3856:
3926:
110:
1699:"All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a
34:
1640:
is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and " ham sandwich is better than nothing" in a mathematical tone. According to
1614:. This issue can be overcome by viewing a set as a bagâan empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all
964:) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.
245:
When writing in languages such as Danish and
Norwegian, where the empty set character may be confused with the alphabetic letter à (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⊰ may be used instead.
262:, two sets are equal if they have the same elements (that is, neither of them has an element not in the other). As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".
952:
876:
1349:
1211:
730:, every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the
1287:
1470:
notation was utilized in definitions; for example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed
1691:
was undoubtedly an important landmark in the history of mathematics, ⊠we should not assume that its utility in calculation is dependent upon its actually denoting some object.
1382:
881:
805:
466:
1237:
800:
768:
1679:
1659:
1094:
624:
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552:
528:
504:
394:
363:
236:
135:
1425:
1511:
1468:
711:
1117:
1531:
1488:
1445:
1074:
1051:
2235:
4390:
2910:
2993:
2134:
1636:
Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness
1580:(which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the
634:
1292:
3307:
1569:
exists, and in the language of set theory, that thing must be a set. Now the existence of the empty set follows easily from the
4540:
3465:
2035:
2253:
649:
of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (the
4079:
3892:
3320:
2643:
1169:
4407:
3325:
3315:
3052:
2905:
2258:
2249:
3461:
2054:
2027:
1945:
174:) was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.
2803:
1355:, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers,
1242:
3558:
3302:
2127:
4385:
3979:
2863:
2556:
4265:
2297:
713:), and it is vacuously true that no element (of the empty set) can be found that retains its original position.
3819:
3521:
3284:
3279:
3104:
2525:
2209:
2077:
1902:
1885:
1822:
1147:
1143:
269:
of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its
4159:
4038:
3814:
3597:
3514:
3227:
3158:
3035:
2277:
4402:
3739:
3565:
3251:
2885:
2484:
734:, every real number is both an upper and lower bound for the empty set. When considered as a subset of the
4395:
4033:
3996:
3617:
3612:
3222:
2961:
2890:
2219:
2120:
3546:
3136:
2530:
2498:
2189:
4050:
1710:
argued that much of what has been heretofore obtained by set theory can just as easily be obtained by
1641:
Darling, the former is equivalent to "The set of all things that are better than eternal happiness is
1358:
4084:
3969:
3957:
3952:
3836:
3785:
3682:
3180:
3141:
2618:
2263:
2069:
1984:
681:
442:
312:
2292:
1216:
3885:
3677:
3607:
3146:
2998:
2981:
2704:
2184:
1997:
1684:
777:
745:
1664:
1644:
1079:
609:
585:
565:
537:
513:
489:
379:
348:
4504:
4422:
4297:
4249:
4063:
3986:
3509:
3486:
3447:
3333:
3274:
2920:
2840:
2684:
2628:
2241:
2018:
1550:
988:
259:
87:
In some textbooks and popularizations, the empty set is referred to as the "null set". However,
684:. The empty set can be considered a derangement of itself, because it has only one permutation (
221:
120:
4456:
4337:
4149:
3962:
3799:
3526:
3504:
3471:
3364:
3210:
3195:
3168:
3119:
3003:
2938:
2763:
2729:
2724:
2598:
2429:
2406:
1877:
1404:
1054:
70:
27:
2022:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
1493:
1450:
77:, while in other theories, its existence can be deduced. Many possible properties of sets are
4372:
4342:
4286:
4206:
4186:
4164:
3729:
3582:
3374:
3092:
2828:
2734:
2593:
2578:
2459:
2434:
1711:
1151:
1015:
335:
167:
58:
20:
4446:
4436:
4270:
4201:
4154:
4094:
3974:
3702:
3664:
3541:
3345:
3185:
3109:
3087:
2915:
2873:
2772:
2739:
2603:
2391:
2302:
1703:
which has no members. We cannot conjure such an entity into existence by mere stipulation."
255:
1593:
While the empty set is a standard and widely accepted mathematical concept, it remains an
687:
8:
4545:
4441:
4352:
4260:
4255:
4069:
4011:
3942:
3878:
3831:
3722:
3707:
3687:
3644:
3531:
3481:
3407:
3352:
3289:
3082:
3077:
3025:
2793:
2782:
2454:
2354:
2282:
2273:
2269:
2204:
2199:
1937:
1570:
1163:
1099:
947:{\displaystyle \inf \varnothing =\max(\{-\infty ,+\infty \}\cup \mathbb {R} )=+\infty .}
871:{\displaystyle \sup \varnothing =\min(\{-\infty ,+\infty \}\cup \mathbb {R} )=-\infty ,}
4364:
4359:
4144:
4099:
4006:
3860:
3629:
3592:
3577:
3570:
3553:
3357:
3339:
3205:
3131:
3114:
3067:
2880:
2789:
2623:
2608:
2568:
2520:
2505:
2493:
2449:
2424:
2194:
2143:
1870:
1681:". The first compares elements of sets, while the second compares the sets themselves.
1546:
1542:
1516:
1473:
1430:
1059:
1036:
1022:
298:
163:
74:
2813:
802:
which is defined to be greater than every other extended real number), we have that:
630:. This is often paraphrased as "everything is true of the elements of the empty set."
4221:
4058:
4021:
3991:
3915:
3855:
3795:
3602:
3412:
3402:
3294:
3175:
3010:
2986:
2767:
2751:
2656:
2633:
2510:
2479:
2444:
2339:
2174:
2095:
2073:
2050:
2031:
2023:
1941:
1881:
1818:
1581:
1558:
1352:
1135:
973:
771:
739:
323:
54:
1919:", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.
4509:
4499:
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3809:
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3437:
3432:
3417:
3243:
3200:
3097:
2895:
2845:
2419:
2381:
1597:
curiosity, whose meaning and usefulness are debated by philosophers and logicians.
731:
654:
1715:
4378:
4316:
4134:
3947:
3790:
3780:
3734:
3717:
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3634:
3536:
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3263:
3190:
3163:
3151:
3057:
2971:
2945:
2900:
2868:
2669:
2471:
2414:
2364:
2329:
2287:
1993:
1128:
960:) of the empty set is negative infinity, while the greatest lower bound (inf or
95:, in which it describes a set of measure zero (which is not necessarily empty).
