
5619:
5605:
5585:
5562:
5545:
5527:
5521:
5518:
5515:
5509:
5506:
5503:
5497:
5491:
5485:
5474:
5460:
5440:
5420:
5398:
5389:
5386:
5383:
5377:
5374:
5371:
5365:
5350:
5347:
5344:
5333:
5319:
5299:
5279:
5271:
5255:
5245:
5235:
5222:
5210:
5203:
5176:
5172:
5165:
5162:
5156:
5150:
5144:
5141:
5138:
5132:
5126:
5120:
5117:
5114:
5111:
5105:
5102:
5099:
5089:
5080:
5073:
5069:
5058:
5037:
5031:
4999:
4972:
4945:
4942:
4939:
4936:
4933:
4910:
4887:
4867:
4861:
4855:
4852:
4829:
4823:
4809:
4806:
4803:
4800:
4796:
4792:
4789:
4786:
4783:
4779:
4775:
4772:
4769:
4766:
4762:
4758:
4755:
4752:
4749:
4745:
4741:
4738:
4735:
4732:
4728:
4721:
4711:
4709:
4693:
4673:
4653:
4645:
4629:
4626:
4623:
4620:
4597:
4591:
4583:
4567:
4547:
4527:
4518:
4504:
4479:
4459:
4456:
4453:
4450:
4447:
4441:
4432:
4429:
4426:
4416:
4407:
4395:
4391:
4388:
4379:
4376:
4373:
4367:
4352:
4348:
4341:
4336:
4332:
4323:
4319:
4312:
4301:
4287:
4267:
4247:
4242:
4238:
4234:
4231:
4228:
4223:
4219:
4215:
4212:
4209:
4206:
4203:
4200:
4192:
4188:
4172:
4163:
4161:
4157:
4151:
4147:
4124:
4108:
4105:
4102:
4099:
4096:
4093:
4087:
4081:
4078:
4075:
4072:
4066:
4056:
4048:
4034:
4007:
3980:
3958:
3952:
3949:
3946:
3930:
3927:
3924:
3921:
3918:
3915:
3906:
3899:
3892:
3870:
3832:
3799:
3786:
3780:
3777:
3774:
3771:
3768:
3739:
3736:
3733:
3727:
3721:
3718:
3715:
3701:
3695:
3685:
3683:
3675:
3656:
3647:
3644:
3641:
3635:
3629:
3626:
3623:
3614:
3608:
3602:
3591:
3577:
3557:
3537:
3517:
3507:
3497:
3483:
3475:
3459:
3451:
3433:
3424:
3421:
3418:
3409:
3403:
3400:
3397:
3391:
3388:
3385:
3382:
3376:
3373:
3365:
3351:
3331:
3311:
3303:
3298:
3296:
3292:
3287:
3273:
3253:
3233:
3213:
3190:
3184:
3164:
3144:
3124:
3121:
3118:
3110:
3109:
3092:
3089:
3086:
3078:
3065:
3058:
3041:
3033:
3017:
3014:
3011:
3005:
3002:
2999:
2976:
2973:
2970:
2967:
2964:
2958:
2955:
2947:
2929:
2920:
2914:
2911:
2908:
2897:
2881:
2869:
2866:
2861:
2857:
2853:
2850:
2847:
2844:
2841:
2836:
2832:
2828:
2823:
2819:
2815:
2812:
2806:
2803:
2797:
2794:
2791:
2779:
2776:
2773:
2767:
2761:
2753:
2749:
2742:
2737:
2733:
2724:
2720:
2713:
2702:
2688:
2668:
2646:
2642:
2638:
2635:
2632:
2627:
2623:
2619:
2616:
2613:
2610:
2590:
2570:
2562:
2561:
2541:
2532:
2526:
2523:
2520:
2517:
2514:
2503:
2489:
2466:
2460:
2440:
2418:
2408:
2404:
2396:
2393:
2390:
2387:
2379:
2376:
2365:
2347:
2343:
2335:
2332:
2329:
2321:
2295:
2289:
2279:
2277:
2274:
2269:
2256:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2217:
2205:
2202:
2195:
2170:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2134:
2125:
2122:
2119:
2116:
2110:
2098:
2095:
2092:
2086:
2077:
2066:
2064:
2063:disjoint sets
2048:
2028:
2008:
1988:
1978:
1968:
1954:
1934:
1912:
1903:
1900:
1897:
1891:
1888:
1885:
1879:
1867:
1864:
1861:
1855:
1852:
1849:
1843:
1834:
1828:
1817:
1814:
1801:
1795:
1792:
1789:
1783:
1780:
1777:
1771:
1765:
1759:
1756:
1753:
1747:
1744:
1741:
1735:
1712:
1709:
1706:
1686:
1678:
1660:
1654:
1651:
1648:
1639:
1636:
1633:
1627:
1624:
1621:
1615:
1606:
1600:
1589:
1585:
1575:
1573:
1569:
1565:
1561:
1542:
1539:
1536:
1530:
1519:
1513:
1512:for axiom 9.
