1021:, thusâas FerreirĂłs observesâ"by forbidding 'circular' and 'ungrounded' sets, it incorporated one of the crucial motivations of TT âthe principle of the types of arguments". This 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy. FerreirĂłs writes that "Zermelo's 'layers' are essentially the same as the types in the contemporary versions of simple TT offered by Gödel and Tarski. One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which transfinite types are allowed. (Once we have adopted an impredicative standpoint, abandoning the idea that classes are constructed, it is not unnatural to accept transfinite types.) Thus, simple TT and ZFC could now be regarded as systems that 'talk' essentially about the same intended objects. The main difference is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order 'description' of the cumulative hierarchy is much weaker, as is shown by the existence of countable models (Skolem's paradox), but it enjoys some important advantages."
1542:
a true catalogue of those that list themselves. However, just as the librarian cannot go wrong with the first master catalogue, he is doomed to fail with the second. When it comes to the 'catalogue of all catalogues that do not list themselves', the librarian cannot include it in its own listing, because then it would include itself, and so belong in the other catalogue, that of catalogues that do include themselves. However, if the librarian leaves it out, the catalogue is incomplete. Either way, it can never be a true master catalogue of catalogues that do not list themselves.
6089:
916:. Therefore, the presence of contradictions like Russell's paradox in an axiomatic set theory is disastrous; since if any formula can be proved true it destroys the conventional meaning of truth and falsity. Further, since set theory was seen as the basis for an axiomatic development of all other branches of mathematics, Russell's paradox threatened the foundations of mathematics as a whole. This motivated a great deal of research around the turn of the 20th century to develop a consistent (contradiction-free) set theory.
2721:
39:
2224:"; this seems to contradict the contemporary notion of a "function in extension"; see Frege's wording at page 128: "Incidentally, it seems to me that the expression 'a predicate is predicated of itself' is not exact. ...Therefore I would prefer to say that 'a concept is predicated of its own extension' ". But he waffles at the end of his suggestion that a function-as-concept-in-extension can be written as predicated of its function. van Heijenoort cites Quine: "For a late and thorough study of Frege's "way out", see
3741:
3671:
2962:
1370:
3661:
1436:
1329:
1538:
of the library's books is self evident. Now imagine that all these catalogues are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master cataloguesâone of all the catalogues that list themselves, and one of all those that do not.
1541:
The question is: should these master catalogues list themselves? The 'catalogue of all catalogues that list themselves' is no problem. If the librarian does not include it in its own listing, it remains a true catalogue of those catalogues that do include themselves. If he does include it, it remains
1518:
An easy refutation of the "layman's versions" such as the barber paradox seems to be that no such barber exists, or that the barber is not a man, and so can exist without paradox. The whole point of
Russell's paradox is that the answer "such a set does not exist" means the definition of the notion of
2174:
van
Heijenoort's commentary, cf van Heijenoort 1967:126; Frege starts his analysis by this exceptionally honest comment : "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position
1537:
One way that the paradox has been dramatised is as follows: Suppose that every public library has to compile a catalogue of all its books. Since the catalogue is itself one of the library's books, some librarians include it in the catalogue for completeness; while others leave it out as it being one
1208:
to
Hilbert; as noted above, Frege's last volume mentioned the paradox that Russell had communicated to Frege. After receiving Frege's last volume, on 7 November 1903, Hilbert wrote a letter to Frege in which he said, referring to Russell's paradox, "I believe Dr. Zermelo discovered it three or four
891:
In particular, there was no distinction between sets and proper classes as collections of objects. Additionally, the existence of each of the elements of a collection was seen as sufficient for the existence of the set of said elements. However, paradoxes such as
Russell's and Burali-Forti's showed
1192:
in Cantor's naive set theory. He states: "And yet, even the elementary form that
Russell gave to the set-theoretic antinomies could have persuaded them that the solution of these difficulties is not to be sought in the surrender of well-ordering but only in a suitable restriction of the notion of
1001:
has a notion of layers that resemble types. Zermelo himself never accepted Skolem's formulation of ZFC using the language of first-order logic. As José Ferreirós notes, Zermelo insisted instead that "propositional functions (conditions or predicates) used for separating off subsets, as well as the
2065:
I had an intellectual set-back . Cantor had a proof that there is no greatest number, and it seemed to me that the number of all the things in the world ought to be the greatest possible. Accordingly, I examined his proof with some minuteness, and endeavoured to apply it to the class of all the
251:
is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is
Russell's paradox. In symbols:
886:
A set is an arbitrary collection of objects, absolutely no restriction being placed on the nature and number of these objects, the elements of the set in question. The elements constitute and determine the set as such, without any ordering or relationship of any kind between
1228:
The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition
1164:
Before taking leave of fundamental questions, it is necessary to examine more in detail the singular contradiction, already mentioned, with regard to predicates not predicable of themselves. ... I may mention that I was led to it in the endeavour to reconcile Cantor's
1689:
with "denote": The denoter (number) that denotes all denoters (numbers) that do not denote themselves. (In this paradox, all descriptions of numbers get an assigned number. The term "that denotes all denoters (numbers) that do not denote themselves" is here called
367:). The main difference between Russell's and Zermelo's solution to the paradox is that Zermelo modified the axioms of set theory while maintaining a standard logical language, while Russell modified the logical language itself. The language of ZFC, with the
1173:. Frege responded to Russell very quickly; his letter dated 22 June 1902 appeared, with van Heijenoort's commentary in Heijenoort 1967:126â127. Frege then wrote an appendix admitting to the paradox, and proposed a solution that Russell would endorse in his
383:
Most sets commonly encountered are not members of themselves. Let us call a set "normal" if it is not a member of itself, and "abnormal" if it is a member of itself. Clearly every set must be either normal or abnormal. For example, consider the set of all
1136:
There is just one point where I have encountered a difficulty. You state (p. 17 ) that a function too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let
2200:
that determine equal classes must be equivalent. As it seems very likely that this is the true solution, the reader is strongly recommended to examine Frege's argument on the point" (Russell 1903:522); The abbreviation Gg. stands for Frege's
2214:
Livio states that "While Frege did make some desperate attempts to remedy his axiom system, he was unsuccessful. The conclusion appeared to be disastrous ..." Livio 2009:188. But van
Heijenoort in his commentary before Frege's (1902)
1149:
is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection does not form a
330:
2066:
things there are. This led me to consider those classes which are not members of themselves, and to ask whether the class of such classes is or is not a member of itself. I found that either answer implies its contradictory".
550:
2219:
describes Frege's proposed "way out" in some detailâthe matter has to do with the " 'transformation of the generalization of an equality into an equality of courses-of-values. For Frege a function is something incomplete,
2196:. The second volume of Gg., which appeared too late to be noticed in the Appendix, contains an interesting discussion of the contradiction (pp. 253â265), suggesting that the solution is to be found by denying that two
1514:
supposes a barber who shaves all men who do not shave themselves and only men who do not shave themselves. When one thinks about whether the barber should shave himself or not, a similar paradox begins to emerge.
2300:
in van
Heijenoort 1967:183â198. Livio 2009:191 reports that Zermelo "discovered Russell's paradox independently as early as 1900"; Livio in turn cites Ewald 1996 and van Heijenoort 1967 (cf Livio 2009:268).
621:
784:
2057:, George Allen and Unwin Ltd., 1971, page 147: "At the end of the Lent Term , I went back to Fernhurst, where I set to work to write out the logical deduction of mathematics which afterwards became
1281:
by employing a theory of types they devised for this purpose. While they succeeded in grounding arithmetic in a fashion, it is not at all evident that they did so by purely logical means. While
1530:, in which words and meaning are the elements of the scenario rather than people and hair-cutting. Though it is easy to refute the barber's paradox by saying that such a barber does not (and
861:
392:. This set is not itself a square in the plane, thus it is not a member of itself and is therefore normal. In contrast, the complementary set that contains everything which is
1071:
operation. It is thus now possible again to reason about sets in a non-axiomatic fashion without running afoul of
Russell's paradox, namely by reasoning about the elements of
2600:
723:
694:
668:
644:
1200:, pp. 366â368. I had, however, discovered this antinomy myself, independently of Russell, and had communicated it prior to 1903 to Professor Hilbert among others.
456:
1315:
In 2001 A Centenary
International Conference celebrating the first hundred years of Russell's paradox was held in Munich and its proceedings have been published.
811:
258:
4468:
468:
157:
2593:
1510:
There are some versions of this paradox that are closer to real-life situations and may be easier to understand for non-logicians. For example, the
892:
the impossibility of this conception of set, by examples of collections of objects that do not form sets, despite all said objects being existent.
2916:
5143:
4205:
983:
927:
of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his
1249:. Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of
1868:
Remarkably, this letter was unpublished until van Heijenoort 1967âit appears with van Heijenoort's commentary at van Heijenoort 1967:124â125.
1177:, but was later considered by some to be unsatisfactory. For his part, Russell had his work at the printers and he added an appendix on the
1118:, he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal". In a 1902 letter, he announced the discovery to
415:
were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that
5226:
4367:
2948:
2586:
962:
ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set
228:â considered the founder of modern set theory â had already realized that his theory would lead to a contradiction, as he told Hilbert and
85:
3107:
2631:
1916:
Irvine, A. D., H. Deutsch (2021). "Russell's Paradox". Stanford Encyclopedia of Philosophy (Spring 2021 Edition), E. N. Zalta (ed.),
70:
2175:
I was placed in by a letter of Mr Bertrand Russell, just when the printing of this volume was nearing its completion" (Appendix of
1519:
set within a given theory is unsatisfactory. Note the difference between the statements "such a set does not exist" and "it is an
5540:
3255:
2898:
2776:
1676:
The original Russell's paradox with "contain": The container (Set) that contains all (containers) that do not contain themselves.