4514:
4311:
4292:
4196:
4181:
4138:
4074:
4016:
3775:
3754:
3712:
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3442:
3040:
3030:
3020:
3015:
2949:
2823:
2699:
2588:
2583:
2561:
2162:
1838:
1553:. However, the axiom of empty set can be shown redundant in at least two ways:
1166:, 0 is defined as the empty set, and the successor of an ordinal is defined as
1124:
1120:
735:
658:
138:
92:
78:
2098:
1773:
4534:
4519:
4321:
4235:
4230:
3749:
3427:
2934:
2719:
2709:
2679:
2664:
2334:
1979:
1736:
1707:
1615:
1562:
662:
627:
370:
4489:
265:
The only subset of the empty set is the empty set itself; equivalently, the
142:
4469:
4464:
4282:
4211:
4169:
4028:
3925:
3649:
3496:
3397:
3389:
3269:
3217:
3126:
3062:
3045:
2976:
2835:
2694:
2396:
2179:
1623:
1385:
1138:, called the empty space, in just one way: by defining the empty set to be
726:
Since the empty set has no member when it is considered as a subset of any
273:) is zero. The empty set is the only set with either of these properties.
2049:. Springer Monographs in Mathematics (3rd millennium ed.). Springer.
4494:
4129:
3759:
3639:
2818:
2808:
2755:
2439:
2359:
2344:
2224:
2169:
2042:
2013:
1812:
1594:
1004:
999:
are complements of each other, the empty set is also closed, making it a
727:
677:
673:
270:
62:
42:
4474:
4245:
3901:
2689:
2544:
2515:
2321:
1916:
1577:
1008:
1000:
992:
770:
which is defined to be less than every other extended real number, and
738:
formed by adding two "numbers" or "points" to the real numbers (namely
666:
4277:
4191:
4089:
3841:
3744:
2797:
2714:
2638:
2574:
2386:
2376:
2349:
2112:
2103:
1966:
1797:
1748:
1630:
650:
646:
266:
104:
3826:
3624:
3072:
2777:
2371:
1661:" and the latter to "The set {ham sandwich} is better than the set
1142:. This empty topological space is the unique initial object in the
1139:
980:
957:
109:
88:
2030:(Springer-Verlag edition). Reprinted by Martino Fine Books, 2011.
3422:
2214:
1742:
1602:
1019:
961:
276:
178:
1745: â Complete absence of anything; the opposite of everything
4302:
4124:
1344:{\displaystyle 2=1\cup \{1\}=\{\varnothing ,\{\varnothing \}\}}
471:
288:
1751: â Mathematical set containing all subsets of a given set
4174:
3934:
3870:
2966:
2312:
2157:
1619:
1018:
of the empty set is empty. This is known as "preservation of
211:
1718:
sets as singular entities having other entities as members.
1351:, and so on. The von Neumann construction, along with the
195:
171:
1900:
Bruckner, A.N., Bruckner, J.B., and
Thomson, B.S. (2008).
146:
137:", and "â
". The latter two symbols were introduced by the
33:
1739: â Property of sets used in constructive mathematics
1490:
as an existent set on its own, or if Cantor merely used
669:, since one is the identity element for multiplication.
1533:
itself as a set, but considered it an "improper set".
1154:: only the empty set has a function to the empty set.
84:
Any set other than the empty set is called non-empty.
1727:
1667:
1647:
1519:
1496:
1476:
1453:
1433:
1407:
1361:
1295:
1245:
1219:
1172:
1102:
1082:
1062:
1039:
884:
808:
780:
748:
690:
612:
588:
568:
540:
516:
492:
445:
382:
351:
224:
123:
66:
1401:
In the context of sets of real numbers, Cantor used
510:. Indeed, if it were not true that every element of
1545:, the existence of the empty set is assured by the
653:) is zero. The reason for this is that zero is the
117:
Common notations for the empty set include "{ }", "
2093:
1982:(1984), "To be is to be the value of a variable",
1917:The Empty Set, the Singleton, and the Ordered Pair
1869:
1798:"Earliest Uses of Symbols of Set Theory and Logic"
1732:Pages displaying short descriptions with no spaces
1673:
1653:
1525:
1505:
1482:
1462:
1439:
1419:
1376:
1343:
1281:
1231:
1206:{\displaystyle S(\alpha )=\alpha \cup \{\alpha \}}
1205:
1111:
1088:
1068:
1045:
946:
870:
794:
762:
705:
618:
606:. Any statement that begins "for every element of
594:
574:
546:
522:
498:
460:
388:
357:
230:
129:
787:
755:
73:ensure that the empty set exists by including an
4532:
894:
885:
818:
809:
716:
37:The empty set is the set containing no elements.
626:" is not making any substantive claim; it is a
1867:
1282:{\displaystyle 1=0\cup \{0\}=\{\varnothing \}}
640:
534:, then there would be at least one element of
3886:
2128:
1931:
19:"â
" redirects here. For similar symbols, see
1513:as an emptiness predicate. Zermelo accepted
1338:
1335:
1329:
1320:
1314:
1308:
1276:
1270:
1264:
1258:
1200:
1194:
918:
900:
842:
824:
1123:. As a result, the empty set is the unique
635:set-theoretic definition of natural numbers
91:is a distinct notion within the context of
3893:
3879:
2320:
2135:
2121:
1960:
1817:(3rd ed.). McGraw-Hill. p. 300.
1053:is a set, then there exists precisely one
1927:
1925:
1364:
925:
849:
788:
756:
170:alphabets. In the past, "0" (the numeral
1164:von Neumann construction of the ordinals
721:
108:
32:
1600:The empty set is not the same thing as
1588:
1536:
1391:
956:That is, the least upper bound (sup or
474:, the empty set is a subset of any set
16:Mathematical set containing no elements
4533:
2142:
2063:
1922:
1549:, and its uniqueness follows from the
661:of the elements of the empty set (the
3874:
2116:
2094:
1810:
1771:
637:, zero is modelled by the empty set.
412:, the following two statements hold:
2041:
1954:
1856:Fonetik og Fonologi: Almen og dansk.