1511:
1507:
1503:
1498:
1496:
1492:
1486:
1476:
1462:
1439:
1396:
1376:
1353:
1350:
1340:
1326:
1323:
1313:
1296:
1293:
1290:
1280:
1263:
1260:
1257:
1247:
1233:
1223:
1222:
1206:
1186:
1166:
1158:
1157:
1142:
1139:
1136:
1129:
1115:
1112:
1109:
1102:
1101:
1097:
1093:
1092:metavariables
1077:
1057:
1049:
1048:
1047:
1045:
1037:
1036:
1021:
1013:
1012:
997:
989:
988:
945:
944:
929:
909:
881:
880:
876:
875:
874:
871:
869:
863:
848:
828:
821:
806:
786:
779:
765:
762:
759:
752:
748:
732:
724:
720:
716:
710:
700:
698:
694:
690:
686:
682:
678:
674:
670:
666:
662:
620:
616:
595:
589:
586:
583:
580:
566:
562:
528:
517:
513:
489:
481:
477:
463:
458:
454:
428:
415:
411:
407:
403:
399:
398:Ernst Zermelo
394:
392:
388:
384:
380:
376:
372:
366:
356:
354:
350:
346:
342:
338:
333:
331:
315:
295:
272:
269:
266:
243:
223:
215:
210:
208:
204:
200:
196:
195:universal set
191:
187:
183:
179:
175:
171:
167:
163:
159:
156:
153:
148:
146:
142:
138:
134:
130:
126:
122:
118:
114:
110:
109:Ernst Zermelo
106:
102:
91:
88:
80:
77:February 2024
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
10258:
10056:Ultraproduct
9903:Model theory
9868:Independence
9804:Formal proof
9796:Proof theory
9779:
9752:
9709:real numbers
9681:second-order
9592:Substitution
9469:Metalanguage
9410:conservative
9383:Axiom schema
9327:Constructive
9297:Morse–Kelley
9269:
9263:Set theories
9242:Aleph number
9235:inaccessible
9141:Grothendieck
9025:intersection
8962:
8912:Higher-order
8900:Second-order
8846:Truth tables
8803:Venn diagram
8586:Formal proof
8458:Georg Cantor
8453:Paul Bernays
8384:Morse–Kelley
8373:
8359:
8292:
8291:Subset
8238:hereditarily
8200:Venn diagram
8158:ordered pair
8073:Intersection
8017:Axiom schema
7855:
7812:
7797:
7790:"Set Theory"
7779:Joan Bagaria
7777:articles by
7761:
7712:
7706:
7680:
7653:
7647:
7628:
7611:
7602:
7583:
7577:
7554:. Springer.
7549:
7534:
7522:
7496:
7472:
7442:
7431:
7409:
7390:
7370:
7341:
7320:. Springer.
7317:
7314:Jech, Thomas
7291:
7283:
7261:
7248:
7228:
7213:
7210:Lévy, Azriel
7181:
7153:. Springer.