1051:, the structure of what some see as the "natural" objects described by ZFC eventually became clear: they are the elements of the
565:
6129:
2808:
150:
731:
5698:
2668:
2454:
2404:
2131:
2090:
2039:
1969:
1391:
1342:
1102:. Yet another approach is to define multiple membership relation with appropriately modified comprehension scheme, as in the
882:), a common conception of the idea of set was the "extensional concept of set", as recounted by von Neumann and Morgenstern:
75:
4486:
1792:
1285:
avoided the known paradoxes and allows the derivation of a great deal of mathematics, its system gave rise to new problems.
935:). (Avoiding paradox was not Zermelo's original intention, but instead to document which assumptions he used in proving the
5553:
4876:
3894:
3707:
2784:
411:
were normal, it would be contained in the set of all normal sets (itself), and therefore be abnormal; on the other hand if
205:
396:
a square in the plane is itself not a square in the plane, and so it is one of its own members and is therefore abnormal.
6139:
2998:
2517:
2498:
355:. In particular, Zermelo's axioms restricted the unlimited comprehension principle. With the additional contributions of
1646:
5558:
5548:
5285:
5138:
4491:
4222:
3316:
2966:
1044:
by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything.
4482:
1297:
5694:
3285:
3122:
3047:
2943:
2429:
1948:
1497:
1479:
1417:
1356:
143:
79:
5036:
1461:
1399:
5791:
5535:
4360:
3541:
2880:
2354:
208:
leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician
1128:
and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of
5096:
4789:
4200:
3794:
2892:
2886:
2824:
2330:
368:
360:
247:
be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If
17:
4530:
4080:
1188:(published at the same time he published "the first axiomatic set theory") laid claim to prior discovery of the
6052:
5754:
5517:
5512:
5337:
4758:
4442:
2910:
2874:
2676:
1446:
1395:
1259:
436:
110:
6047:
5830:
5747:
5460:
5391:
5268:
4510:
3974:
3853:
3534:
2736:
1903:
1786:
1156:
6124:
5972:
5798:
5484:
5118:
4717:
4217:
3358:
2832:
2816:
2639:
823:
105:
5850:
5845:
5455:
5194:
5123:
4452:
4353:
4210:
3848:
3811:
3188:
1680:
1527:
1103:
1099:
5779:
5369:
4763:
4731:
4422:
3310:
3250:
2760:
2701:
1798:
1759:
1708:
556:
431:" is used in various ways. In one usage, naive set theory is a formal theory, that is formulated in a
100:
3865:
3305:
1986:
221:
6119:
6069:
6018:
5915:
5413:
5374:
4851:
4496:
3899:
3784:
3772:
3767:
3378:
3353:
3193:
2933:
1927:
Bernhard Rang, Wolfgang Thomas: Zermelo's Discovery of the "Russell Paradox", Historia Mathematica 8.
1735:
1348:
1080:
790:
90:
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5231:
5214:
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2244:
and incorporated in the revised edition (1959), 253â260" (cf REFERENCES in van Heijenoort 1967:649)
1523:". It is like the difference between saying "There is no bucket" and saying "The bucket is empty".
1380:
878:
Prior to Russell's paradox (and to other similar paradoxes discovered around the time, such as the
5742:
5719:
5680:
5566:
5507:
5153:
5073:
4917:
4861:
4474:
4319:
4237:
4112:
4064:
3878:
3801:
3601:
3561:
3438:
3235:
3153:
3057:
3052:
2991:
2396:
1814:
1683:
with "describer": The describer (word) that describes all words, that do not describe themselves.
1457:
1384:
1091:
814:
699:
459:
343:, hence undermining Frege's attempt to reduce mathematics to logic and calling into question the
183:
2279:
van Heijenoort 1967:190â191. In the section before this he objects strenuously to the notion of
1145:
be predicated of itself? From each answer its opposite follows. Therefore we must conclude that
6032:
5759:
5737:
5704:
5597:
5443:
5428:
5401:
5352:
5236:
5171:
4996:
4962:
4957:
4831:
4662:
4639:
4271:
4152:
3964:
3777:
3478:
3368:
2654:
2359:
2197:
2004:
1268:
1129:
901:
673:
236:
125:
2192:
cf van Heijenoort's commentary, cf van Heijenoort 1967:126. The added text reads as follows: "
2121:
2080:
1169:
Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his
653:
629:
5962:
5815:
5607:
5325:
5061:
4967:
4826:
4811:
4692:
4667:
4187:
4157:
4101:
4021:
4001:
3979:
3636:
3621:
3596:
3591:
3519:
3463:
3443:
3348:
3183:
3077:
2752:
1660:
1453:
1273:
1064:
936:
2017:
1555:
As illustrated above for the barber paradox, Russell's paradox is not hard to extend. Take:
1079:
to think of sets in this way is a point of contention among the rival points of view on the
5935:
5897:
5774:
5578:
5418:
5342:
5320:
5148:
5106:
5005:
4972:
4836:
4624:
4261:
4085:
4016:
3969:
3909:
3789:
3641:
3626:
3616:
3581:
3529:
3458:
3373:
3240:
3163:
3158:
3087:
2768:
2545:
1721:
1686:
1630:
1277:
hoping to achieve what Frege had been unable to do. They sought to banish the paradoxes of
1052:
998:
956:
879:
2418:
From Frege to Gödel: A Source Book in Mathematical Logic, 1879â1931, (third printing 1976)
2253:
Russell mentions this fact to Frege, cf van Heijenoort's commentary before Frege's (1902)
441:
325:{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}
8:
6064:
5955:
5940:
5920:
5877:
5764:
5714:
5640:
5585:
5522:
5315:
5310:
5258:
5026:
5015:
4687:
4587:
4515:
4506:
4502:
4437:
4432:
4256:
4167:
4075:
4070:
3884:
3826:
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3363:
3327:
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3127:
3097:
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3022:
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2800:
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2446:
2413:
1907:. 2d. ed. Reprint, New York: W. W. Norton & Company, 1996. (First published in 1903.)
1765:
1651:
1221:
1018:
1014:
928:
347:. Two influential ways of avoiding the paradox were both proposed in 1908: Russell's own
235:
According to the unrestricted comprehension principle, for any sufficiently well-defined
120:
6093:
5862:
5825:
5810:
5803:
5786:
5590:
5572:
5438:
5364:
5347:
5300:
5113:
5022:
4856:
4841:
4801:
4753:
4738:
4726:
4682:
4657:
4427:
4376:
4179:
4174:
3959:
3914:
3821:
3674:
3631:
3611:
3576:
3551:
3524:
3322:
3275:
3265:
3223:
3218:
3178:
3173:
3082:
3042:
2984:
1819:
1745:
1702:
1010:
1009:"; the modern interpretation given to this statement is that Zermelo wanted to include
796:
352:
171:
5046:
2530:
2508:
1917:
1534:) exist, it is impossible to say something similar about a meaningfully defined word.
1017:. Around 1930, Zermelo also introduced (apparently independently of von Neumann), the
6088:
6028:
5835:
5645:
5635:
5527:
5408:
5243:
5219:
5000:
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4889:
4866:
4743:
4712:
4677:
4572:
4407:
4036:
3873:
3836:
3806:
3730:
3664:
3646:
3586:
3566:
3556:
3483:
3468:
3388:
3383:
3213:
3208:
3168:
3037:
2527:
2450:
2425:
2400:
2318:
2127:
2086:
2035:
1965:
1944:
971:
372:
240:
1086:
Other solutions to Russell's paradox, with an underlying strategy closer to that of
6042:
6037:
5930:
5887:
5709:
5670:
5665:
5650:
5476:
5433:
5330:
5128:
5078:
4652:
4614:
4324:
4314:
4299:
4294:
4162:
3816:
3606:
3571:
3546:
3498:
3423:
3398:
3393:
3295:
3270:
2904:
2862:
2681:
2609:
2314:
1278:
1114:
Russell discovered the paradox in May or June 1901. By his own account in his 1919
1048:
940:
924:
428:
389:
356:
336:
229:
197:
115:
53:
2720:
1832: â Cyclic structure that goes through several levels in a hierarchical system
1209:
years ago". A written account of Zermelo's actual argument was discovered in the
545:{\displaystyle \forall x\,\forall y\,(\forall z\,(z\in x\iff z\in y)\implies x=y)}
38:
6023:
6013:
5967:
5950:
5905:
5867:
5769:
5689:
5496:
5423:
5396:
5384:
5290:
5204:
5178:
5133:
5101:
4902:
4704:
4647:
4597:
4562:
4520:
4193:
4131:
3949:
3762:
3514:
3493:
3488:
3448:
3290:
3280:
3117:
3112:
3102:
3067:
2644:
2512:
2392:
2280:
1739:
1560:
1124:
1095:
952:
905:
432:
364:
130:
2032:
Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics
978:
defined by Russell's paradox above cannot be constructed as a subset of any set
6008:
5987:
5945:
5925:
5820:
5675:
5273:
5263:
5253:
5248:
5182:
5056:
4932:
4821:
4816:
4794:
4395:
4329:
4126:
4107:
4011:
3996:
3953:
3889:
3831:
3453:
3428:
3418:
3413:
3343:
3143:
3032:
1936:
1824:
1670:
1656:
1511:
1214:
944:
217:
187:
2472:
1663:", concluding that the lyrics present a musical example of Russell's Paradox.