1767:
1765:
1396:
1814:Principles of Mathematical Analysis
1606:; rather, it is a set with nothing
1134:The empty set can be turned into a
319:with the empty set is the empty set
13:
2007:
1028:
938:
915:
906:
862:
839:
830:
784:
752:
330:and the empty set is the empty set
225:
160:LATIN CAPITAL LETTER O WITH STROKE
145:) in 1939, inspired by the letter
124:
14:
4557:
2087:
1934:The Universal Book of Mathematics
1762:
1332:
1323:
1273:
1226:
888:
812:
452:
404:Conversely, if for some property
3924:
3854:
1687:argues that while the empty set
1447:contains no single point". This
1377:{\displaystyle \mathbb {N} _{0}}
65:(count of elements in a set) is
1988:91: 430â49. Reprinted in 1998,
1973:
1854:e.g. Nina GrĂžnnum (2005, 2013)
582:at all, there is no element of
461:{\displaystyle V=\varnothing .}
26:For other uses of "Empty", see
3900:
1909:
1894:
1861:
1848:
1831:
1804:
1790:
1232:{\displaystyle 0=\varnothing }
1182:
1176:
1144:category of topological spaces
929:
897:
853:
821:
1:
3815:History of mathematical logic
2066:Modern Elementary Mathematics
1858:Akademisk forlag, Copenhagen.
1755:
1388:of arithmetic are satisfied.
1157:
1003:. Moreover, the empty set is
795:{\displaystyle +\infty \!\,,}
763:{\displaystyle -\infty \!\,,}
717:In other areas of mathematics
665:) should be considered to be
657:for addition. Similarly, the
249:
177:The symbol â
is available at
4541:Basic concepts in set theory
3740:Primitive recursive function
1674:{\displaystyle \varnothing }
1654:{\displaystyle \varnothing }
1089:{\displaystyle \varnothing }
619:{\displaystyle \varnothing }
595:{\displaystyle \varnothing }
575:{\displaystyle \varnothing }
547:{\displaystyle \varnothing }
523:{\displaystyle \varnothing }
499:{\displaystyle \varnothing }
389:{\displaystyle \varnothing }
358:{\displaystyle \varnothing }
7:
1872:Linear Algebra and Geometry
1721:
967:
641:Operations on the empty set
260:principle of extensionality
98:
10:
4562:
4391:von NeumannâBernaysâGödel
2804:SchröderâBernstein theorem
2531:Monadic predicate calculus
2190:Foundations of mathematics
1714:over individuals, without
1695:it is also the case that:
231:{\displaystyle \emptyset }
130:{\displaystyle \emptyset }
113:A symbol for the empty set
102:
25:
18:
4455:
4418:
4330:
4220:
4192:One-to-one correspondence
4108:
4049:
3933:
3922:
3908:
3850:
3837:Philosophy of mathematics
3786:Automated theorem proving
3768:
3663:
3495:
3388:
3240:
2957:
2933:
2911:Von NeumannâBernaysâGödel
2856:
2750:
2654:
2552:
2543:
2470:
2405:
2311:
2233:
2150:
2070:Harcourt Brace Jovanovich
1996:, and Burgess, J., eds.)
1985:The Journal of Philosophy
1906:, 2nd edition, p. 9.
1561:implies, merely from the
1420:{\displaystyle P\equiv O}
239:
215:
207:
203:
199:
2064:Graham, Malcolm (1975).
1998:Harvard University Press
1903:Elementary Real Analysis
1506:{\displaystyle \equiv O}
1463:{\displaystyle \equiv O}
3487:Self-verifying theories
3308:Tarski's axiomatization
2259:Tarski's undefinability
2254:incompleteness theorems
1868:David M. Bloom (1979).
1610:it and a set is always
1551:axiom of extensionality
1131:of sets and functions.
1007:by the fact that every
554:that is not present in
431:for which the property
427:There is no element of
396:for which the property
376:There is no element of
4150:Constructible universe
3970:Constructibility (V=L)
3861:Mathematics portal
3472:Proof of impossibility
3120:propositional variable
2430:Propositional calculus
1990:Logic, Logic and Logic
1932:D. J. Darling (2004).
1839:"Unicode Standard 5.2"
1811:Rudin, Walter (1976).
1675:
1655:
1527:
1507:
1484:
1464:
1441:
1421:
1378:
1345:
1283:
1233:
1207:
1113:
1090:
1070:
1047:
995:and the empty set and
948:
872:
796:
764:
707:
620:
596:
576:
548:
524:
500:
462:
390:
359:
305:with the empty set is
232:
131:
114:
71:axiomatic set theories
38:
28:Empty (disambiguation)
4373:Principia Mathematica
4207:Transfinite induction
4066:(i.e. set difference)
3730:Kolmogorov complexity
3683:Computably enumerable
3583:Model complete theory
3375:Principia Mathematica
2435:Propositional formula
2264:BanachâTarski paradox
1778:mathworld.wolfram.com
1712:plural quantification
1676:
1656:
1528:
1508:
1485:
1465:
1442:
1422:
1379:
1346:
1284:
1234:
1208:
1152:strict initial object
1114:
1091:
1071:
1048:
983:by definition, as is
949:
873:
797:
765:
722:Extended real numbers
708:
645:When speaking of the
621:
597:
577:
549:
525:
501:
470:By the definition of
463:
416:For every element of
391:
360:
345:For every element of
238:is coded in LaTeX as
233:
210:. It can be coded in
194:. It can be coded in
132:
112:
36:
4447:Burali-Forti paradox
4202:Set-builder notation
4155:Continuum hypothesis
4095:Symmetric difference
3678:ChurchâTuring thesis
3665:Computability theory
2874:continuum hypothesis
2392:Square of opposition
2250:Gödel's completeness
2038:(paperback edition).
1665:
1645:
1589:Philosophical issues
1537:Axiomatic set theory
1517:
1494:
1474:
1451:
1431:
1405:
1392:Questioned existence
1359:
1293:
1243:
1217:
1170:
1100:
1080:
1060:
1037:
882:
806:
778:
746:
706:{\displaystyle 0!=1}
688:
610:
586:
566:
538:
514:
490:
443:
380:
349:
256:axiomatic set theory
222:
121:
4408:TarskiâGrothendieck
3832:Mathematical object
3723:P versus NP problem
3688:Computable function
3482:Reverse mathematics
3408:Logical consequence
3285:primitive recursive
3280:elementary function
3053:Free/bound variable
2906:TarskiâGrothendieck
2425:Logical connectives
2355:Logical equivalence
2205:Logical consequence
1961:E. J. Lowe (2005).