7150:
7134:
7111:
7099:
7078:
7074:
7059:
7046:Bibliography
7031:
7019:
7007:
6996:
6984:
6979:
6967:
6955:
6943:
6931:
6924:Hatcher 1982
6919:
6907:
6896:
6884:
6872:
6867:, p. 10
6860:
6848:
6804:
6766:Mizar system
6735:
6696:
6667:
6660:
6641:
6628:
6617:
6584:inner models
6577:
6545:
6537:inner models
6515:
6512:Independence
6505:
6501:
6475:
6461:
6422:
6398:
6366:
6362:
6358:
6354:
6351:Quine (1969)
6346:
6343:
6324:
6320:
6316:
6312:
6308:
6304:
6300:
6292:
6290:
6275:
6270:
6266:
6262:
6258:
6254:
6250:
6246:
6243:well-founded
6234:
6232:
6227:
6223:
6219:
6205:
6181:
6155:
6151:Zorn's lemma
6084:, called a "
6001:
5997:
5993:
5991:
5865:linear order
5775:Kunen (1980)
5764:
5761:Zorn's lemma
5742:
5689:
5687:
5541:
5412:
5247:
5190:
4815:
4519:
4493:
4164:
4153:
3972:
3800:
3697:
3676:, or by the
3670:
3509:
3447:
3299:
3288:
3246:that leaves
3107:
3106:
3056:
3055:
2943:
2895:
2559:
2555:
2432:
2291:
2270:
2184:
1980:
1926:
1815:
1674:
1587:
1571:
1567:
1559:
1514:
1502:Kunen (1980)
1499:
1491:Kunen (1980)
1488:
1368:
1041:
1038:Brackets ( )
872:
864:
712:
692:
443:and the set
395:
375:Georg Cantor
368:
334:
211:
155:well-founded
149:
144:
140:
104:
98:
83:
74:
55:
10166:Type theory
10114:undecidable
10046:Truth value
9933:equivalence
9612:non-logical
9225:Enumeration
9215:Isomorphism
9162:cardinality
9146:Von Neumann
9111:Ultrafilter
9076:Uncountable
9010:equivalence
8927:Quantifiers
8917:Fixed-point
8886:First-order
8766:Consistency
8751:Proposition
8728:Traditional
8699:Lindström's
8689:Compactness
8631:Type theory
8576:Cardinality
8483:Thomas Jech
8326:Alternative
8305:Uncountable
8259:Ultrafilter
8118:Cardinality
8022:replacement
7970:Determinacy
7400:048642079-5
7282:bearing on
7177:Hájek, Petr
7012:Tarski 1939
6972:Hinman 2005
6795:Inner model
6518:independent
6436:consistency
6419:Consistency
6375:Levy (2002)
5966:well-orders
5822:well-orders
5771:well-orders
5694:Jech (2003)
4845:abbreviate
3266:outside so
778:means that
349:consistency
69:introducing
10297:Z notation
10281:Categories
9977:elementary
9670:arithmetic
9538:Quantifier
9516:functional
9388:Expression
9106:Transitive
9050:identities
9035:complement
8968:hereditary
8951:Set theory
8478:Kurt Gödel
8463:Paul Cohen
8300:Transitive
8068:Identities
8052:Complement
8039:Operations
8000:Regularity
7938:Adjunction
7897:Set theory
7835:derivation
7586:: 176–83.
7501:A K Peters
7296:A K Peters
7081:: 155–58.
6912:Kunen 1980
6865:Kunen 2007
6748:, and the
6717:(NBG) and
6655:See also:
6651:Criticisms
6645:multiverse
6465:studied a
6357:∈ {
6178:finite set
4804:{0,1,2,3}
4193:are among
4000:, the set
3448:Thus, the
3304:, denoted
2322:predicate
2021:such that
1564:free logic
1483:See also:
707:See also:
389:, such as
371:set theory
182:urelements
152:hereditary
101:set theory
52:references
10248:Supertask
10151:Recursion
10109:decidable
9943:saturated
9921:of models
9844:deductive
9839:axiomatic
9759:Hilbert's
9746:Euclidean
9727:canonical
9650:axiomatic
9582:Signature
9511:Predicate
9400:Extension
9322:Ackermann
9247:Operation
9126:Universal
9116:Recursive
9091:Singleton
9086:Inhabited
9071:Countable
9061:Types of
9045:power set
9015:partition
8932:Predicate
8878:Predicate
8793:Syllogism
8783:Soundness
8756:Inference
8746:Tautology
8648:paradoxes
8411:Paradoxes
8331:Axiomatic
8310:Universal
8286:Singleton
8281:Recursive
8224:Countable
8219:Amorphous
8078:Power set
7995:Power set
7953:dependent
7948:countable
7857:MathWorld
7768:EMS Press
7731:0016-2736
7715:: 29–47.
7670:120085563
7521:(1972) .
7495:(2001) .
7212:(1973) .
7133:(1996) .