6113:
5982:
5660:
5167:
4952:
4942:
4912:
4897:
4567:
4334:
4136:
4050:
4045:
3433:
3300:
3148:
1835:
1749:
1729:
1141:
be the predicate: to be a predicate that cannot be predicated of itself. Can
1119:
920:
913:
340:
213:
209:
194:
4304:
2565:
1289:
5882:
5729:
5630:
5622:
5502:
5450:
5359:
5295:
5278:
5209:
5068:
4927:
4629:
4412:
4284:
4279:
4097:
4026:
3984:
3843:
3740:
3230:
3203:
3198:
2938:
2856:
2550:
1829:
1698:
1301:
991:
982:, and is therefore not a set in ZFC. In some extensions of ZFC, notably in
225:
2283:
as defined by Poincaré (and soon to be taken by Russell, too, in his 1908
1308:. This is very widelyâthough not universallyâregarded as having shown the
335:
Russell also showed that a version of the paradox could be derived in the
212:. However, Zermelo did not publish the idea, which remained known only to
5992:
5872:
5051:
5041:
4988:
4672:
4592:
4577:
4457:
4402:
4309:
3944:
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1781:
1769:
1567:
1178:
1087:
867:
348:
190:
2034:(2nd ed.). Springer. § Zermelo's cumulative hierarchy pp. 374-378.
1892:. Translated by Hans Kaal., University of Chicago Press, Chicago, 1980.
4922:
4777:
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4554:
4289:
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2309:
B. Rang and W. Thomas, "Zermelo's discovery of the 'Russell Paradox'",
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Appendix. Completeness of quantification theory. Loewenheim's theorem
2007:(1967), pp.176â178. Ph.D dissertation, University of British Columbia
1520:
1068:
1060:
947:, and by Zermelo himself resulted in the axiomatic set theory called
2976:
2205:. Begriffsschriftlich abgeleitet. Vol. I. Jena, 1893. Vol. II. 1903.
1464:. Statements consisting only of original research should be removed.
1369:
6059:
5857:
5305:
5010:
4604:
2578:
2240:, enclosed as a pamphlet with part of the third printing (1955) of
1753:
1309:
1189:
939:.) Modifications to this axiomatic theory proposed in the 1920s by
344:
2493:
5655:
4447:
3008:
1959:
1888:
Russell, Bertrand, "Correspondence with Frege}. In Gottlob Frege
2473:"Russell's Paradox - a simple explanation of a profound problem"
4117:
3939:
2525:
1241:
must have different meanings, since the inner one has the form
955:
ceased to be controversial, and ZFC has remained the canonical
385:
1855:
In the following, p. 17 refers to a page in the original
5199:
4545:
4390:
3989:
3749:
3685:
2119:
1859:, and page 23 refers to the same page in van Heijenoort 1967
3072:
1160:, where he repeated his first encounter with the paradox:
616:{\displaystyle \exists y\forall x(x\in y\iff \varphi (x))}
948:
243:
of all and only the objects that have that property. Let
779:{\displaystyle \exists y\forall x(x\in y\iff x\notin x)}
339:
constructed by the German philosopher and mathematician
1803:
Pages displaying short descriptions of redirect targets
1224:
proposed to "dispose" of Russell's paradox as follows:
1742:
is inconsistent, by means of a self-negating statement
1154:
Russell would go on to cover it at length in his 1903
866:
a contradiction. Therefore, this naive set theory is
359:, Zermelo set theory developed into the now-standard
1953:
826:
799:
734:
702:
676:
656:
632:
568:
471:
444:
261:
951:. This theory became widely accepted once Zermelo's
2266:van Heijenoort's commentary before Zermelo (1908a)
1047:Through the work of Zermelo and others, especially
419:is neither normal nor abnormal: Russell's paradox.
2630:
2285:Mathematical logic as based on the theory of types
1545:
1292:in 1930â31 proved that while the logic of much of
855:
805:
778:
717:
688:
662:
638:
615:
544:
450:
324:
2298:A new proof of the possibility of a well-ordering
1186:A new proof of the possibility of a well-ordering
6111:
2618:British philosopher, logician, and social critic
1795: â Limitative results in mathematical logic
2917:Henrietta Stanley, Baroness Stanley of Alderley
2667:
2268:Investigations in the foundations of set theory
1838: â Mathematical set containing all objects
1312:program of Frege to be impossible to complete.
1193:set". Footnote 9 is where he stakes his claim:
2412:
2355:"Play That Funky Music Was No. 1 40 Years Ago"
2161:cf van Heijenoort's commentary before Frege's
2029:
1960:A.A. Fraenkel; Y. Bar-Hillel; A. Levy (1973).
4361:
3701:
2992:
2594:
2570:
2078:
2061:. I thought the work was nearly finished but
2023:
1985:Irvine, Andrew David; Deutsch, Harry (2014).
1984:
1890:Philosophical and Mathematical Correspondence
873:
151:
2183:, p. 279, translation by Michael Beaney
1666:Paradoxes that fall in this scheme include:
1526:A notable exception to the above may be the
399:Now we consider the set of all normal sets,
291:
273:
200:in 1901. Russell's paradox shows that every
1398:. Unsourced material may be challenged and
1357:Learn how and when to remove these messages
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363:(commonly known as ZFC when including the
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2020:" (1988). Association for Symbolic Logic.
1801: â Proposition in mathematical logic
1550:
1498:Learn how and when to remove this message
1480:Learn how and when to remove this message
1418:Learn how and when to remove this message
495:
485:
478:
1594:Sometimes the "all" is replaced by "all
1002:replacement functions, can be 'entirely
997:ZFC is silent about types, although the
378:
2899:Katharine Russell, Viscountess Amberley
2777:Introduction to Mathematical Philosophy
2074:
2072:
1991:The Stanford Encyclopedia of Philosophy
1257:. That disposes of Russell's paradox. (
1116:Introduction to Mathematical Philosophy
14:
6112:
4375:
2809:In Praise of Idleness and Other Essays
2506:
2470:
2386:
2120:Gottlob Frege, Michael Beaney (1997),
2082:One hundred years of Russell's paradox
422:
4349:
3689:
3006:
2980:
2582:
2569:
2526:
2437:
2055:The Autobiography of Bertrand Russell
1296:, now known as first-order logic, is
1032:that consists of exactly the sets in
856:{\displaystyle y\in y\iff y\notin y,}
2785:Free Thought and Official Propaganda
2608:
2177:Grundgesetze der Arithmetik, vol. II
2069:
2010:
1714:
1429:
1396:adding citations to reliable sources
1363:
1322:
1036:that are not members of themselves.
984:von NeumannâBernaysâGödel set theory
206:unrestricted comprehension principle
2967:Category: Works by Bertrand Russell
2518:Stanford Encyclopedia of Philosophy
2499:Internet Encyclopedia of Philosophy
2387:Potter, Michael (15 January 2004),
1613:s all (and only those) that do not
1318:
1304:is necessarily incomplete if it is
24:
3317:What the Tortoise Said to Achilles
2143:. Also van Heijenoort 1967:124â125
2085:, Walter de Gruyter, p. 350,
1586:s all (and only those) who do not
741:
735:
575:
569:
489:
479:
472:
25:
6151:
2464:
1338:This section has multiple issues.
1028:, it is possible to define a set
912:proposition can be proved from a
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3739:
3670:
3669:
3659:
2961:
2960:
2881:Conrad Russell, 5th Earl Russell
2719:
2287:cf van Heijenoort 1967:150â182).
1434:
1368:
1327:
37:
2893:John Russell, Viscount Amberley
2887:Frank Russell, 2nd Earl Russell
2825:A History of Western Philosophy
2347:
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1941:Vita Mathematica - Georg Cantor
1862:
1849:
1793:Gödel's incompleteness theorems
1546:Applications and related topics
1346:or discuss these issues on the
1245:and the outer one has the form
1122:of the paradox in Frege's 1879
403:, and try to determine whether
3715:
2911:John Russell, 1st Earl Russell
2875:John Russell, 4th Earl Russell
2313:, v. 8 n. 1, 1981, pp. 15â22.
1997:
1978:
1964:. Elsevier. pp. 156â157.
1930:
1921:
1910:
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1882:
1760:smallest uninteresting integer
1260:Tractatus Logico-Philosophicus
1233:, in which the outer function
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6130:Paradoxes of naive set theory
6048:History of mathematical logic
2737:The Principles of Mathematics
2507:Irvine, Andrew David (2016).
2389:Set Theory and its Philosophy
1989:. In Zalta, Edward N. (ed.).
1904:The Principles of Mathematics
1875:
1655:, "The Skywalker Intrusion",
1601:An example would be "paint":
1566:, that can be applied to its
1157:The Principles of Mathematics
220:, and other academics at the
5973:Primitive recursive function
2833:My Philosophical Development
2817:Power: A New Social Analysis
2420:, Cambridge, Massachusetts:
2319:10.1016/0315-0860(81)90002-1
2270:I in van Heijenoort 1967:199
1738:, showing that the original
1184:Ernst Zermelo in his (1908)
7:
2949:Professorship of Philosophy
2203:Grundgezetze der Arithmetik
2165:in van Heijenoort 1964:126.