1938:John Wiley and Sons
1772:Weisstein, Eric W.
1571:axiom of separation
1150:. In fact, it is a
979:, the empty set is
287:The empty set is a
81:for the empty set.
3997:Limitation of size
3630:Transfer principle
3593:Semantics of logic
3578:Categorical theory
3554:Non-standard model
3068:Logical connective
2195:Information theory
2144:Mathematical logic
2096:Weisstein, Eric W.
1671:
1651:
1547:axiom of empty set
1543:Zermelo set theory
1523:
1503:
1480:
1460:
1437:
1417:
1374:
1341:
1279:
1229:
1203:
1112:{\displaystyle A,}
1109:
1086:
1066:
1043:
991:of an open set is
944:
868:
792:
760:
703:
616:
592:
572:
558:. Since there are
544:
520:
496:
458:
386:
355:
228:
127:
115:
75:axiom of empty set
39:
21:Ă (disambiguation)
4528:
4527:
4437:Russell's paradox
4386:ZermeloâFraenkel
4287:Dedekind-infinite
4160:Diagonal argument
4059:Cartesian product
3916:Set (mathematics)
3868:
3867:
3800:Abstract category
3603:Theories of truth
3413:Rule of inference
3403:Natural deduction
3384:
3383:
2929:
2928:
2634:Cartesian product
2539:
2538:
2445:Many-valued logic
2420:Boolean functions
2303:Russell's paradox
2278:diagonal argument
2175:First-order logic
2036:978-1-61427-131-4
1582:axiom of infinity
1559:first-order logic
1526:{\displaystyle O}
1483:{\displaystyle O}
1440:{\displaystyle P}
1397:Historical issues
1353:axiom of infinity
1136:topological space
1069:{\displaystyle f}
1046:{\displaystyle A}
974:topological space
772:positive infinity
740:negative infinity
680:of a set without
324:Cartesian product
4553:
4510:Bertrand Russell
4500:John von Neumann
4485:Abraham Fraenkel
4480:Richard Dedekind
4442:Suslin's problem
4353:Cantor's theorem
4070:De Morgan's laws
3928:
3895:
3888:
3881:
3872:
3871:
3859:
3858:
3810:History of logic
3805:Category of sets
3698:Decision problem
3477:Ordinal analysis
3418:Sequent calculus
3316:Boolean algebras
3256:
3255:
3230:
3201:logical/constant
2955:
2954:
2941:
2864:ZermeloâFraenkel
2615:Set operations:
2550:
2549:
2487:
2318:
2317:
2298:LöwenheimâSkolem
2185:Formal semantics
2137:
2130:
2123:
2114:
2113:
2109:
2108:
2083:
2068:(2nd ed.).
2060:
2019:Naive Set Theory
2001:
1977:
1971:
1970:
1958:
1952:
1951:
1929:
1920:
1913:
1907:
1898:
1892:
1891:
1875:
1865:
1859:
1852:
1846:
1845:
1843:
1835:
1829:
1828:
1808:
1802:
1801:
1794:
1788:
1787:
1785:
1784:
1769:
1733:
1680:
1678:
1677:
1672:
1660:
1658:
1657:
1652:
1532:
1530:
1529:
1524:
1512:
1510:
1509:
1504:
1489:
1487:
1486:
1481:
1469:
1467:
1466:
1461:
1446:
1444:
1443:
1438:
1426:
1424:
1423:
1418:
1384:, such that the
1383:
1381:
1380:
1375:
1373:
1372:
1367:
1350:
1348:
1347:
1342:
1288:
1286:
1285:
1280:
1238:
1236:
1235:
1230:
1213:. Thus, we have
1212:
1210:
1209:
1204:
1118:
1116:
1115:
1110:
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1052:
1050:
1049:
1044:
953:
951:
950:
945:
928:
877:
875:
874:
869:
852:
801:
799:
798:
793:
769:
767:
766:
761:
732:real number line
712:
710:
709:
704:
655:identity element
625:
623:
622:
617:
601:
599:
598:
593:
581:
579:
578:
573:
553:
551:
550:
545:
529:
527:
526:
521:
505:
503:
502:
497:
467:
465:
464:
459:
395:
393:
392:
387:
364:
362:
361:
356:
241:
237:
235:
234:
229:
217:
209:
205:
201:
193:
190:
187:
185:
161:
158:
155:
153:
136:
134:
133:
128:
4561:
4560:
4556:
4555:
4554:
4552:
4551:
4550:
4531:
4530:
4529:
4524:
4451:
4430:
4414:
4379:New Foundations
4326:
4216:
4135:Cardinal number
4118:
4104:
4045:
3929:
3920:
3904:
3899:
3869:
3864:
3853:
3846:
3791:Category theory
3781:Algebraic logic
3764:
3735:Lambda calculus
3673:Church encoding
3659:
3635:Truth predicate
3491:
3457:Complete theory
3380:
3249:
3245:
3241:
3236:
3228:
2948: and
2944:
2939:
2925:
2901:New Foundations
2869:axiom of choice
2852:
2814:Gödel numbering
2754: and
2746:
2650:
2535:
2485:
2466:
2415:Boolean algebra
2401:
2365:Equiconsistency
2330:Classical logic
2307:
2288:Halting problem
2276: and
2252: and
2240: and
2239:
2234:Theorems (
2229:
2146:
2141:
2090:
2080:
2057:
2010:
2008:Further reading
2005:
2004:
1994:Richard Jeffrey
1978:
1974:
1959:
1955:
1948:
1940:. p. 106.