7001:Link 2014
6713:. Unlike
6578:Remarks:
6494:power set
6467:subtheory
6134:∈
6099:∈
5951:∃
5945:∀
5665:⊆
5653:∈
5618:exactly:
5544:power set
5519:∈
5513:⇒
5507:⊆
5495:∀
5489:∃
5483:∀
5387:∈
5381:⇒
5375:∈
5363:∀
5357:⇔
5348:⊆
5272:of a set
5163:∈
5148:⇒
5142:∈
5130:∀
5127:∧
5118:∈
5112:∧
5103:∈
5094:¬
5087:∀
5078:∃
5067:∃
5000:∅
4856:∪
4642:then the
4624:∈
4528:φ
4502:∃
4460:φ
4457:∧
4451:∈
4439:∃
4436:⇒
4430:∈
4414:∀
4405:∃
4402:⇒
4396:φ
4386:∃
4383:⇒
4377:∈
4365:∀
4345:∀
4342:…
4329:∀
4316:∀
4310:∀
4288:φ
4232:…
4173:φ
4109:∈
4103:∧
4097:∈
4085:∃
4076:∈
4057:∪
4008:∪
3981:∪
3950:∈
3944:⇒
3931:∈
3925:∧
3919:∈
3904:∀
3897:∀
3890:∃
3879:∀
3645:∈
3636:∧
3627:∈
3612:∃
3606:∀
3600:∀
3484:∅
3422:∈
3413:¬
3410:∧
3401:∈
3392:∣
3386:∈
3374:∅
3332:φ
3312:∅
3302:empty set
3185:φ
3122:⊆
3090:∈
3015:∉
3009:⇔
3003:∈
2974:∉
2915:φ
2807:φ
2804:∧
2795:∈
2783:⇔
2777:∈
2765:∀
2759:∃
2746:∀
2743:…
2730:∀
2717:∀
2711:∀
2701:). Then:
2689:φ
2636:…
2591:φ
2571:φ
2527:φ
2518:∈
2461:φ
2394:≡
2380:∈
2333:≡
2248:∅
2239:∩
2233:∧
2227:∈
2215:∃
2212:⇒
2209:∅
2206:≠
2193:∀
2156:∈
2150:∧
2144:∈
2132:∃
2129:¬
2126:∧
2120:∈
2108:∃
2105:⇒
2096:∈
2084:∃
2075:∀
1901:∈
1895:⇔
1889:∈
1877:∀
1874:⇒
1865:∈
1859:⇔
1853:∈
1841:∀
1832:∀
1826:∀
1793:∈
1787:⇔
1781:∈
1769:∀
1766:∧
1757:∈
1751:⇔
1745:∈
1733:∀
1646:⇒
1637:∈
1631:⇔
1625:∈
1613:∀
1604:∀
1598:∀
1528:∃
1460:∃
1437:∀
1417:¬
1397:∨
1377:∧
1354:ϕ
1348:∃
1327:ϕ
1321:∀
1297:ψ
1294:∨
1291:ϕ
1264:ψ
1261:∧
1258:ϕ
1234:ϕ
1231:¬
1187:ψ
1167:ϕ
1140:∈
1022:∈
974:∃
954:∀
930:∨
910:∧
890:¬
763:∈
733:∈
723:signature
679:with the
661:power set
429:ω
425:ℵ
396:In 1908,
383:paradoxes
174:pure sets
10233:Logicism
10226:timeline
10202:Concrete
10061:Validity
10031:T-schema
10024:Kripke's
10019:Tarski's
10014:semantic
10004:Strength
9953:submodel
9948:spectrum
9916:function
9764:Tarski's
9753:Elements
9740:geometry
9696:Robinson
9617:variable
9602:function
9575:spectrum
9565:Sentence
9521:variable
9464:Language
9417:Relation
9378:Automata
9368:Alphabet
9352:language
9206:-jection
9184:codomain
9170:Function
9131:Universe
9101:Infinite
9005:Relation
8788:Validity
8778:Argument
8676:theorem,
8415:Problems
8319:Theories
8295:Superset
8271:Infinite
8100:Concepts
7980:Infinity
7904:Overview
7839:Metamath
7701:(1930).
7642:(1908).
7572:(1939).
7379:Archived
7348:Elsevier
7338:(1980).
7316:(2003).
7276:Fraenkel
7171:(1996).
7149:(2007).
7139:Springer
7056:(1965).
6805:Related
6784:See also
6770:metamath
6707:universe
6490:infinity
6383:Metamath
6114:one has
5773:, as in
4787:{0,1,2}
4776:{∅,{∅}}
4300:. Then:
4160:function
3680:and the
3293:and the
1677:equality
1568:infinite
162:entities
115:, is an
10175:Related
9972:Diagram
9870: (
9849:Hilbert
9834:Systems
9829:Theorem
9707:of the
9652:systems
9432:Formula
9427:Grammar
9343: (
9287:General
9000:Forcing
8985:Element
8905:Monadic
8680:paradox
8621:Theorem
8557:General
8353:General
8348:Zermelo
8254:subbase
8236: (
8175:Forcing
8153:Element
8125: (
8103:Methods
7990:Pairing
7796:(ed.).