1789: â Proof in set theory
1775:
1705:, whose origins are ancient
1460:the claims made and adding
1255:(do) : F(Ou) . Ou = Fu
1206:Grundgesetze der Arithmetik
1171:Grundgesetze der Arithmetik
1104:Double extension set theory
1011:higher-order quantification
718:{\displaystyle \varphi (x)}
371:, turned out to be that of
361:ZermeloâFraenkel set theory
224:. At the end of the 1890s,
86:Professorship of Philosophy
10:
6156:
6140:Self-referential paradoxes
5037:SchröderâBernstein theorem
4764:Monadic predicate calculus
4423:Foundations of mathematics
4206:von NeumannâBernaysâGödel
2761:The Problems of Philosophy
2677:RussellâEinstein Manifesto
2379:
2257:in van Heijenoort 1967:126
1787:Cantor's diagonal argument
1752:), which does not require
1697:"I am lying.", namely the
1109:
874:Philosophical implications
650:as a free variable inside
557:unrestricted comprehension
435:with a binary non-logical
407:is normal or abnormal. If
111:RussellâEinstein Manifesto
6083:
6070:Philosophy of mathematics
6019:Automated theorem proving
6001:
5896:
5728:
5621:
5473:
5190:
5166:
5144:Von NeumannâBernaysâGödel
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4007:One-to-one correspondence
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2623:
2616:
2576:
2571:Links to related articles
2228:": "On Frege's way out",
2005:Infinite Sets and Numbers
1962:Foundations of Set Theory
1271:wrote their three-volume
1204:Frege sent a copy of his
1175:Principles of Mathematics
1081:philosophy of mathematics
959:down to the present day.
791:existential instantiation
689:{\displaystyle x\notin x}
2793:Why I Am Not a Christian
2640:CoplestonâRussell debate
2471:Kaplan, Jeffrey (2022).
2422:Harvard University Press
2232:, 145â159; reprinted in
1842:
663:{\displaystyle \varphi }
639:{\displaystyle \varphi }
555:and the axiom schema of
458:, and that includes the
106:CoplestonâRussell debate
46:This article is part of
5720:Self-verifying theories
5541:Tarski's axiomatization
4492:Tarski's undefinability
4487:incompleteness theorems
3236:Paradoxes of set theory
2443:Is God a Mathematician?
2397:Oxford University Press
2198:propositional functions
2030:José Ferreirós (2008).
1815:Paradoxes of set theory
1799:Hilbert's first problem
1681:GrellingâNelson paradox
1671:The barber with "shave"
1528:GrellingâNelson paradox
1237:and the inner function
1100:ScottâPotter set theory
1065:transfinitely iterating
896:Set-theoretic responses
815:universal instantiation
460:axiom of extensionality
222:University of Göttingen
6094:Mathematics portal
5705:Proof of impossibility
5353:propositional variable
4663:Propositional calculus
3965:Constructible universe
3785:Constructibility (V=L)
2934:Appointment court case
2919:(maternal grandmother)
2913:(paternal grandfather)
2702:PeanoâRussell notation
2655:Theory of descriptions
2360:Minnesota Public Radio
2126:, Wiley, p. 253,
2079:Godehard Link (2004),
2018:Believing the Axioms I
1709:RussellâMyhill paradox
1551:Russell-like paradoxes
1269:Alfred North Whitehead
1265:
1202:
1167:
1152:
902:principle of explosion
889:
857:
807:
780:
719:
690:
664:
640:
617:
546:
452:
369:help of Thoralf Skolem
326:
126:Theory of descriptions
101:PeanoâRussell notation
91:Appointment court case
5963:Kolmogorov complexity
5916:Computably enumerable
5816:Model complete theory
5608:Principia Mathematica
4668:Propositional formula
4497:BanachâTarski paradox
4188:Principia Mathematica
4022:Transfinite induction
3881:(i.e. set difference)
2753:Principia Mathematica
2296:Ernst Zermelo (1908)
2059:Principia Mathematica
1736:KleeneâRosser paradox
1661:Play That Funky Music
1294:Principia Mathematica
1283:Principia Mathematica
1274:Principia Mathematica
1226:
1195:
1162:
1134:
937:well-ordering theorem
884:
858:
808:
781:
720:
691:
665:
641:
618:
547:
453:
379:Informal presentation
327:
184:set-theoretic paradox
27:Paradox in set theory
5911:ChurchâTuring thesis
5898:Computability theory
5107:continuum hypothesis
4625:Square of opposition
4483:Gödel's completeness
4262:Burali-Forti paradox
4017:Set-builder notation
3970:Continuum hypothesis
3910:Symmetric difference
3602:Kavka's toxin puzzle
3374:Income and fertility
2531:"Russell's Antinomy"
2447:Simon & Schuster
2414:van Heijenoort, Jean
2363:. September 27, 2016
2311:Historia Mathematica
1943:, BirkhÀuser, 1986,
1722:Burali-Forti paradox
1392:improve this section
1059:, built up from the
1053:von Neumann universe
1024:In ZFC, given a set
999:cumulative hierarchy
957:axiomatic set theory
880:Burali-Forti paradox
824:
797:
793:(reusing the symbol
732:
700:
674:
654:
630:
566:
469:
451:{\displaystyle \in }
442:
433:first-order language
259:
6125:Eponymous paradoxes
6065:Mathematical object
5956:P versus NP problem
5921:Computable function
5715:Reverse mathematics
5641:Logical consequence
5518:primitive recursive
5513:elementary function
5286:Free/bound variable
5139:TarskiâGrothendieck
4658:Logical connectives
4588:Logical equivalence
4438:Logical consequence
4223:TarskiâGrothendieck
3261:Temperature paradox
3184:Free choice paradox
3048:Fitch's knowability
2869:Edith Finch Russell
2851:Alys Pearsall Smith
2801:Marriage and Morals
2632:Views on philosophy
2546:"Russell's Paradox"
2509:"Russell's Paradox"
2494:"Russell's Paradox"
2063:in the month of May
1987:"Russell's Paradox"
1939:, Hans J. Ilgauds:
1901:Russell, Bertrand.
1659:analyzes the song "
1652:The Big Bang Theory
1574:Form the sentence:
1222:Ludwig Wittgenstein
1019:axiom of foundation
974:exists. The object
929:axiom of separation
423:Formal presentation
71:Views on philosophy
5863:Transfer principle
5826:Semantics of logic
5811:Categorical theory
5787:Non-standard model
5301:Logical connective
4428:Information theory
4377:Mathematical logic
3812:Limitation of size
3637:Prisoner's dilemma
3323:Heat death paradox
3311:Unexpected hanging
3276:Chicken or the egg
2528:Weisstein, Eric W.
2441:(6 January 2009),
1703:Epimenides paradox
1637:s all that do not
1445:possibly contains
1013:in order to avoid
853:
803:
776:
715:
686:
660:
636:
626:for any predicate
613:
542:
448:
353:Zermelo set theory
345:logicist programme
322:
180:Russell's antinomy
172:mathematical logic
6107:
6106:
6101:
6100:
6033:Abstract category
5836:Theories of truth
5646:Rule of inference
5636:Natural deduction
5617:
5616:
5162:
5161:
4867:Cartesian product
4772:
4771:
4678:Many-valued logic
4653:Boolean functions
4536:Russell's paradox
4511:diagonal argument
4408:First-order logic
4343:
4342:
4252:Russell's paradox
4201:ZermeloâFraenkel
4102:Dedekind-infinite
3975:Diagonal argument
3874:Cartesian product
3731:Set (mathematics)
3683:
3682:
3354:Arrow information
2974:
2973:
2853:(wife, 1894â1921)
2715:
2714:
2707:Russell's paradox
2690:
2689:
2663:
2662:
2456:978-0-7432-9405-8
2406:978-0-19-926973-0
2255:Letter to Russell
2217:Letter to Russell
2163:Letter to Russell
2133:978-0-631-19445-3
2092:978-3-11-017438-0
2041:978-3-7643-8350-3
1971:978-0-08-088705-0
1715:Related paradoxes
1687:Richard's paradox
1596:⟨V⟩
1588:⟨V⟩
1584:⟨V⟩
1580:⟨V⟩
1564:⟨V⟩
1508:
1507:
1500:
1490:
1489:
1482:
1447:original research
1428:
1427:
1420:
1361:
1179:doctrine of types
972:first-order logic
806:{\displaystyle y}
373:first-order logic
297:
265:
204:that contains an
186:published by the
176:Russell's paradox
168:
167:
96:Russell's paradox
63:
62:
16:(Redirected from
6147:
6120:Bertrand Russell
6092:
6091:
6043:History of logic
6038:Category of sets
5931:Decision problem
5710:Ordinal analysis
5651:Sequent calculus
5549:Boolean algebras
5489:
5488:
5463:
5434:logical/constant
5188:
5187:
5174:
5097:ZermeloâFraenkel
4848:Set operations:
4783:
4782:
4720:
4551:
4550:
4531:LöwenheimâSkolem
4418:Formal semantics
4370:
4363:
4356:
4347:
4346:
4325:Bertrand Russell
4315:John von Neumann
4300:Abraham Fraenkel
4295:Richard Dedekind
4257:Suslin's problem
4168:Cantor's theorem
3885:De Morgan's laws
3743:
3710:
3703:
3696:
3687:
3686:
3673:
3672:
3663:
3662:
3474:Service recovery
3328:Olbers's paradox
3028:Buridan's bridge
3001:
2994:
2987:
2978:
2977:
2964:
2963:
2944:Peace Foundation
2905:John Stuart Mill
2863:Patricia Russell
2723:
2692:
2691:
2682:Russell Tribunal
2669:Views on society
2665:
2664:
2650:Russell's teapot
2628:
2627:
2610:Bertrand Russell
2603:
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2589:
2580:
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2513:Zalta, Edward N.
