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1710:"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
121:
45:
1651:
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
1625:. 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
975:) 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.
256:
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.
273:, 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".
963:
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741:, 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
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1481:
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
1702:
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.
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Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness
1591:(which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the
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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
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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
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185:) was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.
1366:, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers,
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1995:
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724:), and it is vacuously true that no element (of the empty set) can be found that retains its original position.
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2008:
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of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its
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745:, every real number is both an upper and lower bound for the empty set. When considered as a subset of the
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argued that much of what has been heretofore obtained by set theory can just as easily be obtained by
1652:
Darling, the former is equivalent to "The set of all things that are better than eternal happiness is
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In some textbooks and popularizations, the empty set is referred to as the "null set". However,
85:
3368:
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695:. The empty set can be considered a derangement of itself, because it has only one permutation (
232:
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2033:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
1952:
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88:, while in other theories, its existence can be deduced. Many possible properties of sets are
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which has no members. We cannot conjure such an entity into existence by mere stipulation."
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2004:
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While the empty set is a standard and widely accepted mathematical concept, it remains an
1139:
698:
1110:
958:{\displaystyle \inf \varnothing =\max(\{-\infty ,+\infty \}\cup \mathbb {R} )=+\infty .}
882:{\displaystyle \sup \varnothing =\min(\{-\infty ,+\infty \}\cup \mathbb {R} )=-\infty ,}
4518:
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1692:". The first compares elements of sets, while the second compares the sets themselves.
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which is defined to be greater than every other extended real number), we have that:
641:. This is often paraphrased as "everything is true of the elements of the empty set."
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1930:", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.
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curiosity, whose meaning and usefulness are debated by philosophers and logicians.
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971:) of the empty set is negative infinity, while the greatest lower bound (inf or
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688:
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281:
106:, in which it describes a set of measure zero (which is not necessarily empty).
73:
53:
1849:
1564:. However, the axiom of empty set can be shown redundant in at least two ways:
1177:, 0 is defined as the empty set, and the successor of an ordinal is defined as
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The only subset of the empty set is the empty set itself; equivalently, the
115:
1149:, called the empty space, in just one way: by defining the empty set to be
737:
Since the empty set has no member when it is considered as a subset of any
284:) is zero. The empty set is the only set with either of these properties.
2060:. Springer Monographs in Mathematics (3rd millennium ed.). Springer.
3837:
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are complements of each other, the empty set is also closed, making it a
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which is defined to be less than every other extended real number, and
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formed by adding two "numbers" or "points" to the real numbers (namely
287:
189:
4306:
4128:
2123:
1808:
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299:
1672:" and the latter to "The set {ham sandwich} is better than the set
1153:. This empty topological space is the unique initial object in the
44:
4178:
3945:
2977:
2323:
2168:
2041:(Springer-Verlag edition). Reprinted by Martino Fine Books, 2011.
1630:
222:
119:
43:
1756: â Complete absence of anything; the opposite of everything
206:
182:
157:
3885:
2127:
1355:{\displaystyle 2=1\cup \{1\}=\{\varnothing ,\{\varnothing \}\}}
1762: â Mathematical set containing all subsets of a given set
3881:
1029:
of the empty set is empty. This is known as "preservation of
1729:
sets as singular entities having other entities as members.
1362:, and so on. The von Neumann construction, along with the
1911:
Bruckner, A.N., Bruckner, J.B., and
Thomson, B.S. (2008).
148:", and "â
". The latter two symbols were introduced by the
1750: â Property of sets used in constructive mathematics
1738:
77:
1501:
as an existent set on its own, or if Cantor merely used
680:, since one is the identity element for multiplication.
1544:
itself as a set, but considered it an "improper set".
1165:: only the empty set has a function to the empty set.
95:
Any set other than the empty set is called non-empty.
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In the context of sets of real numbers, Cantor used
521:. Indeed, if it were not true that every element of
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1556:, the existence of the empty set is assured by the
664:) is zero. The reason for this is that zero is the
250:
226:
218:
214:
210:
128:
Common notations for the empty set include "{ }", "
1993:(1984), "To be is to be the value of a variable",
1928:The Empty Set, the Singleton, and the Ordered Pair
1880:
1809:"Earliest Uses of Symbols of Set Theory and Logic"
1743:Pages displaying short descriptions with no spaces
1684:
1664:
1536:
1516:
1493:
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1217:{\displaystyle S(\alpha )=\alpha \cup \{\alpha \}}
1216:
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957:
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629:
617:. Any statement that begins "for every element of
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471:
399:
368:
241:
140:
798:
766:
84:ensure that the empty set exists by including an
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829:
820:
48:The empty set is the set containing no elements.
637:" is not making any substantive claim; it is a
1293:{\displaystyle 1=0\cup \{0\}=\{\varnothing \}}
545:, then there would be at least one element of
3897:
2139:
30:"â
" redirects here. For similar symbols, see
8:
1524:as an emptiness predicate. Zermelo accepted
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1319:
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1134:. As a result, the empty set is the unique
646:set-theoretic definition of natural numbers
102:is a distinct notion within the context of
3904:
3890:
3882:
2965:
2560:
2328:
2146:
2132:
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1828:(3rd ed.). McGraw-Hill. p. 300.