1930:
1923:
1914:
1910:
1899:
1895:
1888:
1866:
1862:
1853:
1849:
1841:
1837:
1836:
1832:
1825:
1809:
1805:
1796:
1795:
1791:
1782:
1780:
1770:
1763:
1758:
1731:
1724:
1666:
1663:
1662:
1646:
1643:
1642:
1622:that involve a
1591:
1539:
1518:
1515:
1514:
1495:
1492:
1491:
1475:
1472:
1471:
1452:
1449:
1448:
1432:
1429:
1428:
1406:
1403:
1402:
1399:
1394:
1368:
1363:
1362:
1360:
1357:
1356:
1294:
1291:
1290:
1244:
1241:
1240:
1218:
1215:
1214:
1171:
1168:
1167:
1160:
1148:continuous maps
1101:
1098:
1097:
1081:
1078:
1077:
1061:
1058:
1057:
1038:
1035:
1034:
1031:
1029:Category theory
970:
924:
883:
880:
879:
848:
807:
804:
803:
779:
776:
775:
747:
744:
743:
724:
719:
689:
686:
685:
643:
611:
608:
607:
602:that is not in
587:
584:
583:
567:
564:
563:
539:
536:
535:
515:
512:
511:
491:
488:
487:
444:
441:
440:
381:
378:
377:
365:, the property
350:
347:
346:
252:
223:
220:
219:
191:
188:
183:
182:
159:
156:
151:
150:
122:
119:
118:
107:
101:
31:
24:
17:
12:
11:
5:
4559:
4549:
4548:
4543:
4526:
4525:
4523:
4522:
4517:
4515:Thoralf Skolem
4512:
4507:
4502:
4497:
4492:
4487:
4482:
4477:
4472:
4467:
4461:
4459:
4453:
4452:
4450:
4449:
4444:
4439:
4433:
4431:
4429:
4428:
4425:
4419:
4416:
4415:
4413:
4412:
4411:
4410:
4405:
4400:
4399:
4398:
4383:
4382:
4381:
4369:
4368:
4367:
4356:
4355:
4350:
4345:
4340:
4334:
4332:
4328:
4327:
4325:
4324:
4319:
4314:
4309:
4300:
4295:
4290:
4280:
4275:
4274:
4273:
4268:
4263:
4253:
4243:
4238:
4233:
4227:
4225:
4218:
4217:
4215:
4214:
4209:
4204:
4199:
4197:Ordinal number
4194:
4189:
4184:
4179:
4178:
4177:
4172:
4162:
4157:
4152:
4147:
4142:
4132:
4127:
4121:
4119:
4117:
4116:
4113:
4109:
4106:
4105:
4103:
4102:
4097:
4092:
4087:
4082:
4077:
4075:Disjoint union
4072:
4067:
4061:
4055:
4053:
4047:
4046:
4044:
4043:
4042:
4041:
4036:
4025:
4024:
4022:Martin's axiom
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3987:Extensionality
3984:
3983:
3982:
3972:
3967:
3966:
3965:
3960:
3955:
3945:
3939:
3937:
3931:
3930:
3923:
3921:
3919:
3918:
3912:
3910:
3906:
3905:
3898:
3897:
3890:
3883:
3875:
3866:
3865:
3851:
3848:
3847:
3845:
3844:
3839:
3834:
3829:
3824:
3823:
3822:
3812:
3807:
3802:
3793:
3788:
3783:
3778:
3776:Abstract logic
3772:
3770:
3766:
3765:
3763:
3762:
3757:
3755:Turing machine
3752:
3747:
3742:
3737:
3732:
3727:
3726:
3725:
3720:
3715:
3710:
3705:
3695:
3693:Computable set
3690:
3685:
3680:
3675:
3669:
3667:
3661:
3660:
3658:
3657:
3652:
3647:
3642:
3637:
3632:
3627:
3622:
3621:
3620:
3615:
3610:
3600:
3595:
3590:
3588:Satisfiability
3585:
3580:
3575:
3574:
3573:
3563:
3562:
3561:
3551:
3550:
3549:
3544:
3539:
3534:
3529:
3519:
3518:
3517:
3512:
3505:Interpretation
3501:
3499:
3493:
3492:
3490:
3489:
3484:
3479:
3474:
3469:
3459:
3454:
3453:
3452:
3451:
3450:
3440:
3435:
3425:
3420:
3415:
3410:
3405:
3400:
3394:
3392:
3386:
3385:
3382:
3381:
3379:
3378:
3370:
3369:
3368:
3367:
3362:
3361:
3360:
3355:
3350:
3330:
3329:
3328:
3326:minimal axioms
3323:
3312:
3311:
3310:
3299:
3298:
3297:
3292:
3287:
3282:
3277:
3272:
3259:
3257:
3238:
3237:
3235:
3234:
3233:
3232:
3220:
3215:
3214:
3213:
3208:
3203:
3198:
3188:
3183:
3178:
3173:
3172:
3171:
3166:
3156:
3155:
3154:
3149:
3144:
3139:
3129:
3124:
3123:
3122:
3117:
3112:
3102:
3101:
3100:
3095:
3090:
3085:
3080:
3075:
3065:
3060:
3055:
3050:
3049:
3048:
3043:
3038:
3033:
3023:
3018:
3016:Formation rule
3013:
3008:
3007:
3006:
3001:
2991:
2990:
2989:
2979:
2974:
2969:
2964:
2958:
2952:
2935:Formal systems
2931:
2930:
2927:
2926:
2924:
2923:
2918:
2913:
2908:
2903:
2898:
2893:
2888:
2883:
2878:
2877:
2876:
2871:
2860:
2858:
2854:
2853:
2851:
2850:
2849:
2848:
2838:
2833:
2832:
2831:
2824:Large cardinal
2821:
2816:
2811:
2806:
2801:
2787:
2786:
2785:
2780:
2775:
2760:
2758:
2748:
2747:
2745:
2744:
2743:
2742:
2737:
2732:
2722:
2717:
2712:
2707:
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2661:
2659:
2652:
2651:
2649:
2648:
2647:
2646:
2641:
2636:
2631:
2626:
2621:
2613:
2612:
2611:
2606:
2596:
2591:
2589:Extensionality
2586:
2584:Ordinal number
2581:
2571:
2566:
2565:
2564:
2553:
2547:
2541:
2540:
2537:
2536:
2534:
2533:
2528:
2523:
2518:
2513:
2508:
2503:
2502:
2501:
2491:
2490:
2489:
2476:
2474:
2468:
2467:
2465:
2464:
2463:
2462:
2457:
2452:
2442:
2437:
2432:
2427:
2422:
2417:
2411:
2409:
2403:
2402:
2400:
2399:
2394:
2389:
2384:
2379:
2374:
2369:
2368:
2367:
2357:
2352:
2347:
2342:
2337:
2332:
2326:
2324:
2315:
2309:
2308:
2306:
2305:
2300:
2295:
2290:
2285:
2280:
2268:Cantor's
2266:
2261:
2256:
2246:
2244:
2231:
2230:
2228:
2227:
2222:
2217:
2212:
2207:
2202:
2197:
2192:
2187:
2182:
2177:
2172:
2167:
2166:
2165:
2154:
2152:
2148:
2147:
2140:
2139:
2132:
2125:
2117:
2111:
2110:
2089:
2088:External links
2086:
2085:
2084:
2078:
2061:
2055:
2039:
2009:
2006:
2003:
2002:
1972:
1953:
1946:
1921:
1915:A. Kanamori, "
1908:
1893:
1886:
1860:
1847:
1830:
1823:
1803:
1789:
1760:
1759:
1757:
1754:
1753:
1752:
1746:
1740:
1734:
1730: â Number
1723:
1720:
1705:
1704:
1702:
1693:
1692:
1670:
1650:
1638:
1637:
1613:
1609:
1605:
1590:
1587:
1586:
1585:
1574:
1568:
1563:logical axioms
1538:
1535:
1522:
1502:
1499:
1479:
1459:
1456:
1436:
1416:
1413:
1410:
1398:
1395:
1393:
1390:
1371:
1366:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1228:
1225:
1222:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1159:
1156:
1125:initial object
1121:empty function
1108:
1105:
1085:
1065:
1042:
1030:
1027:
969:
966:
943:
940:
937:
934:
931:
927:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
867:
864:
861:
858:
855:
851:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
791:
786:
783:
759:
754:
751:
736:extended reals
723:
720:
718:
715:
702:
699:
696:
693:
642:
639:
615:
591:
571:
561:
543:
519:
495:
481:
457:
454:
451:
448:
437:
436:
425:
402:
401:
385:
374:
354:
332:
331:
320:
309:
295:
251:
248:
227:
141:(specifically
139:Bourbaki group
126:
103:Main article:
100:
97:
93:measure theory
79:vacuously true
61:; its size or
53:is the unique
15:
9:
6:
4:
3:
2:
4558:
4547:
4544:
4542:
4539:
4538:
4536:
4521:
4520:Ernst Zermelo
4518:
4516:
4513:
4511:
4508:
4506:
4505:Willard Quine
4503:
4501:
4498:
4496:
4493:
4491:
4488:
4486:
4483:
4481:
4478:
4476:
4473:
4471:
4468:
4466:
4463:
4462:
4460:
4458:
4457:Set theorists
4454:
4448:
4445:
4443:
4440:
4438:
4435:
4434:
4432:
4426:
4424:
4421:
4420:
4417:
4409:
4406:
4404:
4403:KripkeâPlatek
4401:
4397:
4394:
4393:
4392:
4389:
4388:
4387:
4384:
4380:
4377:
4376:
4375:
4374:
4370:
4366:
4363:
4362:
4361:
4358:
4357:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4335:
4333:
4329:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4306:
4301:
4299:
4296:
4294:
4291:
4288:
4284:
4281:
4279:
4276:
4272:
4269:
4267:
4264:
4262:
4259:
4258:
4257:
4254:
4251:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4229:
4228:
4226:
4223:
4219:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4176:
4173:
4171:
4168:
4167:
4166:
4163:
4161:
4158:
4156:
4153:
4151:
4148:
4146:
4143:
4140:
4136:
4133:
4131:
4128:
4126:
4123:
4122:
4120:
4114:
4111:
4110:
4107:
4101:
4098:
4096:
4093:
4091:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4071:
4068:
4065:
4062:
4060:
4057:
4056:
4054:
4052:
4048:
4040:
4039:specification
4037:
4035:
4032:
4031:
4030:
4027:
4026:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3981:
3978:
3977:
3976:
3973:
3971:
3968:
3964:
3961:
3959:
3956:
3954:
3951:
3950:
3949:
3946:
3944:
3941:
3940:
3938:
3936:
3932:
3927:
3917:
3914:
3913:
3911:
3907:
3903:
3896:
3891:
3889:
3884:
3882:
3877:
3876:
3873:
3863:
3862:
3857:
3849:
3843:
3840:
3838:
3835:
3833:
3830:
3828:
3825:
3821:
3818:
3817:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3773:
3771:
3767:
3761:
3758:
3756:
3753:
3751:
3750:Recursive set
3748:
3746:
3743:
3741:
3738:
3736:
3733:
3731:
3728:
3724:
3721:
3719:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3700:
3699:
3696:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3670:
3668:
3666:
3662:
3656:
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3619:
3616:
3614:
3611:
3609:
3606:
3605:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3576:
3572:
3569:
3568:
3567:
3564:
3560:
3559:of arithmetic
3557:
3556:
3555:
3552:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3524:
3523:
3520:
3516:
3513:
3511:
3508:
3507:
3506:
3503:
3502:
3500:
3498:
3494:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3467:
3466:from ZFC
3463:
3460:
3458:
3455:
3449:
3446:
3445:
3444:
3441:
3439:
3436:
3434:
3431:
3430:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3395:
3393:
3391:
3387:
3377:
3376:
3372:
3371:
3366:
3365:non-Euclidean
3363:
3359:
3356:
3354:
3351:
3349:
3348:
3344:
3343:
3341:
3338:
3337:
3335:
3331:
3327:
3324:
3322:
3319:
3318:
3317:
3313:
3309:
3306:
3305:
3304:
3300:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3267:
3265:
3261:
3260:
3258:
3253:
3247:
3242:Example
3239:
3231:
3226:
3225:
3224:
3221:
3219:
3216:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3193:
3192:
3189:
3187:
3184:
3182:
3179:
3177:
3174:
3170:
3167:
3165:
3162:
3161:
3160:
3157:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3134:
3133:
3130:
3128:
3125:
3121:
3118:
3116:
3113:
3111:
3108:
3107:
3106:
3103:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3074:
3071:
3070:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3028:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3005:
3002:
3000:
2999:by definition
2997:
2996:
2995:
2992:
2988:
2985:
2984:
2983:
2980:
2978:
2975:
2973:
2970:
2968:
2965:
2963:
2960:
2959:
2956:
2953:
2951:
2947:
2942:
2936:
2932:
2922:
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2902:
2899:
2897:
2894:
2892:
2889:
2887:
2886:KripkeâPlatek
2884:
2882:
2879:
2875:
2872:
2870:
2867:
2866:
2865:
2862:
2861:
2859:
2855:
2847:
2844:
2843:
2842:
2839:
2837:
2834:
2830:
2827:
2826:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2799:
2795:
2791:
2788:
2784:
2781:
2779:
2776:
2774:
2771:
2770:
2769:
2765:
2762:
2761:
2759:
2757:
2753:
2749:
2741:
2738:
2736:
2733:
2731:
2730:constructible
2728:
2727:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
2706:
2703:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2662:
2660:
2658:
2653:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2616:
2614:
2610:
2607:
2605:
2602:
2601:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2576:
2572:
2570:
2567:
2563:
2560:
2559:
2558:
2555:
2554:
2551:
2548:
2546:
2542:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2500:
2497:
2496:
2495:
2492:
2488:
2483:
2482:
2481:
2478:
2477:
2475:
2473:
2469:
2461:
2458:
2456:
2453:
2451:
2448:
2447:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2412:
2410:
2408:
2407:Propositional
2404:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2366:
2363:
2362:
2361:
2358:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2335:Logical truth
2333:
2331:
2328:
2327:
2325:
2323:
2319:
2316:
2314:
2310:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2275:
2271:
2267:
2265:
2262:
2260:
2257:
2255:
2251:
2248:
2247:
2245:
2243:
2237:
2232:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2181:
2178:
2176:
2173:
2171:
2168:
2164:
2161:
2160:
2159:
2156:
2155:
2153:
2149:
2145:
2138:
2133:
2131:
2126:
2124:
2119:
2118:
2115:
2106:
2105:
2100:
2097:
2092:
2091:
2081:
2075:
2071:
2067:
2062:
2058:
2056:3-540-44085-2
2052:
2048:
2044:
2040:
2037:
2033:
2029:
2028:0-387-90092-6
2025:
2021:
2020:
2015:
2012:
2011:
1999:
1995:
1991:
1987:
1986:
1981:
1980:George Boolos
1976:
1969:. p. 87.
1968:
1964:
1957:
1949:
1947:0-471-27047-4
1943:
1939:
1935:
1928:
1926:
1918:
1912:
1905:
1904:
1897:
1889:
1883:
1879:
1874:
1873:
1864:
1857:
1851:
1840:
1834:
1826:
1820:
1816:
1815:
1807:
1799:
1793:
1779:
1775:
1768:
1766:
1761:
1750:
1747:
1744:
1741:
1738:
1737:Inhabited set
1735:
1729:
1726:
1725:
1719:
1717:
1713:
1709:
1708:George Boolos
1700:
1698:
1697:
1696:
1690:
1689:
1688:
1686:
1685:Jonathan Lowe
1682:
1668:
1648:
1635:
1634:
1633:
1632:
1627:
1625:
1621:
1617:
1616:opening moves
1611:
1607:
1604:
1601:
1598:
1596:
1583:
1579:
1575:
1572:
1566:
1564:
1560:
1556:
1555:
1554:
1552:
1548:
1544:
1534:
1520:
1500:
1497:
1477:
1457:
1454:
1434:
1414:
1411:
1408:
1389:
1387:
1369:
1354:
1326:
1317:
1311:
1305:
1302:
1299:
1296:
1267:
1261:
1255:
1252:
1249:
1246:
1223:
1220:
1197:
1191:
1188:
1185:
1179:
1173:
1165:
1155:
1153:
1149:
1145:
1141:
1137:
1132:
1130:
1126:
1122:
1106:
1103:
1083:
1063:
1056:
1040:
1026:
1024:
1021:
1017:
1012:
1010:
1006:
1002:
998:
994:
990:
986:
982:
978:
975:
965:
963:
959:
954:
941:
935:
932:
921:
912:
909:
903:
891:
865:
859:
856:
845:
836:
833:
827:
815:
789:
781:
773:
757:
749:
741:
737:
733:
729:
714:
700:
697:
694:
691:
683:
679:
675:
670:
668:
664:
663:empty product
660:
656:
652:
648:
638:
636:
633:In the usual
631:
629:
628:vacuous truth
613:
605:
589:
569:
559:
557:
541:
533:
517:
509:
493:
485:
479:
477:
473:
468:
455:
449:
446:
434:
430:
426:
423:
420:the property
419:
415:
414:
413:
411:
408:and some set
407:
399:
383:
375:
372:
371:vacuous truth
368:
352:
344:
343:
342:
340:
337:
329:
325:
321:
318:
314:
310:
308:
304:
300:
296:
294:
290:
286:
285:
284:
282:
278:
274:
272:
268:
263:
261:
257:
247:
243:
218:. The symbol
213:
197:
180:
175:
173:
169:
165:
148:
144:
140:
111:
106:
96:
94:
90:
85:
82:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
35:
29:
22:
4470:Georg Cantor
4465:Paul Bernays
4396:MorseâKelley
4371:
4304:
4303:Subset
4250:hereditarily
4240:
4212:Venn diagram
4170:ordered pair
4085:Intersection
4029:Axiom schema
3852:
3650:Ultraproduct
3497:Model theory
3462:Independence
3398:Formal proof
3390:Proof theory
3373:
3346:
3303:real numbers
3275:second-order
3186:Substitution
3063:Metalanguage
3004:conservative
2977:Axiom schema
2921:Constructive
2891:MorseâKelley
2857:Set theories
2836:Aleph number
2829:inaccessible
2735:Grothendieck
2674:
2619:intersection
2506:Higher-order
2494:Second-order
2440:Truth tables
2397:Venn diagram
2180:Formal proof
2102:
2065:
2046:
2043:Jech, Thomas
2017:
2014:Halmos, Paul
1989:
1983:
1975:
1962:
1956:
1933:
1911:
1901:
1896:
1871:
1863:
1855:
1850:
1833:
1813:
1806:
1792:
1781:. Retrieved
1777:
1706:
1694:
1683:
1639:
1629:The popular
1628:
1599:
1592:
1540:
1400:
1386:Peano axioms
1161:
1133:
1032:
1013:
1011:is compact.
