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In a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with
1662:
Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the
1035:
will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order.
2614:
This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory. Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic may be found in any of these:
2211:, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a
1306:(−π/2, π/2) was made larger to include an integer multiple of π/2, then this function would no longer be onto (surjective), since there is no real number which could be paired with the multiple of π/2 by this arctan function.
841:, up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same
1844:
1509:
1402:
1036:
Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list.
1006:). With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto".
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Earliest Uses of Some of the Words of
Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.
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John Meakin (2007). "Groups and semigroups: connections and contrasts". In C.M. Campbell; M.R. Quick; E.F. Robertson; G.C. Smith (eds.).
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and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category
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The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.
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team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set
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1599:, yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain
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There are names associated to properties (1) and (2) as well. A relation which satisfies property (1) is called a
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1595:(by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not,
1020:
880:
2285:
2215:, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the
4900:
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4019:
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2115:
459:
226:
151:
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813:)—meaning that each element of the codomain is mapped to from at least one element of the domain. The term
511:
between two sets such that each element of either set is paired with exactly one element of the other set.
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59:
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the seat they are sitting in. What the instructor observed in order to reach this conclusion was that:
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805:)—meaning that each element in the codomain is mapped to from at most one element of the domain—and
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971:. It is more common to see properties (1) and (2) written as a single statement: Every element of
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simply defined on the complex plane, rather than its completion to the extended complex plane.
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is not onto (surjective). However, if the codomain is restricted to the positive real numbers
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exists and is also a bijection. Functions that have inverse functions are said to be
868:
2480:
Mathematics across the Iron
Curtain: A History of the Algebraic Theory of Semigroups
859:
Some bijections with further properties have received specific names, which include
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will be the players on the team (of size nine in the case of baseball) and the set
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Another way of defining the same notion is to say that a partial bijection from
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1983:
to itself, together with the operation of functional composition (∘), form a
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2014:
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batting order and outputs the player who will be batting in that position.
2453:"Bijection, Injection, And Surjection | Brilliant Math & Science Wiki"
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1303:
827:
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427:
166:
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When the partial bijection is on the same set, it is sometimes called a
4813:
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1079:
1024:
783:
432:
2191:
since they must preserve the group structure, so the isomorphisms are
1675:
A bijection composed of an injection (X → Y) and a surjection (Y → Z).
48:, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example,
32:
3503:
3275:
3097:
2259:
1515:
would be bijective; its inverse is the positive square root function.
1055:
Every seat had someone sitting there (there were no empty seats), and
2507:
Handbook of
Categorical Algebra: Volume 2, Categories and Structures
2845:
2733:
Sets, Functions, and Logic: An
Introduction to Abstract Mathematics
4357:
3703:
3548:
3147:
2914:
2234:(which turns out to be a partial function) with the property that
1839:{\displaystyle (g\,\circ \,f)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})}
1069:
31:
2336:
A non-injective surjective function (surjection, not a bijection)
2308:
An injective non-surjective function (injection, not a bijection)
2147:
is the same as the number of total orderings of that set—namely,
1504:{\displaystyle \mathbb {R} _{0}^{+}\equiv \left[0,\infty \right)}
1935:, this is taken as the definition of "same number of elements" (
980:
3507:
2854:
2348:
A non-injective non-surjective function (also not a bijection)
1603:. Moreover, properties (1) and (2) then say that this inverse
2850:
1615:. A function is invertible if and only if it is a bijection.
1397:{\displaystyle \mathbb {R} ^{+}\equiv \left(0,\infty \right)}
960:
Satisfying properties (1) and (2) means that a pairing is a
503:) is the image of exactly one element of the first set (the
213:
1947:, a way to distinguish the various sizes of infinite sets.
848:
A bijective function from a set to itself is also called a
2661:
Chapter Zero: Fundamental
Notions of Abstract Mathematics
1049:
Every student was in a seat (there was no one standing),
994:). Functions which satisfy property (4) are said to be "
823:, which means injective but not necessarily surjective.
2203:
The notion of one-to-one correspondence generalizes to
1626:
is bijective if and only if it satisfies the condition
979:. Functions which satisfy property (3) are said to be "
2634:
Proof, Logic and
Conjecture: A Mathematician's Toolbox
1972:
meets every horizontal and vertical line exactly once.
1913:, then there exists a bijection between the two sets
1855:
1760:
1732:
1688:
1461:
1359:
852:, and the set of all permutations of a set forms its
759:
739:
695:
675:
655:
611:
560:
528:
1877:
of two functions is bijective, it only follows that
1618:
Stated in concise mathematical notation, a function
5159:
5054:
4886:
4779:
4631:
4324:
4247:
4141:
4045:
3934:
3861:
3796:
3711:
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3624:
3541:
3428:
3391:
3303:
3193:
3081:
3022:
2913:
2888:
2534:
2127:, there is a bijection between the set of possible
1408:would be bijective; its inverse (see below) is the
1333:) = e, is not bijective: for instance, there is no
142:
2753:
1869:
1838:
1746:
1702:
1503:
1396:
1239:is obtained from (or paired with) the real number
797:A function is bijective if and only if it is both
768:
745:
725:
681:
661:
641:
581:
546:
2742:Mathematical Thinking: Problem Solving and Proofs
2670:The Structure of Proof: With Logic and Set Theory
138:
554:is bijective if and only if there is a function
258:
2131:of the elements and the set of bijections from
1607:is a surjection and an injection, that is, the
1451:is not one-to-one (injective). However, if the
2652:A Transition to Advanced Mathematics (6th Ed.)