1064:is a set, then there exists precisely one
1938:
1936:
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181:alphabets. In the past, "0" (the numeral
133:
1175:von Neumann construction of the ordinals
1772:
1679:
1659:
1611:The empty set is not the same thing as
1343:
1334:
1284:
1237:
1094:
967:That is, the least upper bound (sup or
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823:
624:
600:
580:
552:
528:
504:
485:, the empty set is a subset of any set
463:
394:
363:
27:Mathematical set containing no elements
1560:, and its uniqueness follows from the
672:of the elements of the empty set (the
648:, zero is modelled by the empty set.
423:, the following two statements hold:
7:
1867:Fonetik og Fonologi: Almen og dansk.
1778:
1776:
1825:Principles of Mathematical Analysis
1617:; rather, it is a set with nothing
1145:The empty set can be turned into a
330:with the empty set is the empty set
949:
926:
917:
873:
850:
841:
795:
763:
341:and the empty set is the empty set
236:
171:LATIN CAPITAL LETTER O WITH STROKE
156:) in 1939, inspired by the letter
135:
25:
1945:The Universal Book of Mathematics
415:Conversely, if for some property
3935:
3865:
1698:argues that while the empty set
1458:contains no single point". This
1388:{\displaystyle \mathbb {N} _{0}}
76:(count of elements in a set) is
1999:91: 430â49. Reprinted in 1998,
1865:e.g. Nina GrĂžnnum (2005, 2013)
593:at all, there is no element of
472:{\displaystyle V=\varnothing .}
37:For other uses of "Empty", see
1243:{\displaystyle 0=\varnothing }
1193:
1187:
1155:category of topological spaces
940:
908:
864:
832:
1:
3826:History of mathematical logic
2077:Modern Elementary Mathematics
1869:Akademisk forlag, Copenhagen.
1399:of arithmetic are satisfied.
1014:. Moreover, the empty set is
806:{\displaystyle +\infty \!\,,}
774:{\displaystyle -\infty \!\,,}
728:In other areas of mathematics
676:) should be considered to be
668:for addition. Similarly, the
188:The symbol â
is available at
4545:Basic concepts in set theory
3751:Primitive recursive function
1685:{\displaystyle \varnothing }
1665:{\displaystyle \varnothing }
1100:{\displaystyle \varnothing }
630:{\displaystyle \varnothing }
606:{\displaystyle \varnothing }
586:{\displaystyle \varnothing }
558:{\displaystyle \varnothing }
534:{\displaystyle \varnothing }
510:{\displaystyle \varnothing }
400:{\displaystyle \varnothing }
369:{\displaystyle \varnothing }
1883:Linear Algebra and Geometry
652:Operations on the empty set
271:principle of extensionality
4566:
4395:von NeumannâBernaysâGödel
2815:SchröderâBernstein theorem
2542:Monadic predicate calculus
2201:Foundations of mathematics
1725:over individuals, without
1706:it is also the case that:
242:{\displaystyle \emptyset }
141:{\displaystyle \emptyset }
124:A symbol for the empty set
113:
36:
29:
4196:One-to-one correspondence
3933:
3861:
3848:Philosophy of mathematics
3797:Automated theorem proving
2968:
2922:Von NeumannâBernaysâGödel
2563:
2081:Harcourt Brace Jovanovich
2007:, and Burgess, J., eds.)
1996:The Journal of Philosophy
1917:, 2nd edition, p. 9.
1572:implies, merely from the
1431:{\displaystyle P\equiv O}
2075:Graham, Malcolm (1975).
2009:Harvard University Press
1914:Elementary Real Analysis
1517:{\displaystyle \equiv O}
1474:{\displaystyle \equiv O}
3498:Self-verifying theories
3319:Tarski's axiomatization
2270:Tarski's undefinability
2265:incompleteness theorems
1879:David M. Bloom (1979).
1621:it and a set is always
1562:axiom of extensionality
1142:of sets and functions.
1018:by the fact that every
565:that is not present in
442:for which the property
438:There is no element of
407:for which the property
387:There is no element of
4154:Constructible universe
3981:Constructibility (V=L)
3872:Mathematics portal
3483:Proof of impossibility
3131:propositional variable
2441:Propositional calculus
2001:Logic, Logic and Logic
1943:D. J. Darling (2004).
1850:"Unicode Standard 5.2"
1822:Rudin, Walter (1976).
1686:
1666:
1538:
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1432:
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1356:
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1006:and the empty set and
959:
883:
807:
775:
718:
631:
607:
587:
559:
535:
511:
473:
401:
370:
316:with the empty set is
243:
142:
125:
82:axiomatic set theories
49:
39:Empty (disambiguation)
4377:Principia Mathematica
4211:Transfinite induction
4070:(i.e. set difference)
3741:Kolmogorov complexity
3694:Computably enumerable
3594:Model complete theory
3386:Principia Mathematica
2446:Propositional formula
2275:BanachâTarski paradox
1789:mathworld.wolfram.com
1723:plural quantification
1687:
1667:
1539:
1519:
1496:
1476:
1453:
1433:
1390:
1357:
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1219:
1163:strict initial object
1125:
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994:by definition, as is
960:
884:
808:
776:
733:Extended real numbers
719:
656:When speaking of the
632:
608:
588:
560:
536:
512:
481:By the definition of
474:
427:For every element of
402:
371:
356:For every element of
249:is coded in LaTeX as
244:
221:. It can be coded in
205:. It can be coded in
143:
123:
47:
4451:Burali-Forti paradox
4206:Set-builder notation
4159:Continuum hypothesis
4099:Symmetric difference
3689:ChurchâTuring thesis
3676:Computability theory
2885:continuum hypothesis
2403:Square of opposition
2261:Gödel's completeness
2049:(paperback edition).