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2331:"barber paradox"
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2181:The Frege Reader
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2153:
2152:Russell 1903:101
2150:
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2142:
2141:
2140:
2123:The Frege reader
2117:
2111:
2110:Russell 1920:136
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1319:Applied versions
1302:Peano arithmetic
1279:naive set theory
1075:. Whether it is
1049:John von Neumann
1015:Skolem's paradox
1008:
970:definable using
966:, any subset of
941:Abraham Fraenkel
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357:Abraham Fraenkel
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230:Richard Dedekind
198:Bertrand Russell
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121:Russell's teapot
116:Russell Tribunal
80:Peace Foundation
76:Views on society
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6135:1901 in science
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6079:
6024:Category theory
6014:Algebraic logic
5997:
5968:Lambda calculus
5906:Church encoding
5892:
5868:Truth predicate
5724:
5690:Complete theory
5613:
5482:
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5461:
5181: and
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5134:New Foundations
5102:axiom of choice
5085:
5047:Gödel numbering
4987: and
4979:
4883:
4768:
4718:
4699:
4648:Boolean algebra
4634:
4598:Equiconsistency
4563:Classical logic
4540:
4521:Halting problem
4509: and
4485: and
4473: and
4472:
4467:Theorems (
4462:
4379:
4374:
4344:
4339:
4266:
4245:
4229:
4194:New Foundations
4141:
4031:
3950:Cardinal number
3933:
3919:
3860:
3744:
3735:
3719:
3714:
3684:
3679:
3651:
3562:Decision-making
3508:Decision theory
3503:
3332:
3256:Hilbert's Hotel
3189:GrellingâNelson
3132:
3011:
3005:
2975:
2970:
2953:
2922:
2871:(wife, 1952â70)
2865:(wife, 1936â51)
2859:(wife, 1921â35)
2839:
2724:
2711:
2686:
2659:
2645:Logical atomism
2619:
2612:
2607:
2572:
2556:
2554:
2544:
2492:
2481:
2479:
2467:
2462:
2457:
2432:
2407:
2393:Clarendon Press
2382:
2377:
2376:
2366:
2364:
2353:
2352:
2348:
2339:
2337:
2329:
2328:
2324:
2308:
2304:
2295:
2291:
2281:impredicativity
2278:
2274:
2265:
2261:
2252:
2248:
2221:
2213:
2209:
2191:
2187:
2173:
2169:
2160:
2156:
2151:
2147:
2138:
2136:
2134:
2118:
2114:
2109:
2105:
2097:
2095:
2093:
2077:
2070:
2053:
2049:
2042:
2028:
2024:
2015:
2011:
2002:
1998:
1983:
1979:
1972:
1958:
1954:
1935:
1931:
1926:
1922:
1915:
1911:
1900:
1896:
1887:
1883:
1878:
1873:
1872:
1867:
1863:
1857:Begriffsschrift
1854:
1850:
1845:
1820:Quine's paradox
1802:
1778:
1746:Curry's paradox
1740:lambda calculus
1717:
1595:
1587:
1583:
1579:
1563:
1561:transitive verb
1553:
1548:
1504:
1493:
1492:
1491:
1486:
1475:
1469:
1466:
1451:
1439:
1435:
1424:
1413:
1407:
1404:
1389:
1373:
1332:
1328:
1321:
1125:Begriffsschrift
1112:
1096:New Foundations
1006:
986:, objects like
953:axiom of choice
906:classical logic
898:
876:
825:
822:
821:
798:
795:
794:
733:
730:
729:
701:
698:
697:
675:
672:
671:
655:
652:
651:
647:
631:
628:
627:
567:
564:
563:
470:
467:
466:
443:
440:
439:
425:
381:
365:axiom of choice
294:
262:
260:
257:
256:
239:, there is the
178:(also known as
164:
135:
131:Logical atomism
54:
52:
51:
50:
47:
45:
28:
23:
22:
18:Russell paradox
15:
12:
11:
5:
6153:
6143:
6142:
6137:
6132:
6127:
6122:
6105:
6104:
6099:
6098:
6084:
6081:
6080:
6078:
6077:
6072:
6067:
6062:
6057:
6056:
6055:
6045:
6040:
6035:
6026:
6021:
6016:
6011:
6009:Abstract logic
6005:
6003:
5999:
5998:
5996:
5995:
5990:
5988:Turing machine
5985:
5980:
5975:
5970:
5965:
5960:
5959:
5958:
5953:
5948:
5943:
5938:
5928:
5926:Computable set
5923:
5918:
5913:
5908:
5902:
5900:
5894:
5893:
5891:
5890:
5885:
5880:
5875:
5870:
5865:
5860:
5855:
5854:
5853:
5848:
5843:
5833:
5828:
5823:
5821:Satisfiability
5818:
5813:
5808:
5807:
5806:
5796:
5795:
5794:
5784:
5783:
5782:
5777:
5772:
5767:
5762:
5752:
5751:
5750:
5745:
5738:Interpretation
5734:
5732:
5726:
5725:
5723:
5722:
5717:
5712:
5707:
5702:
5692:
5687:
5686:
5685:
5684:
5683:
5673:
5668:
5658:
5653:
5648:
5643:
5638:
5633:
5627:
5625:
5619:
5618:
5615:
5614:
5612:
5611:
5603:
5602:
5601:
5600:
5595:
5594:
5593:
5588:
5583:
5563:
5562:
5561:
5559:minimal axioms
5556:
5545:
5544:
5543:
5532:
5531:
5530:
5525:
5520:
5515:
5510:
5505:
5492:
5490:
5471:
5470:
5468:
5467:
5466:
5465:
5453:
5448:
5447:
5446:
5441:
5436:
5431:
5421:
5416:
5411:
5406:
5405:
5404:
5399:
5389:
5388:
5387:
5382:
5377:
5372:
5362:
5357:
5356:
5355:
5350:
5345:
5335:
5334:
5333:
5328:
5323:
5318:
5313:
5308:
5298:
5293:
5288:
5283:
5282:
5281:
5276:
5271:
5266:
5256:
5251:
5249:Formation rule
5246:
5241:
5240:
5239:
5234:
5224:
5223:
5222:
5212:
5207:
5202:
5197:
5191:
5185:
5168:Formal systems
5164:
5163:
5160:
5159:
5157:
5156:
5151:
5146:
5141:
5136:
5131:
5126:
5121:
5116:
5111:
5110:
5109:
5104:
5093:
5091:
5087:
5086:
5084:
5083:
5082:
5081:
5071:
5066:
5065:
5064:
5057:Large cardinal
5054:
5049:
5044:
5039:
5034:
5020:
5019:
5018:
5013:
5008:
4993:
4991:
4981:
4980:
4978:
4977:
4976:
4975:
4970:
4965:
4955:
4950:
4945:
4940:
4935:
4930:
4925:
4920:
4915:
4910:
4905:
4900:
4894:
4892:
4885:
4884:
4882:
4881:
4880:
4879:
4874:
4869:
4864:
4859:
4854:
4846:
4845:
4844:
4839:
4829:
4824:
4822:Extensionality
4819:
4817:Ordinal number
4814:
4804:
4799:
4798:
4797:
4786:
4780:
4774:
4773:
4770:
4769:
4767:
4766:
4761:
4756:
4751:
4746:
4741:
4736:
4735:
4734:
4724:
4723:
4722:
4709:
4707:
4701:
4700:
4698:
4697:
4696:
4695:
4690:
4685:
4675:
4670:
4665:
4660:
4655:
4650:
4644:
4642:
4636:
4635:
4633:
4632:
4627:
4622:
4617:
4612:
4607:
4602:
4601:
4600:
4590:
4585:
4580:
4575:
4570:
4565:
4559:
4557:
4548:
4542:
4541:
4539:
4538:
4533:
4528:
4523:
4518:
4513:
4501:Cantor's
4499:
4494:
4489:
4479:
4477:
4464:
4463:
4461:
4460:
4455:
4450:
4445:
4440:
4435:
4430:
4425:
4420:
4415:
4410:
4405:
4400:
4399:
4398:
4387:
4385:
4381:
4380:
4373:
4372:
4365:
4358:
4350:
4341:
4340:
4338:
4337:
4332:
4330:Thoralf Skolem
4327:
4322:
4317:
4312:
4307:
4302:
4297:
4292:
4287:
4282:
4276:
4274:
4268:
4267:
4265:
4264:
4259:
4254:
4248:
4246:
4244:
4243:
4240:
4234:
4231:
4230:
4228:
4227:
4226:
4225:
4220:
4215:
4214:
4213:
4198:
4197:
4196:
4184:
4183:
4182:
4171:
4170:
4165:
4160:
4155:
4149:
4147:
4143:
4142:
4140:
4139:
4134:
4129:
4124:
4115:
4110:
4105:
4095:
4090:
4089:
4088:
4083:
4078:
4068:
4058:
4053:
4048:
4042:
4040:
4033:
4032:
4030:
4029:
4024:
4019:
4014:
4012:Ordinal number
4009:
4004:
3999:
3994:
3993:
3992:
3987:
3977:
3972:
3967:
3962:
3957:
3947:
3942:
3936:
3934:
3932:
3931:
3928:
3924:
3921:
3920:
3918:
3917:
3912:
3907:
3902:
3897:
3892:
3890:Disjoint union
3887:
3882:
3876:
3870:
3868:
3862:
3861:
3859:
3858:
3857:
3856:
3851:
3840:
3839:
3837:Martin's axiom
3834:
3829:
3824:
3819:
3814:
3809:
3804:
3802:Extensionality
3799:
3798:
3797:
3787:
3782:
3781:
3780:
3775:
3770:
3760:
3754:
3752:
3746:
3745:
3738:
3736:
3734:
3733:
3727:
3725:
3721:
3720:
3713:
3712:
3705:
3698:
3690:
3681:
3680:
3678:
3677:
3667:
3656:
3653:
3652:
3650:
3649:
3644:
3639:
3634:
3629:
3624:
3619:
3614:
3609:
3604:
3599:
3594:
3589:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3544:
3539:
3538:
3537:
3532:
3527:
3517:
3511:
3509:
3505:
3504:
3502:
3501:
3496:
3491:
3486:
3481:
3479:St. Petersburg
3476:
3471:
3466:
3461:
3456:
3451:
3446:
3441:
3436:
3431:
3426:
3421:
3416:
3411:
3406:
3401:
3396:
3391:
3386:
3381:
3376:
3371:
3366:
3361:
3356:
3351:
3346:
3340:
3338:
3334:
3333:
3331:
3330:
3325:
3320:
3313:
3308:
3303:
3298:
3293:
3288:
3283:
3278:
3273:
3268:
3263:
3258:
3253:
3248:
3243:
3238:
3233:
3228:
3227:
3226:
3221:
3216:
3211:
3206:
3196:
3191:
3186:
3181:
3176:
3171:
3166:
3161:
3156:
3151:
3146:
3140:
3138:
3134:
3133:
3131:
3130:
3125:
3120:
3115:
3110:
3108:Rule-following
3105:
3100:
3095:
3090:
3085:
3080:
3075:
3070:
3065:
3060:
3055:
3050:
3045:
3040:
3035:
3033:Dream argument
3030:
3025:
3019:
3017:
3013:
3012:
3004:
3003:
2996:
2989:
2981:
2972:
2971:
2958:
2955:
2954:
2952:
2951:
2946:
2941:
2936:
2930:
2928:
2924:
2923:
2921:
2920:
2914:
2908:
2902:
2896:
2890:
2884:
2878:
2872:
2866:
2860:
2854:
2847:
2845:
2841:
2840:
2838:
2837:
2829:
2821:
2813:
2805:
2797:
2789:
2781:
2773:
2765:
2757:
2749:
2741:
2732:
2730:
2726:
2725:
2718:
2716:
2713:
2712:
2710:
2709:
2704:
2698:
2696:
2688:
2687:
2685:
2684:
2679:
2673:
2671:
2661:
2660:
2658:
2657:
2652:
2647:
2642:
2636:
2634:
2625:
2621:
2620:
2617:
2614:
2613:
2606:
2605:
2598:
2591:
2583:
2577:
2574:
2573:
2564:
2563:
2542:
2523:
2504:
2489:
2488:
2466:
2465:External links
2463:
2461:
2460:
2455:
2435:
2430:
2410:
2405:
2383:
2381:
2378:
2375:
2374:
2346:
2322:
2302:
2289:
2272:
2259:
2246:
2207:
2185:
2167:
2154:
2145:
2132:
2112:
2103:
2091:
2068:
2047:
2040:
2022:
2009:
1996:
1977:
1970:
1952:
1937:Walter Purkert
1929:
1920:
1909:
1894:
1880:
1879:
1877:
1874:
1871:
1870:
1861:
1847:
1846:
1844:
1841:
1840:
1839:
1833:
1827:
1825:Self-reference
1822:
1817:
1812:
1805:
1796:
1790:
1784:
1777:
1774:
1773:
1772:
1763:
1756:
1743:
1732:
1730:well-orderings
1716:
1713:
1712:
1711:
1706:
1695:
1684:
1677:
1674:
1657:Sheldon Cooper
1643:
1642:
1631:representative
1619:
1618:
1592:
1591:
1572:
1571:
1552:
1549:
1547:
1544:
1512:barber paradox
1506:
1505:
1488:
1487:
1442:
1440:
1433:
1426:
1425:
1376:
1374:
1367:
1362:
1336:
1335:
1333:
1326:
1320:
1317:
1288:In any event,
1215:Edmund Husserl
1111:
1108:
992:proper classes
945:Thoralf Skolem
925:axiomatization
897:
894:
875:
872:
864:
863:
852:
849:
846:
843:
839:
835:
832:
829:
802:
787:
786:
775:
772:
769:
766:
762:
758:
755:
752:
749:
746:
743:
740:
737:
714:
711:
708:
705:
685:
682:
679:
659:
635:
624:
623:
612:
609:
606:
603:
600:
596:
592:
589:
586:
583:
580:
577:
574:
571:
553:
552:
541:
538:
535:
532:
528:
524:
521:
518:
515:
511:
507:
504:
501:
498:
494:
491:
488:
484:
481:
477:
474:
447:
424:
421:
380:
377:
333:
332:
321:
318:
315:
311:
307:
304:
301:
293:
290:
287:
284:
281:
278:
275:
272:
269:
218:Edmund Husserl
166:
165:
163:
162:
155:
148:
140:
137:
136:
134:
133:
128:
123:
118:
113:
108:
103:
98:
93:
88:
83:
73:
65:
64:
61:
60:
48:a series about
44:
42:
26:
9:
6:
4:
3:
2:
6152:
6141:
6138:
6136:
6133:
6131:
6128:
6126:
6123:
6121:
6118:
6117:
6115:
6096:
6095:
6090:
6082:
6076:
6073:
6071:
6068:
6066:
6063:
6061:
6058:
6054:
6051:
6050:
6049:
6046:
6044:
6041:
6039:
6036:
6034:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6006:
6004:
6000:
5994:
5991:
5989:
5986:
5984:
5983:Recursive set
5981:
5979:
5976:
5974:
5971:
5969:
5966:
5964:
5961:
5957:
5954:
5952:
5949:
5947:
5944:
5942:
5939:
5937:
5934:
5933:
5932:
5929:
5927:
5924:
5922:
5919:
5917:
5914:
5912:
5909:
5907:
5904:
5903:
5901:
5899:
5895:
5889:
5886:
5884:
5881:
5879:
5876:
5874:
5871:
5869:
5866:
5864:
5861:
5859:
5856:
5852:
5849:
5847:
5844:
5842:
5839:
5838:
5837:
5834:
5832:
5829:
5827:
5824:
5822:
5819:
5817:
5814:
5812:
5809:
5805:
5802:
5801:
5800:
5797:
5793:
5792:of arithmetic
5790:
5789:
5788:
5785:
5781:
5778:
5776:
5773:
5771:
5768:
5766:
5763:
5761:
5758:
5757:
5756:
5753:
5749:
5746:
5744:
5741:
5740:
5739:
5736:
5735:
5733:
5731:
5727:
5721:
5718:
5716:
5713:
5711:
5708:
5706:
5703:
5700:
5699:from ZFC
5696:
5693:
5691:
5688:
5682:
5679:
5678:
5677:
5674:
5672:
5669:
5667:
5664:
5663:
5662:
5659:
5657:
5654:
5652:
5649:
5647:
5644:
5642:
5639:
5637:
5634:
5632:
5629:
5628:
5626:
5624:
5620:
5610:
5609:
5605:
5604:
5599:
5598:non-Euclidean
5596:
5592:
5589:
5587:
5584:
5582:
5581:
5577:
5576:
5574:
5571:
5570:
5568:
5564:
5560:
5557:
5555:
5552:
5551:
5550:
5546:
5542:
5539:
5538:
5537:
5533:
5529:
5526:
5524:
5521:
5519:
5516:
5514:
5511:
5509:
5506:
5504:
5501:
5500:
5498:
5494:
5493:
5491:
5486:
5480:
5475:Example
5472:
5464:
5459:
5458:
5457:
5454:
5452:
5449:
5445:
5442:
5440:
5437:
5435:
5432:
5430:
5427:
5426:
5425:
5422:
5420:
5417:
5415:
5412:
5410:
5407:
5403:
5400:
5398:
5395:
5394:
5393:
5390:
5386:
5383:
5381:
5378:
5376:
5373:
5371:
5368:
5367:
5366:
5363:
5361:
5358:
5354:
5351:
5349:
5346:
5344:
5341:
5340:
5339:
5336:
5332:
5329:
5327:
5324:
5322:
5319:
5317:
5314:
5312:
5309:
5307:
5304:
5303:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5280:
5277:
5275:
5272:
5270:
5267:
5265:
5262:
5261:
5260:
5257:
5255:
5252:
5250:
5247:
5245:
5242:
5238:
5235:
5233:
5232:by definition
5230:
5229:
5228:
5225:
5221:
5218:
5217:
5216:
5213:
5211:
5208:
5206:
5203:
5201:
5198:
5196:
5193:
5192:
5189:
5186:
5184:
5180:
5175:
5169:
5165:
5155:
5152:
5150:
5147:
5145:
5142:
5140:
5137:
5135:
5132:
5130:
5127:
5125:
5122:
5120:
5119:KripkeâPlatek
5117:
5115:
5112:
5108:
5105:
5103:
5100:
5099:
5098:
5095:
5094:
5092:
5088:
5080:
5077:
5076:
5075:
5072:
5070:
5067:
5063:
5060:
5059:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5035:
5032:
5028:
5024:
5021:
5017:
5014:
5012:
5009:
5007:
5004:
5003:
5002:
4998:
4995:
4994:
4992:
4990:
4986:
4982:
4974:
4971:
4969:
4966:
4964:
4963:constructible
4961:
4960:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4939:
4936:
4934:
4931:
4929:
4926:
4924:
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4904:
4901:
4899:
4896:
4895:
4893:
4891:
4886:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4849:
4847:
4843:
4840:
4838:
4835:
4834:
4833:
4830:
4828:
4825:
4823:
4820:
4818:
4815:
4813:
4809:
4805:
4803:
4800:
4796:
4793:
4792:
4791:
4788:
4787:
4784:
4781:
4779:
4775:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4733:
4730:
4729:
4728:
4725:
4721:
4716:
4715:
4714:
4711:
4710:
4708:
4706:
4702:
4694:
4691:
4689:
4686:
4684:
4681:
4680:
4679:
4676:
4674:
4671:
4669:
4666:
4664:
4661:
4659:
4656:
4654:
4651:
4649:
4646:
4645:
4643:
4641:
4640:Propositional
4637:
4631:
4628:
4626:
4623:
4621:
4618:
4616:
4613:
4611:
4608:
4606:
4603:
4599:
4596:
4595:
4594:
4591:
4589:
4586:
4584:
4581:
4579:
4576:
4574:
4571:
4569:
4568:Logical truth
4566:
4564:
4561:
4560:
4558:
4556:
4552:
4549:
4547:
4543:
4537:
4534:
4532:
4529:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4508:
4504:
4500:
4498:
4495:
4493:
4490:
4488:
4484:
4481:
4480:
4478:
4476:
4470:
4465:
4459:
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4397:
4394:
4393:
4392:
4389:
4388:
4386:
4382:
4378:
4371:
4366:
4364:
4359:
4357:
4352:
4351:
4348:
4336:
4335:Ernst Zermelo
4333:
4331:
4328:
4326:
4323:
4321:
4320:Willard Quine
4318:
4316:
4313:
4311:
4308:
4306:
4303:
4301:
4298:
4296:
4293:
4291:
4288:
4286:
4283:
4281:
4278:
4277:
4275:
4273:
4272:Set theorists
4269:
4263:
4260:
4258:
4255:
4253:
4250:
4249:
4247:
4241:
4239:
4236:
4235:
4232:
4224:
4221:
4219:
4218:KripkeâPlatek
4216:
4212:
4209:
4208:
4207:
4204:
4203:
4202:
4199:
4195:
4192:
4191:
4190:
4189:
4185:
4181:
4178:
4177:
4176:
4173:
4172:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4150:
4148:
4144:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4121:
4116:
4114:
4111:
4109:
4106:
4103:
4099:
4096:
4094:
4091:
4087:
4084:
4082:
4079:
4077:
4074:
4073:
4072:
4069:
4066:
4062:
4059:
4057:
4054:
4052:
4049:
4047:
4044:
4043:
4041:
4038:
4034:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3991:
3988:
3986:
3983:
3982:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3955:
3951:
3948:
3946:
3943:
3941:
3938:
3937:
3935:
3929:
3926:
3925:
3922:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3880:
3877:
3875:
3872:
3871:
3869:
3867:
3863:
3855:
3854:specification
3852:
3850:
3847:
3846:
3845:
3842:
3841:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3805:
3803:
3800:
3796:
3793:
3792:
3791:
3788:
3786:
3783:
3779:
3776:
3774:
3771:
3769:
3766:
3765:
3764:
3761:
3759:
3756:
3755:
3753:
3751:
3747:
3742:
3732:
3729:
3728:
3726:
3722:
3718:
3711:
3706:
3704:
3699:
3697:
3692:
3691:
3688:
3676:
3668:
3666:
3658:
3657:
3654:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3607:Morton's fork
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3547:Buridan's ass
3545:
3543:
3540:
3536:
3533:
3531:
3528:
3526:
3523:
3522:
3521:
3520:Apportionment
3518:
3516:
3513:
3512:
3510:
3506:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3395:
3392:
3390:
3387:
3385:
3382:
3380:
3379:DownsâThomson
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3341:
3339:
3335:
3329:
3326:
3324:
3321:
3318:
3314:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3296:Plato's beard
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3259:
3257:
3254:
3252:
3249:
3247:
3244:
3242:
3239:
3237:
3234:
3232:
3229:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3201:
3200:
3197:
3195:
3194:KleeneâRosser
3192:
3190:
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3141:
3139:
3135:
3129:
3126:
3124:
3121:
3119:
3118:Theseus' ship
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3078:Mere addition
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3021:
3020:
3018:
3016:Philosophical
3014:
3010:
3002:
2997:
2995:
2990:
2988:
2983:
2982:
2979:
2969:
2968:
2956:
2950:
2947:
2945:
2942:
2940:
2937:
2935:
2932:
2931:
2929:
2925:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2848:
2846:
2842:
2835:
2834:
2830:
2827:
2826:
2822:
2819:
2818:
2814:
2811:
2810:
2806:
2803:
2802:
2798:
2795:
2794:
2790:
2787:
2786:
2782:
2779:
2778:
2774:
2771:
2770:
2769:Why Men Fight
2766:
2763:
2762:
2758:
2755:
2754:
2750:
2747:
2746:
2742:
2739:
2738:
2734:
2733:
2731:
2727:
2722:
2708:
2705:
2703:
2700:
2699:
2697:
2693:
2683:
2680:
2678:
2675:
2674:
2672:
2670:
2666:
2656:
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2637:
2635:
2633:
2629:
2626:
2622:
2615:
2611:
2604:
2599:
2597:
2592:
2590:
2585:
2584:
2581:
2575:
2568:
2553:
2552:
2547:
2543:
2538:
2537:
2532:
2529:
2524:
2520:
2519:
2514:
2510:
2505:
2501:
2500:
2495:
2491:
2490:
2478:
2474:
2469:
2468:
2458:
2452:
2448:
2444:
2440:
2436:
2433:
2431:0-674-32449-8
2427:
2423:
2419:
2415:
2411:
2408:
2402:
2398:
2394:
2390:
2385:
2384:
2362:
2361:
2356:
2350:
2336:
2332:
2326:
2320:
2316:
2312:
2306:
2299:
2293:
2286:
2282:
2276:
2269:
2263:
2256:
2250:
2243:
2239:
2235:
2231:
2227:
2218:
2211:
2204:
2199:
2195:
2189:
2182:
2178:
2171:
2164:
2158:
2149:
2135:
2129:
2125:
2124:
2116:
2107:
2094:
2088:
2084:
2083:
2075:
2073:
2064:
2060:
2056:
2051:
2043:
2037:
2033:
2026:
2019:
2013:
2006:
2000:
1992:
1988:
1981:
1973:
1967:
1963:
1956:
1950:
1949:3-764-31770-1
1946:
1942:
1938:
1933:
1924:
1918:
1913:
1906:
1905:
1898:
1891:
1885:
1881:
1865:
1858:
1852:
1848:
1837:
1836:Universal set
1834:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1810:
1806:
1800:
1797:
1794:
1791:
1788:
1785:
1783:
1780:
1779:
1771:
1767:
1764:
1761:
1757:
1755:
1751:
1750:Haskell Curry
1748:(named after
1747:
1744:
1741:
1737:
1733:
1731:
1727:
1723:
1719:
1718:
1710:
1707:
1704:
1700:
1696:
1693:
1688:
1685:
1682:
1678:
1675:
1672:
1669:
1668:
1667:
1664:
1662:
1658:
1654:
1653:
1648:
1640:
1636:
1632:
1628:
1624:
1623:
1622:
1616:
1612:
1608:
1604:
1603:
1602:
1599:
1577:
1576:
1575:
1569:
1562:
1558:
1557:
1556:
1543:
1539:
1535:
1533:
1529:
1524:
1522:
1516:
1513:
1502:
1499:
1484:
1481:
1473:
1463:
1459:
1455:
1449:
1448:
1443:This section
1441:
1432:
1431:
1422:
1419:
1411:
1401:
1397:
1393:
1387:
1386:
1382:
1377:This section
1375:
1371:
1366:
1365:
1360:
1358:
1351:
1350:
1345:
1344:
1339:
1334:
1325:
1324:
1316:
1313:
1311:
1307:
1303:
1299:
1295:
1291:
1286:
1284:
1280:
1276:
1275:
1270:
1264:
1262:
1261:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1225:
1223:
1218:
1216:
1212:
1207:
1201:
1199:
1194:
1191:
1187:
1182:
1180:
1176:
1172:
1166:
1161:
1159:
1158:
1151:
1148:
1144:
1140:
1133:
1131:
1127:
1126:
1121:
1120:Gottlob Frege
1117:
1107:
1105:
1101:
1097:
1093:
1089:
1084:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1045:
1043:
1040:cannot be in
1039:
1035:
1031:
1027:
1022:
1020:
1016:
1012:
1005:
1000:
995:
993:
989:
985:
981:
977:
973:
969:
965:
960:
958:
954:
950:
946:
942:
938:
934:
930:
926:
922:
921:Ernst Zermelo
917:
915:
914:contradiction
911:
907:
903:
893:
888:
883:
881:
871:
869:
850:
847:
844:
841:
833:
830:
827:
820:
819:
818:
816:
800:
792:
770:
767:
764:
756:
753:
750:
744:
738:
728:
727:
726:
709:
703:
683:
680:
677:
670:. Substitute
657:
633:
604:
598:
590:
587:
584:
578:
572:
562:
561:
560:
558:
536:
533:
530:
519:
516:
513:
505:
502:
499:
492:
482:
475:
465:
464:
463:
461:
445:
438:
434:
430:
420:
418:
414:
410:
406:
402:
397:
395:
391:
387:
376:
374:
370:
366:
362:
358:
354:
350:
346:
342:
341:Gottlob Frege
338:
319:
316:
313:
305:
302:
299:
288:
285:
282:
279:
276:
270:
267:
255:
254:
253:
250:
246:
242:
238:
233:
231:
227:
223:
219:
215:
214:David Hilbert
211:
210:Ernst Zermelo
207:
203:
199:
196:
195:mathematician
192:
189:
185:
181:
177:
173:
161:
156:
154:
149:
147:
142:
141:
139:
138:
132:
129:
127:
124:
122:
119:
117:
114:
112:
109:
107:
104:
102:
99:
97:
94:
92:
89:
87:
84:
81:
77:
74:
72:
69:
68:
67:
66:
57:
43:
40:
36:
35:
32:
31:
19:
6085:
5883:Ultraproduct
5730:Model theory
5695:Independence
5631:Formal proof
5623:Proof theory
5606:
5579:
5536:real numbers
5508:second-order
5419:Substitution
5296:Metalanguage
5237:conservative
5210:Axiom schema
5154:Constructive
5124:MorseâKelley
5090:Set theories
5069:Aleph number
5062:inaccessible
4968:Grothendieck
4852:intersection
4739:Higher-order
4727:Second-order
4673:Truth tables
4630:Venn diagram
4535:
4413:Formal proof
4285:Georg Cantor
4280:Paul Bernays
4251:
4211:MorseâKelley
4186:
4119:
4118:Subset
4065:hereditarily
4027:Venn diagram
3985:ordered pair
3900:Intersection
3844:Axiom schema
3627:Preparedness
3459:Productivity
3439:Mandeville's
3245:
3231:Opposite Day
3159:Burali-Forti
3154:Bhartrhari's
2959:
2939:Earl Russell
2857:Dora Russell
2831:
2823:
2815:
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1830:Strange loop
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1699:liar paradox
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334:
296:, then
248:
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234:
226:Georg Cantor
179:
175:
169:
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5993:Type theory
5941:undecidable
5873:Truth value
5760:equivalence
5439:non-logical
5052:Enumeration
5042:Isomorphism
4989:cardinality
4973:Von Neumann
4938:Ultrafilter
4903:Uncountable
4837:equivalence
4754:Quantifiers
4744:Fixed-point
4713:First-order
4593:Consistency
4578:Proposition
4555:Traditional
4526:Lindström's
4516:Compactness
4458:Type theory
4403:Cardinality
4310:Thomas Jech
4153:Alternative
4132:Uncountable
4086:Ultrafilter
3945:Cardinality
3849:replacement
3790:Determinacy
3557:Condorcet's
3409:Giffen good
3369:Competition
3123:White horse
3098:Omnipotence
2907:(godfather)
2756:(1910â1913)
2745:On Denoting
2695:Mathematics
2557:25 November
2482:25 November
2367:January 30,
2234:Quine 1955b
2016:P. Maddy, "
1809:On Denoting
1782:Basic Law V
1770:type theory
1649:episode of
1641:themselves.