2483:. American Mathematical Society. p. 251.
1235:is non-zero) is a bijection. Each real number
911:to be a bijection, four properties must hold:
499:such that each element of the second set (the
3519:
2866:
1931:have the same number of elements. Indeed, in
1286:in the interval (−π/2, π/2) so that tan(
1015:Batting line-up of a baseball or cricket team
460:
8:
1979:is a set, then the bijective functions from
952:may be paired with more than one element of
941:must be paired with at least one element of
930:may be paired with more than one element of
919:must be paired with at least one element of
114:
2562:. Cambridge University Press. p. 367.
2510:. Cambridge University Press. p. 289.
4345:
3940:
3708:
3526:
3512:
3504:
2873:
2859:
2851:
1792:
1788:
1726:is a bijection, whose inverse is given by
845:if there exists a bijection between them.
467:
453:
123:
55:
2724:An Introduction to Mathematical Reasoning
2643:Mathematical Reasoning: Writing and Proof
2596:
1863:
1859:
1854:
1824:
1816:
1812:
1800:
1779:
1771:
1767:
1759:
1740:
1736:
1731:
1696:
1692:
1687:
1473:
1468:
1464:
1463:
1460:
1366:
1362:
1361:
1358:
758:
738:
694:
674:
654:
610:
559:
527:
2706:An Introduction to Mathematical Thinking
1670:
1058:No seat had more than one student in it.
2415:
2298:
1939:), and generalising this definition to
1278:) is bijective, since each real number
441:
388:
355:
275:
243:
234:
90:
83:
58:
2697:Analysis with an introduction to proof
975:is paired with exactly one element of
2477:Christopher Hollings (16 July 2014).
2092:, then the following are equivalent:
2040:|, one has the following equalities:
1052:No student was in more than one seat,
891:of apparently very different nature.
597:, such that each of the two ways for
262:
247:
228:
217:
202:
183:
172:
153:
127:
108:
7:
2786:Introduction to Advanced Mathematics
2777:Introduction to Abstract Mathematics
2422:
1546:, there exists a bijective function
198:
189:
2438:and a relation satisfying (2) is a
2384:Bijection, injection and surjection
2199:Generalization to partial functions
2195:which are bijective homomorphisms.
2320:An injective surjective function (
2036:of the codomain with cardinality |
1995:, which is denoted variously by S(
1493:
1386:
830:establishes a bijection from some
507:). Equivalently, a bijection is a
25:
2715:Foundations of Higher Mathematics
2688:Mathematical Thinking and Writing
2139:. That is to say, the number of
1282:is paired with exactly one angle
1175:+ 1 is bijective, since for each
1040:Seats and students of a classroom
5245:
2904:
2795:A Primer of Abstract Mathematics
2735:. Chapman & Hall/ CRC Press.
2363:
2341:
2329:
2313:
2301:
2028:of the domain with cardinality |
1968:is bijective if and only if its
1439:is not bijective: for instance,
168:
159:
2650:Smith; Eggen; St.Andre (2006).
2560:Groups St Andrews 2005 Volume 2
2084:with the same cardinality, and
1849:Conversely, if the composition
887:are bijections between sets of
85:History of the function concept
2679:Bridge to Abstract Mathematics
1833:
1817:
1809:
1793:
1776:
1761:
1530:, and two injective functions
714:
711:
705:
699:
630:
627:
621:
615:
601:the two functions produces an
570:
538:
1:
5206:History of mathematical logic
2160:Bijections are precisely the
782:defines a bijection from the
5277:Basic concepts in set theory
5131:Primitive recursive function
2756:The Nuts and Bolts of Proofs
1870:{\displaystyle g\,\circ \,f}
1747:{\displaystyle g\,\circ \,f}
1703:{\displaystyle g\,\circ \,f}
826:The elementary operation of
2815:Encyclopedia of Mathematics
2690:. Harcourt/ Academic Press.
2581:"The Möbius Inverse Monoid"
2217:symmetric inverse semigroup
5303:
4195:Schröder–Bernstein theorem
3922:Monadic predicate calculus
3581:Foundations of mathematics
3364:von Neumann–Bernays–Gödel
2704:Gilbert; Vanstone (2005).
2537:. CRC Press. p. 228.
2531:Pierre A. Grillet (1995).
1520:Schröder–Bernstein theorem
1066:More mathematical examples
817:must not be confused with
443:List of specific functions
5241:
5228:Philosophy of mathematics
5177:Automated theorem proving
4348:
4302:Von Neumann–Bernays–Gödel
3943:
3165:One-to-one correspondence
2902:
2784:Barnier; Feldman (2000).