1676:
1656:
1600:Philosophical issues
1548:Axiomatic set theory
1528:
1505:
1485:
1462:
1442:
1416:
1403:Questioned existence
1370:
1304:
1254:
1228:
1181:
1111:
1091:
1071:
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893:
817:
789:
757:
717:{\displaystyle 0!=1}
699:
621:
597:
577:
549:
525:
501:
454:
391:
360:
267:axiomatic set theory
233:
132:
4412:TarskiâGrothendieck
3843:Mathematical object
3734:P versus NP problem
3699:Computable function
3493:Reverse mathematics
3419:Logical consequence
3296:primitive recursive
3291:elementary function
3064:Free/bound variable
2917:TarskiâGrothendieck
2436:Logical connectives
2366:Logical equivalence
2216:Logical consequence
1972:E. J. Lowe (2005).
1949:John Wiley and Sons
1783:Weisstein, Eric W.
1582:axiom of separation
1161:. In fact, it is a
990:, the empty set is
298:The empty set is a
92:for the empty set.
4001:Limitation of size
3641:Transfer principle
3604:Semantics of logic
3589:Categorical theory
3565:Non-standard model
3079:Logical connective
2206:Information theory
2155:Mathematical logic
2107:Weisstein, Eric W.
1682:
1662:
1558:axiom of empty set
1554:Zermelo set theory
1534:
1514:
1491:
1471:
1448:
1428:
1385:
1352:
1290:
1240:
1214:
1123:{\displaystyle A,}
1120:
1097:
1077:
1054:
1002:of an open set is
955:
879:
803:
771:
714:
627:
603:
583:
569:. Since there are
555:
531:
507:
469:
397:
366:
239:
138:
126:
86:axiom of empty set
50:
32:Ă (disambiguation)
4532:
4531:
4441:Russell's paradox
4390:ZermeloâFraenkel
4291:Dedekind-infinite
4164:Diagonal argument
4063:Cartesian product
3927:Set (mathematics)
3879:
3878:
3811:Abstract category
3614:Theories of truth
3424:Rule of inference
3414:Natural deduction
3395:
3394:
2940:
2939:
2645:Cartesian product
2550:
2549:
2456:Many-valued logic
2431:Boolean functions
2314:Russell's paradox
2289:diagonal argument
2186:First-order logic
2047:978-1-61427-131-4
1593:axiom of infinity
1570:first-order logic
1537:{\displaystyle O}
1494:{\displaystyle O}
1451:{\displaystyle P}
1408:Historical issues
1364:axiom of infinity
1147:topological space
1080:{\displaystyle f}
1057:{\displaystyle A}
985:topological space
783:positive infinity
751:negative infinity
691:of a set without
335:Cartesian product
16:(Redirected from
4557:
4514:Bertrand Russell
4504:John von Neumann
4489:Abraham Fraenkel
4484:Richard Dedekind
4446:Suslin's problem
4357:Cantor's theorem
4074:De Morgan's laws
3939:
3906:
3899:
3892:
3883:
3870:
3869:
3821:History of logic
3816:Category of sets
3709:Decision problem
3488:Ordinal analysis
3429:Sequent calculus
3327:Boolean algebras
3267:
3266:
3241:
3212:logical/constant
2966:
2952:
2875:ZermeloâFraenkel
2626:Set operations:
2561:
2498:
2329:
2309:LöwenheimâSkolem
2196:Formal semantics
2148:
2141:
2134:
2125:
2120:
2119:
2094:
2079:(2nd ed.).
2071:
2030:Naive Set Theory
2012:
1988:
1982:
1981:
1969:
1963:
1962:
1940:
1931:
1924:
1918:
1909:
1903:
1902:
1886:
1876:
1870:
1863:
1857:
1856:
1854:
1846:
1840:
1839:
1819:
1813:
1812:
1805:
1799:
1798:
1796:
1795:
1780:
1744:
1691:
1689:
1688:
1683:
1671:
1669:
1668:
1663:
1543:
1541:
1540:
1535:
1523:
1521:
1520:
1515:
1500:
1498:
1497:
1492:
1480:
1478:
1477:
1472:
1457:
1455:
1454:
1449:
1437:
1435:
1434:
1429:
1395:, such that the
1394:
1392:
1391:
1386:
1384:
1383:
1378:
1361:
1359:
1358:
1353:
1299:
1297:
1296:
1291:
1249:
1247:
1246:
1241:
1224:. Thus, we have
1223:
1221:
1220:
1215:
1129:
1127:
1126:
1121:
1106:
1104:
1103:
1098:
1086:
1084:
1083:
1078:
1063:
1061:
1060:
1055:
964:
962:
961:
956:
939:
888:
886:
885:
880:
863:
812:
810:
809:
804:
780:
778:
777:
772:
743:real number line
723:
721:
720:
715:
666:identity element
636:
634:
633:
628:
612:
610:
609:
604:
592:
590:
589:
584:
564:
562:
561:
556:
540:
538:
537:
532:
516:
514:
513:
508:
478:
476:
475:
470:
406:
404:
403:
398:
375:
373:
372:
367:
252:
248:
246:
245:
240:
228:
220:
216:
212:
204:
201:
198:
196:
172:
169:
166:
164:
147:
145:
144:
139:
21:
4565:
4564:
4560:
4559:
4558:
4556:
4555:
4554:
4535:
4534:
4533:
4528:
4455:
4434:
4418:
4383:New Foundations
4330:
4220:
4139:Cardinal number
4122:
4108:
4049:
3940:
3931:
3915:
3910:
3880:
3875:
3864:
3857:
3802:Category theory
3792:Algebraic logic
3775:
3746:Lambda calculus
3684:Church encoding
3670:
3646:Truth predicate
3502:
3468:Complete theory
3391:
3260:
3256:
3252:
3247:
3239:
2959: and
2955:
2950:
2936:
2912:New Foundations
2880:axiom of choice
2863:
2825:Gödel numbering
2765: and
2757:
2661:
2546:
2496:
2477:
2426:Boolean algebra
2412:
2376:Equiconsistency
2341:Classical logic
2318:
2299:Halting problem
2287: and
2263: and
2251: and
2250:
2245:Theorems (
2240:
2157:
2152:
2105:
2104:
2101:
2091:
2074:
2068:
2052:
2021:
2019:Further reading
2016:
2015:
2005:Richard Jeffrey
1989:
1985:
1971:
1970:
1966:
1959:
1951:. p. 106.