1621:or "elect"
1617:themselves.
1590:themselves,
1568:substantive
1088:type theory
1077:appropriate
990:are called
349:type theory
232:by letter.
191:philosopher
6114:Categories
5804:elementary
5497:arithmetic
5365:Quantifier
5343:functional
5215:Expression
4933:Transitive
4877:identities
4862:complement
4795:hereditary
4778:Set theory
4305:Kurt Gödel
4290:Paul Cohen
4127:Transitive
3895:Identities
3879:Complement
3866:Operations
3827:Regularity
3795:projective
3758:Adjunction
3717:Set theory
3632:Prevention
3622:Parrondo's
3612:Navigation
3597:Inventor's
3592:Hedgehog's
3552:Chainstore
3535:Population
3530:New states
3464:Prosperity
3444:Mayfield's
3286:Entailment
3266:Barbershop
3179:Epimenides
2624:Philosophy
2340:2024-02-04
2242:Quine 1950
2226:Quine 1956
2139:2016-02-22
2098:2016-02-22
1876:References
1726:order type
1692:Richardian
1470:March 2021
1454:improve it
1408:March 2021
1343:improve it
1306:consistent
1290:Kurt Gödel
1090:, include
427:The term "
202:set theory
6075:Supertask
5978:Recursion
5936:decidable
5770:saturated
5748:of models
5671:deductive
5666:axiomatic
5586:Hilbert's
5573:Euclidean
5554:canonical
5477:axiomatic
5409:Signature
5338:Predicate
5227:Extension
5149:Ackermann
5074:Operation
4953:Universal
4943:Recursive
4918:Singleton
4913:Inhabited
4898:Countable
4888:Types of
4872:power set
4842:partition
4759:Predicate
4705:Predicate
4620:Syllogism
4610:Soundness
4583:Inference
4573:Tautology
4475:paradoxes
4238:Paradoxes
4158:Axiomatic
4137:Universal
4113:Singleton
4108:Recursive
4051:Countable
4046:Amorphous
3905:Power set
3822:Power set
3773:dependent
3768:countable
3647:Willpower
3642:Tolerance
3617:Newcomb's
3582:Fredkin's
3469:Scitovsky
3389:Edgeworth
3384:Easterlin
3349:Antitrust
3246:Russell's
3241:Richard's
3214:Pinocchio
3169:Crocodile
3088:Newcomb's
3058:Goodman's
3053:Free will
3038:Epicurean
3009:paradoxes
2889:(brother)
2536:MathWorld
2003:R. Bunn,
1521:empty set
1458:verifying
1379:does not
1349:talk page
1263:, 3.333)
1253:we write
1220:In 1923,
1165:proof....
1150:totality.
1069:power set
1061:empty set
1004:arbitrary
919:In 1908,
900:From the
845:∉
838:⟺
831:∈
768:∉
761:⟺
754:∈
742:∀
736:∃
704:φ
681:∉
658:φ
634:φ
599:φ
595:⟺
588:∈
576:∀
570:∃
527:⟹
517:∈
510:⟺
503:∈
490:∀
480:∀
473:∀
446:∈
437:predicate
310:⟺
303:∈
280:∣
264:Let
6060:Logicism
6053:timeline
6029:Concrete
5888:Validity
5858:T-schema
5851:Kripke's
5846:Tarski's
5841:semantic
5831:Strength
5780:submodel
5775:spectrum
5743:function
5591:Tarski's
5580:Elements
5567:geometry
5523:Robinson
5444:variable
5429:function
5402:spectrum
5392:Sentence
5348:variable
5291:Language
5244:Relation
5205:Automata
5195:Alphabet
5179:language
5033:-jection
5011:codomain
4997:Function
4958:Universe
4928:Infinite
4832:Relation
4615:Validity
4605:Argument
4503:theorem,
4242:Problems
4146:Theories
4122:Superset
4098:Infinite
3927:Concepts
3807:Infinity
3724:Overview
3675:Category
3572:Ellsberg
3424:Leontief
3404:Gibson's
3399:European
3394:Ellsberg
3364:Braess's
3359:Bertrand
3337:Economic
3271:Catch-22
3251:Socratic
3093:Nihilism
3063:Hedonism
3023:Analysis
3007:Notable
2901:(mother)
2895:(father)
2416:(1967),
1776:See also
1754:negation
1647:Season 8
1633:), that
1609:er that
1582:er that
1310:logicist
1298:complete
1247:Y(O(fx))
1231:F(F(fx))
1211:Nachlass
1190:antinomy
1130:function
817:we have
789:Then by
351:and the
317:∉
286:∉
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6002:Related
5799:Diagram
5697: (
5676:Hilbert
5661:Systems
5656:Theorem
5534:of the
5479:systems
5259:Formula
5254:Grammar
5170: (
5114:General
4827:Forcing
4812:Element
4732:Monadic
4507:paradox
4448:Theorem
4384:General
4180:General
4175:Zermelo
4081:subbase
4063: (
4002:Forcing
3980:Element
3952: (
3930:Methods
3817:Pairing
3577:Fenno's
3542:Arrow's
3525:Alabama
3515:Abilene
3494:Tullock
3449:Metzler
3291:Lottery
3281:Drinker
3224:Yablo's
3219:Quine's
3174:Curry's
3137:Logical
3113:Sorites
3103:Preface
3083:Moore's
3068:Liberal
3043:Fiction
2927:Related
2515:(ed.).
2477:YouTube
2380:Sources
2230:Mind 64
1762:paradox
1728:of all
1645:In the
1452:Please
1400:removed
1385:sources
1110:History
725:to get
386:squares
188:British
182:) is a
5765:finite
5528:Skolem
5481:
5456:Theory
5424:Symbol
5414:String
5397:atomic
5274:ground
5269:closed
5264:atomic
5220:ground
5183:syntax
5079:binary
5006:domain
4923:Finite
4688:finite
4546:Logics
4505:
4453:Theory
4071:Filter
4061:Finite
3997:Family
3940:Almost
3778:global
3763:Choice
3750:Axioms
3484:Thrift
3454:Plenty
3429:Lerner
3419:Jevons
3414:Icarus
3344:Allais
3306:Ross's
3144:Barber
3128:Zeno's
3073:Meno's
2965:
2844:Family
2836:(1959)
2828:(1945)
2820:(1938)
2812:(1935)
2804:(1929)
2796:(1927)
2788:(1922)
2780:(1919)
2772:(1916)
2764:(1912)
2748:(1905)
2740:(1903)
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1532:cannot
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5755:Model
5503:Peano
5360:Proof
5200:Arity
5129:Naive
5016:image
4948:Fuzzy
4908:Empty
4857:union
4802:Class
4443:Model
4433:Lemma
4391:Axiom
4163:Naive
4093:Fuzzy
4056:Empty
4039:types
3990:tuple
3960:Class
3954:large
3915:Union
3832:Union
3587:Green
3567:Downs
3499:Value
3434:Lucas
3301:Raven
3209:No-no
3164:Court
3149:Berry
2883:(son)
2877:(son)
2729:Works
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2179:, in
1843:Notes
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1635:elect
1627:elect
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646:with
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