7770:, 2001
7752:YouTube
7469:(ed.).
7272:Zermelo
7179:(ed.).
6620:forcing
6525:forcing
6413:theorem
6323:=
5688:Axioms
4187:formula
4185:be any
2273:ordinal
751:formula
659:is the
359:History
164:in the
65:improve
9938:finite
9701:Skolem
9654:
9629:Theory
9597:Symbol
9587:String
9570:atomic
9447:ground
9442:closed
9437:atomic
9393:ground
9356:syntax
9252:binary
9179:domain
9096:Finite
8861:finite
8719:Logics
8678:
8626:Theory
8244:Filter
8234:Finite
8170:Family
8113:Almost
7958:global
7943:Choice
7930:Axioms
7729:
7687:
7668:
7558:
7507:
7481:
7451:
7416:
7397:
7354:
7324:
7302:
7280:Skolem
7278:, and
7236:
7189:
7157:
7119:
6744:, the
6521:of ZFC
6498:choice
6496:, and
6471:models
6455:, and
6451:, the
6245:. And
5911:has a
5889:subset
5820:which
5759:, and
5270:subset
5205:ω
4880:where
4770:{0,1}
4584:, and
4582:domain
4411:
3057:within
2560:schema
2558:axiom
1479:Axioms
671:whose
608:where
178:models
170:axioms
54:, but
9928:Model
9676:Peano
9533:Proof
9373:Arity
9302:Naive
9189:image
9121:Fuzzy
9081:Empty
9030:union
8975:Class
8616:Model
8606:Lemma
8564:Axiom
8336:Naive
8266:Fuzzy
8229:Empty
8212:types
8163:tuple
8133:Class
8127:large
8088:Union
8005:Union
7792:. In
7758:"ZFC"
7666:S2CID
7382:(PDF)
7375:(PDF)
7175:. In
6840:Notes
6529:model
6480:that
6176:is a
6044:from
5863:is a
5268:is a
5024:then
4644:range
4156:image
3700:union
2992:then
2948:(let
1485:Axiom
10051:Type
9854:list
9658:list
9635:list
9624:Term
9558:rank
9452:open
9346:list
9158:Maps
9063:sets
8922:Free
8892:list
8642:list
8569:list
8249:base
7727:ISSN
7685:ISBN
7556:ISBN
7505:ISBN
7479:ISBN
7449:ISBN
7414:ISBN
7395:ISBN
7352:ISBN
7322:ISBN
7300:ISBN
7234:ISBN
7187:ISBN
7155:ISBN
7117:ISBN
6768:and
6684:and
6672:and
6595:The
6241:and
6239:pure
4816:Let
4759:{∅}
4753:{0}
4148:and
3698:The
3570:and
3530:and
3105:has
2276:rank
2061:are
2041:and
1947:and
1452:and
1389:and
1179:and
1159:Let
1070:and
1050:Let
377:and
335:The
308:and
236:and
111:and
9738:of
9720:of
9668:of
9200:Sur
9174:Map
8981:Ur-
8963:Set
8210:Set
7837:in
7750:on
7717:doi
7658:doi
7588:doi
7284:ZFC
7083:doi
6756:or
6680:).
6361:| F
6331:".
6182:1–8
5891:of
5867:on
5690:1–8
4962:is
4736:{}
4646:of
3758:is
3510:If
3108:not
2405:mod
2344:mod
1699:",
1560:set
1090:be
862:).
717:in
385:in
158:set
141:ZFC
99:In
18:ZFC
10283::
10124:NP
9748::
9742::
9672::
9349:),
9204:Bi
9196:In
7854:.
7833:A
7781::
7766:,
7760:,
7725:.
7713:16
7711:.
7705:.
7664:.
7654:65
7652:.
7646:.
7627:.
7614:.
7584:32
7582:.
7576:.
7537:.
7503:.
7377:.
7350:.
7346:.
7298:.
7294:.
7274:,
7264:.
7251:.
7216:.
7208:;
7204:;
7137:.
7079:19
7077:.
7073:.
6809::
6665:.
6492:,
6459:.
6447::
6377:,
6288:.
6230:.
6153:.
5935:.
5755:,
5473::
5332::
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