1942:
1941:
1934:
1925:
1921:
1910:
1906:
1899:
1878:
1877:
1873:
1864:
1860:
1852:
1848:
1847:
1843:
1836:
1821:
1820:
1816:
1807:
1806:
1802:
1793:
1791:
1782:
1781:
1774:
1769:
1742:
1735:
1674:
1673:
1654:
1653:
1633:that involve a
1602:
1550:
1526:
1525:
1503:
1502:
1483:
1482:
1460:
1459:
1440:
1439:
1414:
1413:
1410:
1405:
1373:
1368:
1367:
1302:
1301:
1252:
1251:
1226:
1225:
1179:
1178:
1171:
1159:continuous maps
1109:
1108:
1089:
1088:
1069:
1068:
1046:
1045:
1042:
1040:Category theory
981:
891:
890:
815:
814:
787:
786:
755:
754:
735:
730:
697:
696:
654:
619:
618:
613:that is not in
595:
594:
575:
574:
547:
546:
523:
522:
499:
498:
452:
451:
389:
388:
376:, the property
358:
357:
263:
231:
230:
202:
199:
194:
193:
170:
167:
162:
161:
130:
129:
118:
112:
42:
35:
28:
23:
22:
15:
12:
11:
5:
4563:
4561:
4553:
4552:
4547:
4537:
4536:
4530:
4529:
4527:
4526:
4521:
4519:Thoralf Skolem
4516:
4511:
4506:
4501:
4496:
4491:
4486:
4481:
4476:
4471:
4465:
4463:
4457:
4456:
4454:
4453:
4448:
4443:
4437:
4435:
4433:
4432:
4429:
4423:
4420:
4419:
4417:
4416:
4415:
4414:
4409:
4404:
4403:
4402:
4387:
4386:
4385:
4373:
4372:
4371:
4360:
4359:
4354:
4349:
4344:
4338:
4336:
4332:
4331:
4329:
4328:
4323:
4318:
4313:
4304:
4299:
4294:
4284:
4279:
4278:
4277:
4272:
4267:
4257:
4247:
4242:
4237:
4231:
4229:
4222:
4221:
4219:
4218:
4213:
4208:
4203:
4201:Ordinal number
4198:
4193:
4188:
4183:
4182:
4181:
4176:
4166:
4161:
4156:
4151:
4146:
4136:
4131:
4125:
4123:
4121:
4120:
4117:
4113:
4110:
4109:
4107:
4106:
4101:
4096:
4091:
4086:
4081:
4079:Disjoint union
4076:
4071:
4065:
4059:
4057:
4051:
4050:
4048:
4047:
4046:
4045:
4040:
4029:
4028:
4026:Martin's axiom
4023:
4018:
4013:
4008:
4003:
3998:
3993:
3991:Extensionality
3988:
3983:
3978:
3977:
3976:
3971:
3966:
3956:
3950:
3948:
3942:
3941:
3934:
3932:
3930:
3929:
3923:
3921:
3917:
3916:
3911:
3909:
3908:
3901:
3894:
3886:
3877:
3876:
3862:
3859:
3858:
3856:
3855:
3850:
3845:
3840:
3835:
3834:
3833:
3823:
3818:
3813:
3804:
3799:
3794:
3789:
3787:Abstract logic
3783:
3781:
3777:
3776:
3774:
3773:
3768:
3766:Turing machine
3763:
3758:
3753:
3748:
3743:
3738:
3737:
3736:
3731:
3726:
3721:
3716:
3706:
3704:Computable set
3701:
3696:
3691:
3686:
3680:
3678:
3672:
3671:
3669:
3668:
3663:
3658:
3653:
3648:
3643:
3638:
3633:
3632:
3631:
3626:
3621:
3611:
3606:
3601:
3599:Satisfiability
3596:
3591:
3586:
3585:
3584:
3574:
3573:
3572:
3562:
3561:
3560:
3555:
3550:
3545:
3540:
3530:
3529:
3528:
3523:
3516:Interpretation
3512:
3510:
3504:
3503:
3501:
3500:
3495:
3490:
3485:
3480:
3470:
3465:
3464:
3463:
3462:
3461:
3451:
3446:
3436:
3431:
3426:
3421:
3416:
3411:
3405:
3403:
3397:
3396:
3393:
3392:
3390:
3389:
3381:
3380:
3379:
3378:
3373:
3372:
3371:
3366:
3361:
3341:
3340:
3339:
3337:minimal axioms
3334:
3323:
3322:
3321:
3310:
3309:
3308:
3303:
3298:
3293:
3288:
3283:
3270:
3268:
3249:
3248:
3246:
3245:
3244:
3243:
3231:
3226:
3225:
3224:
3219:
3214:
3209:
3199:
3194:
3189:
3184:
3183:
3182:
3177:
3167:
3166:
3165:
3160:
3155:
3150:
3140:
3135:
3134:
3133:
3128:
3123:
3113:
3112:
3111:
3106:
3101:
3096:
3091:
3086:
3076:
3071:
3066:
3061:
3060:
3059:
3054:
3049:
3044:
3034:
3029:
3027:Formation rule
3024:
3019:
3018:
3017:
3012:
3002:
3001:
3000:
2990:
2985:
2980:
2975:
2969:
2963:
2946:Formal systems
2942:
2941:
2938:
2937:
2935:
2934:
2929:
2924:
2919:
2914:
2909:
2904:
2899:
2894:
2889:
2888:
2887:
2882:
2871:
2869:
2865:
2864:
2862:
2861:
2860:
2859:
2849:
2844:
2843:
2842:
2835:Large cardinal
2832:
2827:
2822:
2817:
2812:
2798:
2797:
2796:
2791:
2786:
2771:
2769:
2759:
2758:
2756:
2755:
2754:
2753:
2748:
2743:
2733:
2728:
2723:
2718:
2713:
2708:
2703:
2698:
2693:
2688:
2683:
2678:
2672:
2670:
2663:
2662:
2660:
2659:
2658:
2657:
2652:
2647:
2642:
2637:
2632:
2624:
2623:
2622:
2617:
2607:
2602:
2600:Extensionality
2597:
2595:Ordinal number
2592:
2582:
2577:
2576:
2575:
2564:
2558:
2552:
2551:
2548:
2547:
2545:
2544:
2539:
2534:
2529:
2524:
2519:
2514:
2513:
2512:
2502:
2501:
2500:
2487:
2485:
2479:
2478:
2476:
2475:
2474:
2473:
2468:
2463:
2453:
2448:
2443:
2438:
2433:
2428:
2422:
2420:
2414:
2413:
2411:
2410:
2405:
2400:
2395:
2390:
2385:
2380:
2379:
2378:
2368:
2363:
2358:
2353:
2348:
2343:
2337:
2335:
2326:
2320:
2319:
2317:
2316:
2311:
2306:
2301:
2296:
2291:
2279:Cantor's
2277:
2272:
2267:
2257:
2255:
2242:
2241:
2239:
2238:
2233:
2228:
2223:
2218:
2213:
2208:
2203:
2198:
2193:
2188:
2183:
2178:
2177:
2176:
2165:
2163:
2159:
2158:
2153:
2151:
2150:
2143:
2136:
2128:
2122:
2121:
2100:
2099:External links
2097:
2096:
2095:
2089:
2072:
2066:
2050:
2020:
2017:
2014:
2013:
1983:
1964:
1957:
1932:
1926:A. Kanamori, "
1919:
1904:
1897:
1871:
1858:
1841:
1834:
1814:
1800:
1771:
1770:
1768:
1765:
1764:
1763:
1757:
1751:
1745:
1741: â Number
1734:
1731:
1716:
1715:
1713:
1704:
1703:
1681:
1661:
1649:
1648:
1624:
1620:
1616:
1601:
1598:
1597:
1596:
1585:
1579:
1574:logical axioms
1549:
1546:
1533:
1513:
1510:
1490:
1470:
1467:
1447:
1427:
1424:
1421:
1409:
1406:
1404:
1401:
1382:
1377:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1239:
1236:
1233:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1170:
1167:
1136:initial object
1132:empty function
1119:
1116:
1096:
1076:
1053:
1041:
1038:
980:
977:
954:
951:
948:
945:
942:
938:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
878:
875:
872:
869:
866:
862:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
802:
797:
794:
770:
765:
762:
747:extended reals
734:
731:
729:
726:
713:
710:
707:
704:
653:
650:
626:
602:
582:
572:
554:
530:
506:
492:
468:
465:
462:
459:
448:
447:
436:
413:
412:
396:
385:
365:
343:
342:
331:
320:
306:
262:
259:
238:
152:(specifically
150:Bourbaki group
137:
114:Main article:
111:
108:
104:measure theory
90:vacuously true
72:; its size or
64:is the unique
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4562:
4551:
4548:
4546:
4543:
4542:
4540:
4525:
4524:Ernst Zermelo
4522:
4520:
4517:
4515:
4512:
4510:
4509:Willard Quine
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4466:
4464:
4462:
4461:Set theorists
4458:
4452:
4449:
4447:
4444:
4442:
4439:
4438:
4436:
4430:
4428:
4425:
4424:
4421:
4413:
4410:
4408:
4407:KripkeâPlatek
4405:
4401:
4398:
4397:
4396:
4393:
4392:
4391:
4388:
4384:
4381:
4380:
4379:
4378:
4374:
4370:
4367:
4366:
4365:
4362:
4361:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4339:
4337:
4333:
4327:
4324:
4322:
4319:
4317:
4314:
4312:
4310:
4305:
4303:
4300:
4298:
4295:
4292:
4288:
4285:
4283:
4280:
4276:
4273:
4271:
4268:
4266:
4263:
4262:
4261:
4258:
4255:
4251:
4248:
4246:
4243:
4241:
4238:
4236:
4233:
4232:
4230:
4227:
4223:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4184:
4180:
4177:
4175:
4172:
4171:
4170:
4167:
4165:
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4144:
4140:
4137:
4135:
4132:
4130:
4127:
4126:
4124:
4118:
4115:
4114:
4111:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4085:
4082:
4080:
4077:
4075:
4072:
4069:
4066:
4064:
4061:
4060:
4058:
4056:
4052:
4044:
4043:specification
4041:
4039:
4036:
4035:
4034:
4031:
4030:
4027:
4024:
4022:
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3987:
3984:
3982:
3979:
3975:
3972:
3970:
3967:
3965:
3962:
3961:
3960:
3957:
3955:
3952:
3951:
3949:
3947:
3943:
3938:
3928:
3925:
3924:
3922:
3918:
3914:
3907:
3902:
3900:
3895:
3893:
3888:
3887:
3884:
3874:
3873:
3868:
3860:
3854:
3851:
3849:
3846:
3844:
3841:
3839:
3836:
3832:
3829:
3828:
3827:
3824:
3822:
3819:
3817:
3814:
3812:
3808:
3805:
3803:
3800:
3798:
3795:
3793:
3790:
3788:
3785:
3784:
3782:
3778:
3772:
3769:
3767:
3764:
3762:
3761:Recursive set
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3711:
3710:
3707:
3705:
3702:
3700:
3697:
3695:
3692:
3690:
3687:
3685:
3682:
3681:
3679:
3677:
3673:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3615:
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3590:
3587:
3583:
3580:
3579:
3578:
3575:
3571:
3570:of arithmetic
3568:
3567:
3566:
3563:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3535:
3534:
3531:
3527:
3524:
3522:
3519:
3518:
3517:
3514:
3513:
3511:
3509:
3505:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3481:
3478:
3477:from ZFC
3474:
3471:
3469:
3466:
3460:
3457:
3456:
3455:
3452:
3450:
3447:
3445:
3442:
3441:
3440:
3437:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3410:
3407:
3406:
3404:
3402:
3398:
3388:
3387:
3383:
3382:
3377:
3376:non-Euclidean
3374:
3370:
3367:
3365:
3362:
3360:
3359:
3355:
3354:
3352:
3349:
3348:
3346:
3342:
3338:
3335:
3333:
3330:
3329:
3328:
3324:
3320:
3317:
3316:
3315:
3311:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3278:
3276:
3272:
3271:
3269:
3264:
3258:
3253:Example
3250:
3242:
3237:
3236:
3235:
3232:
3230:
3227:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3204:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3181:
3178:
3176:
3173:
3172:
3171:
3168:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3145:
3144:
3141:
3139:
3136:
3132:
3129:
3127:
3124:
3122:
3119:
3118:
3117:
3114:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3081:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3039:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3016:
3013:
3011:
3010:by definition
3008:
3007:
3006:
3003:
2999:
2996:
2995:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2974:
2971:
2970:
2967:
2964:
2962:
2958:
2953:
2947:
2943:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2897:KripkeâPlatek
2895:
2893:
2890:
2886:
2883:
2881:
2878:
2877:
2876:
2873:
2872:
2870:
2866:
2858:
2855:
2854:
2853:
2850:
2848:
2845:
2841:
2838:
2837:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2810:
2806:
2802:
2799:
2795:
2792:
2790:
2787:
2785:
2782:
2781:
2780:
2776:
2773:
2772:
2770:
2768:
2764:
2760:
2752:
2749:
2747:
2744:
2742:
2741:constructible
2739:
2738:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2687:
2684:
2682:
2679:
2677:
2674:
2673:
2671:
2669:
2664:
2656:
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2627:
2625:
2621:
2618:
2616:
2613:
2612:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2587:
2583:
2581:
2578:
2574:
2571:
2570:
2569:
2566:
2565:
2562:
2559:
2557:
2553:
2543:
2540:
2538:
2535:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2511:
2508:
2507:
2506:
2503:
2499:
2494:
2493:
2492:
2489:
2488:
2486:
2484:
2480:
2472:
2469:
2467:
2464:
2462:
2459:
2458:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2423:
2421:
2419:
2418:Propositional
2415:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2389:
2386:
2384:
2381:
2377:
2374:
2373:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2346:Logical truth
2344:
2342:
2339:
2338:
2336:
2334:
2330:
2327:
2325:
2321:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2290:
2286:
2282:
2278:
2276:
2273:
2271:
2268:
2266:
2262:
2259:
2258:
2256:
2254:
2248:
2243:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2209:
2207:
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2175:
2172:
2171:
2170:
2167:
2166:
2164:
2160:
2156:
2149:
2144:
2142:
2137:
2135:
2130:
2129:
2126:
2117:
2116:
2111:
2108:
2103:
2102:
2098:
2092:
2086:
2082:
2078:
2073:
2069:
2067:3-540-44085-2
2063:
2059:
2055:
2051:
2048:
2044:
2040:
2039:0-387-90092-6
2036:
2032:
2031:
2026:
2023:
2022:
2018:
2010:
2006:
2002:
1998:
1997:
1992:
1991:George Boolos
1987:
1984:
1980:. p. 87.