996:
987:. Since the
984:
976:
971:
955:
725:
682:fixed points
671:
644:
632:
603:
562:elements of
555:
531:
507:
483:
475:
469:
438:
432:
428:
421:
417:
409:
405:
403:
397:
366:
338:
333:
327:
316:
313:intersection
306:
302:
292:
280:
275:
264:
254:In standard
253:
244:
208:∅
176:
116:
86:
83:
50:
46:
40:
4495:Thomas Jech
4338:Alternative
4317:Uncountable
4271:Ultrafilter
4130:Cardinality
4034:replacement
3975:Determinacy
3760:Type theory
3708:undecidable
3640:Truth value
3527:equivalence
3206:non-logical
2819:Enumeration
2809:Isomorphism
2756:cardinality
2740:Von Neumann
2705:Ultrafilter
2670:Uncountable
2604:equivalence
2521:Quantifiers
2511:Fixed-point
2480:First-order
2360:Consistency
2345:Proposition
2322:Traditional
2293:Lindström's
2283:Compactness
2225:Type theory
2170:Cardinality
2099:"Empty Set"
1876:. pp.
1774:"Empty Set"
1595:ontological
1576:Even using
1427:to denote "
728:ordered set
678:permutation
674:derangement
506:belongs to
478:. That is,
271:cardinality
216:\varnothing
204:∅
200:∅
63:cardinality
43:mathematics
4546:0 (number)
4535:Categories
4490:Kurt Gödel
4475:Paul Cohen
4312:Transitive
4080:Identities
4064:Complement
4051:Operations
4012:Regularity
3980:projective
3943:Adjunction
3902:Set theory
3571:elementary
3264:arithmetic
3132:Quantifier
3110:functional
2982:Expression
2700:Transitive
2644:identities
2629:complement
2562:hereditary
2545:Set theory
2079:0155610392
2047:Set Theory
1887:0521293243
1824:007054235X
1783:2020-08-11
1756:References
1578:free logic
1158:Set theory
1009:finite set
1001:clopen set
989:complement
774:, denoted
742:, denoted
250:Properties
143:André Weil
57:having no
4423:Paradoxes
4343:Axiomatic
4322:Universal
4298:Singleton
4293:Recursive
4236:Countable
4231:Amorphous
4090:Power set
4007:Power set
3958:dependent
3953:countable
3842:Supertask
3745:Recursion
3703:decidable
3537:saturated
3515:of models
3438:deductive
3433:axiomatic
3353:Hilbert's
3340:Euclidean
3321:canonical
3244:axiomatic
3176:Signature
3105:Predicate
2994:Extension
2916:Ackermann
2841:Operation
2720:Universal
2710:Recursive
2685:Singleton
2680:Inhabited
2665:Countable
2655:Types of
2639:power set
2609:partition
2526:Predicate
2472:Predicate
2387:Syllogism
2377:Soundness
2350:Inference
2340:Tautology
2242:paradoxes
2104:MathWorld
1967:Routledge
1749:Power set
1669:∅
1649:∅
1631:syllogism
1612:something
1567:something
1557:Standard
1498:≡
1455:≡
1412:≡
1333:∅
1324:∅
1306:∪
1274:∅
1256:∪
1227:∅
1198:α
1192:∪
1189:α
1180:α
1084:∅
939:∞
922:∪
916:∞
907:∞
904:−
889:∅
863:∞
860:−
846:∪
840:∞
831:∞
828:−
813:∅
785:∞
753:∞
750:−
651:empty sum
614:∅
590:∅
570:∅
542:∅
518:∅
494:∅
453:∅
384:∅
353:∅
267:power set
258:, by the
240:\emptyset
226:∅
192:EMPTY SET
168:Norwegian
162:) in the
125:∅
105:Null sign
47:empty set
4427:Problems
4331:Theories
4307:Superset
4283:Infinite
4112:Concepts
3992:Infinity
3909:Overview
3827:Logicism
3820:timeline
3796:Concrete
3655:Validity
3625:T-schema
3618:Kripke's
3613:Tarski's
3608:semantic
3598:Strength
3547:submodel
3542:spectrum
3510:function
3358:Tarski's
3347:Elements
3334:geometry
3290:Robinson
3211:variable
3196:function
3169:spectrum
3159:Sentence
3115:variable
3058:Language
3011:Relation
2972:Automata
2962:Alphabet
2946:language
2800:-jection
2778:codomain
2764:Function
2725:Universe
2695:Infinite
2599:Relation
2382:Validity
2372:Argument
2270:theorem,
2045:(2002).
2000:, 54â72.
1722:See also
1716:reifying
1129:category
1055:function
968:Topology
958:supremum
482:element
336:property
334:For any
189:∅
99:Notation
89:null set
59:elements
51:void set
4365:General
4360:Zermelo
4266:subbase
4248: (
4187:Forcing
4165:Element
4137: (
4115:Methods
4002:Pairing
3769:Related
3566:Diagram
3464: (
3443:Hilbert
3428:Systems
3423:Theorem
3301:of the
3246:systems
3026:Formula
3021:Grammar
2937: (
2881:General
2594:Forcing
2579:Element
2499:Monadic
2274:paradox
2215:Theorem
2151:General
1743:Nothing
1603:nothing
1565:, that
1162:In the
1127:of the
1020:nullary
1016:closure
1005:compact
972:In any
962:infimum
659:product
369:holds (
277:For any
202:and as
179:Unicode
69:. Some
4256:Filter
4246:Finite
4182:Family
4125:Almost
3963:global
3948:Choice
3935:Axioms
3532:finite
3295:Skolem
3248:
3223:Theory
3191:Symbol
3181:String
3164:atomic
3041:ground
3036:closed
3031:atomic
2987:ground
2950:syntax
2846:binary
2773:domain
2690:Finite
2455:finite
2313:Logics
2272:
2220:Theory
2076:
2053:
2034:
2026:
1944:
1884:
1821:
1608:inside
1023:unions
993:closed
530:is in
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