1979:
1975:
1968:
1965:
1960:
1958:0-471-27047-4
1954:
1950:
1946:
1939:
1937:
1933:
1929:
1923:
1920:
1916:
1915:
1908:
1905:
1900:
1894:
1890:
1885:
1884:
1875:
1872:
1868:
1862:
1859:
1851:
1845:
1842:
1837:
1831:
1827:
1826:
1818:
1815:
1810:
1804:
1801:
1790:
1786:
1779:
1777:
1773:
1766:
1761:
1758:
1755:
1752:
1749:
1748:Inhabited set
1746:
1740:
1737:
1736:
1732:
1730:
1728:
1724:
1720:
1719:George Boolos
1711:
1709:
1708:
1707:
1701:
1700:
1699:
1697:
1696:Jonathan Lowe
1693:
1646:
1645:
1644:
1643:
1638:
1636:
1632:
1628:
1627:opening moves
1622:
1618:
1615:
1612:
1609:
1607:
1599:
1594:
1590:
1586:
1583:
1577:
1575:
1571:
1567:
1566:
1565:
1563:
1559:
1555:
1547:
1545:
1531:
1511:
1508:
1488:
1468:
1465:
1445:
1425:
1422:
1419:
1407:
1402:
1400:
1398:
1380:
1365:
1337:
1328:
1322:
1316:
1313:
1310:
1307:
1278:
1272:
1266:
1263:
1260:
1257:
1234:
1231:
1208:
1202:
1199:
1196:
1190:
1184:
1176:
1168:
1166:
1164:
1160:
1156:
1152:
1148:
1143:
1141:
1137:
1133:
1117:
1114:
1074:
1067:
1051:
1039:
1037:
1035:
1032:
1028:
1023:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
989:
986:
978:
976:
974:
970:
965:
952:
946:
943:
932:
923:
920:
914:
902:
876:
870:
867:
856:
847:
844:
838:
826:
800:
792:
784:
768:
760:
752:
748:
744:
740:
732:
727:
725:
711:
708:
705:
702:
694:
690:
686:
681:
679:
675:
674:empty product
671:
667:
663:
659:
651:
649:
647:
644:In the usual
642:
640:
639:vacuous truth
616:
570:
568:
544:
520:
496:
490:
488:
484:
479:
466:
460:
457:
445:
441:
437:
434:
431:the property
430:
426:
425:
424:
422:
419:and some set
418:
410:
386:
383:
382:vacuous truth
379:
355:
354:
353:
351:
348:
340:
336:
332:
329:
325:
321:
319:
315:
311:
307:
305:
301:
297:
296:
295:
293:
289:
285:
283:
279:
274:
272:
268:
260:
258:
254:
229:. The symbol
224:
208:
191:
186:
184:
180:
176:
159:
155:
151:
122:
117:
109:
107:
105:
101:
96:
93:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
46:
40:
33:
19:
4474:Georg Cantor
4469:Paul Bernays
4400:MorseâKelley
4375:
4308:
4307:Subset
4254:hereditarily
4244:
4216:Venn diagram
4174:ordered pair
4089:Intersection
4033:Axiom schema
3863:
3661:Ultraproduct
3508:Model theory
3473:Independence
3409:Formal proof
3401:Proof theory
3384:
3357:
3314:real numbers
3286:second-order
3197:Substitution
3074:Metalanguage
3015:conservative
2988:Axiom schema
2932:Constructive
2902:MorseâKelley
2868:Set theories
2847:Aleph number
2840:inaccessible
2746:Grothendieck
2685:
2630:intersection
2517:Higher-order
2505:Second-order
2451:Truth tables
2408:Venn diagram
2191:Formal proof
2113:
2076:
2057:
2054:Jech, Thomas
2028:
2025:Halmos, Paul
2000:
1994:
1986:
1973:
1967:
1944:
1922:
1912:
1907:
1882:
1874:
1866:
1861:
1844:
1824:
1817:
1803:
1792:. Retrieved
1788:
1717:
1705:
1694:
1650:
1640:The popular
1639:
1610:
1603:
1551:
1411:
1397:Peano axioms
1172:
1144:
1043:
1024:
1022:is compact.
1007:
998:. Since the
995:
987:
982:
966:
736:
693:fixed points
682:
655:
643:
614:
573:elements of
566:
542:
518:
494:
486:
480:
449:
443:
439:
432:
428:
420:
416:
414:
408:
377:
349:
344:
338:
327:
324:intersection
317:
313:
303:
291:
286:
275:
265:In standard
264:
255:
219:∅
187:
127:
97:
94:
61:
57:
51:
4499:Thomas Jech
4342:Alternative
4321:Uncountable
4275:Ultrafilter
4134:Cardinality
4038:replacement
3986:Determinacy
3771:Type theory
3719:undecidable
3651:Truth value
3538:equivalence
3217:non-logical
2830:Enumeration
2820:Isomorphism
2767:cardinality
2751:Von Neumann
2716:Ultrafilter
2681:Uncountable
2615:equivalence
2532:Quantifiers
2522:Fixed-point
2491:First-order
2371:Consistency
2356:Proposition
2333:Traditional
2304:Lindström's
2294:Compactness
2236:Type theory
2181:Cardinality
2110:"Empty Set"
1887:. pp.
1785:"Empty Set"
1606:ontological
1587:Even using
1438:to denote "
739:ordered set
689:permutation
685:derangement
517:belongs to
489:. That is,
282:cardinality
227:\varnothing
215:∅
211:∅
74:cardinality
54:mathematics
4550:0 (number)
4539:Categories
4494:Kurt Gödel
4479:Paul Cohen
4316:Transitive
4084:Identities
4068:Complement
4055:Operations
4016:Regularity
3954:Adjunction
3913:Set theory
3582:elementary
3275:arithmetic
3143:Quantifier
3121:functional
2993:Expression
2711:Transitive
2655:identities
2640:complement
2573:hereditary
2556:Set theory
2090:0155610392
2058:Set Theory
1898:0521293243
1835:007054235X
1794:2020-08-11
1767:References
1589:free logic
1169:Set theory
1020:finite set
1012:clopen set
1000:complement
785:, denoted
753:, denoted
261:Properties
154:André Weil
68:having no
4427:Paradoxes
4347:Axiomatic
4326:Universal
4302:Singleton
4297:Recursive
4240:Countable
4235:Amorphous
4094:Power set
4011:Power set
3969:dependent
3964:countable
3853:Supertask
3756:Recursion
3714:decidable
3548:saturated
3526:of models
3449:deductive
3444:axiomatic
3364:Hilbert's
3351:Euclidean
3332:canonical
3255:axiomatic
3187:Signature
3116:Predicate
3005:Extension
2927:Ackermann
2852:Operation
2731:Universal
2721:Recursive
2696:Singleton
2691:Inhabited
2676:Countable
2666:Types of
2650:power set
2620:partition
2537:Predicate
2483:Predicate
2398:Syllogism
2388:Soundness
2361:Inference
2351:Tautology
2253:paradoxes
2115:MathWorld
1978:Routledge
1760:Power set
1680:∅
1660:∅
1642:syllogism
1623:something
1578:something
1568:Standard
1509:≡
1466:≡
1423:≡
1344:∅
1335:∅
1317:∪
1285:∅
1267:∪
1238:∅
1209:α
1203:∪
1200:α
1191:α
1095:∅
950:∞
933:∪
927:∞
918:∞
915:−
900:∅
874:∞
871:−
857:∪
851:∞
842:∞
839:−
824:∅
796:∞
764:∞
761:−
662:empty sum
625:∅
601:∅
581:∅
553:∅
529:∅
505:∅
464:∅
395:∅
364:∅
278:power set
269:, by the
251:\emptyset
237:∅
203:EMPTY SET
179:Norwegian
173:) in the
136:∅
116:Null sign
58:empty set
4431:Problems
4335:Theories
4311:Superset
4287:Infinite
4116:Concepts
3996:Infinity
3920:Overview
3838:Logicism
3831:timeline
3807:Concrete
3666:Validity
3636:T-schema
3629:Kripke's
3624:Tarski's
3619:semantic
3609:Strength
3558:submodel
3553:spectrum
3521:function
3369:Tarski's
3358:Elements
3345:geometry
3301:Robinson
3222:variable
3207:function
3180:spectrum
3170:Sentence
3126:variable
3069:Language
3022:Relation
2983:Automata
2973:Alphabet
2957:language
2811:-jection
2789:codomain
2775:Function
2736:Universe
2706:Infinite
2610:Relation
2393:Validity
2383:Argument
2281:theorem,
2056:(2002).
2011:, 54â72.
1733:See also
1727:reifying
1140:category
1066:function
979:Topology
969:supremum
493:element
347:property
345:For any
200:∅
110:Notation
100:null set
70:elements
62:void set
18:Nonempty
4369:General
4364:Zermelo
4270:subbase
4252: (
4191:Forcing
4169:Element
4141: (
4119:Methods
4006:Pairing
3780:Related
3577:Diagram
3475: (
3454:Hilbert
3439:Systems
3434:Theorem
3312:of the
3257:systems
3037:Formula
3032:Grammar
2948: (
2892:General
2605:Forcing
2590:Element
2510:Monadic
2285:paradox
2226:Theorem
2162:General
1754:Nothing
1614:nothing
1576:, that
1173:In the
1138:of the
1031:nullary
1027:closure
1016:compact
983:In any
973:infimum
670:product
380:holds (
288:For any
213:and as
190:Unicode
80:. Some
4260:Filter
4250:Finite
4186:Family
4129:Almost
3974:global
3959:Choice
3946:Axioms
3543:finite
3306:Skolem
3259:
3234:Theory
3202:Symbol
3192:String
3175:atomic
3052:ground
3047:closed
3042:atomic
2998:ground
2961:syntax
2857:binary
2784:domain
2701:Finite
2466:finite
2324:Logics
2283:
2231:Theory
2087:
2064:
2045:
2037:
1955:
1895:
1832:
1619:inside
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