881:geometric transformations
815:one-to-one correspondence
794:as its inverse function.
726:{\displaystyle f(g(y))=y}
642:{\displaystyle g(f(x))=x}
582:{\displaystyle g:Y\to X,}
491:between two mathematical
489:one-to-one correspondence
27:One-to-one correspondence
2708:. Pearson Prentice-Hall.
2654:. Thomson (Brooks/Cole).
2504:Francis Borceux (1994).
2207:, where they are called
2187:, the morphisms must be
1943:leads to the concept of
1266:→ (−π/2, π/2), given by
903:pairing elements of set
547:{\displaystyle f:X\to Y}
514:A function is bijective
4878:Self-verifying theories
4699:Tarski's axiomatization
3650:Tarski's undefinability
3645:incompleteness theorems
2740:D'Angelo; West (2000).
2379:Ax–Grothendieck theorem
5282:Mathematical relations
5272:Functions and mappings
5252:Mathematics portal
4863:Proof of impossibility
4511:propositional variable
3821:Propositional calculus
3123:Constructible universe
2950:Constructibility (V=L)
2731:Devlin, Keith (2004).
2598:10.1006/jabr.1997.7242
2579:Lawson, M. V. (1998).
2440:single valued relation
2024:of sets: for a subset
1871:
1840:
1748:
1704:
1676:
1505:
1447:(1) = 1, showing that
1398:
1199:. More generally, any
1103:
885:Galois correspondences
770:
747:
727:
683:
663:
643:
583:
548:
522:; that is, a function
53:
36:A bijective function,
5121:Kolmogorov complexity
5074:Computably enumerable
4974:Model complete theory
4766:Principia Mathematica
3826:Propositional formula
3655:Banach–Tarski paradox
3346:Principia Mathematica
3180:Transfinite induction
3039:(i.e. set difference)
2286:Möbius transformation
1872:
1841:
1749:
1705:
1674:
1522:, given any two sets
1506:
1399:
1349:) = −1, showing that
1074:A bijection from the
1073:
907:with elements of set
780:multiplication by two
771:
748:
728:
684:
664:
644:
584:
549:
35:
5069:Church–Turing thesis
5056:Computability theory
4265:continuum hypothesis
3783:Square of opposition
3641:Gödel's completeness
3420:Burali-Forti paradox
3175:Set-builder notation
3128:Continuum hypothesis
3068:Symmetric difference
2625:The Theory of Groups
2404:Multivalued function
2389:Bijective numeration
2284:. An example is the
2020:Bijections preserve
1933:axiomatic set theory
1853:
1758:
1730:
1686:
1459:
1357:
1311:exponential function
996:one-to-one functions
992:surjective functions
889:mathematical objects
757:
737:
693:
673:
653:
609:
558:
526:
5223:Mathematical object
5114:P versus NP problem
5079:Computable function
4873:Reverse mathematics
4799:Logical consequence
4676:primitive recursive
4671:elementary function
4444:Free/bound variable
4297:Tarski–Grothendieck
3816:Logical connectives
3746:Logical equivalence
3596:Logical consequence
3381:Tarski–Grothendieck
2659:Schumacher (1996).
2279:one-to-one partial
1581:functional notation
1478:
1004:injective functions
820:one-to-one function
5287:Types of functions
5021:Transfer principle
4984:Semantics of logic
4969:Categorical theory
4945:Non-standard model
4459:Logical connective
3586:Information theory
3535:Mathematical logic
2970:Limitation of size
2829:Weisstein, Eric W.
2722:Iglewicz; Stoyle.
2641:Sundstrom (2003).
2621:Hall, Marshall Jr.
2585:Journal of Algebra
2371:Mathematics portal
2209:partial bijections
2193:group isomorphisms
1867:
1836:
1744:
1710:of two bijections
1700:
1677:
1638:there is a unique
1501:
1462:
1394:
1179:there is a unique
1104:
877:permutation groups
769:{\displaystyle Y.}
766:
743:
723:
679:
659:
639:
579:
544:
485:bijective function
277:Classes/properties
54:
5259:
5258:
5191:Abstract category
4994:Theories of truth
4804:Rule of inference
4794:Natural deduction
4775:
4774:
4320:
4319:
4025:Cartesian product
3930:
3929:
3836:Many-valued logic
3811:Boolean functions
3694:Russell's paradox
3669:diagonal argument
3566:First-order logic
3501:
3500:
3410:Russell's paradox
3359:Zermelo–Fraenkel
3260:Dedekind-infinite
3133:Diagonal argument
3032:Cartesian product
2896:Set (mathematics)
2713:Fletcher; Patty.
2663:. Addison-Wesley.
2569:978-0-521-69470-4
2544:978-0-8247-9662-4
2517:978-0-521-44179-7
2490:978-1-4704-1493-1
2230:is any relation
2205:partial functions
2123:For a finite set
1585:converse relation
1583:) also defines a
1455:is restricted to
1410:natural logarithm
1187:− 1)/2 such that
1113:identity function
1023:of a baseball or
998:" and are called
986:" and are called
778:For example, the
746:{\displaystyle y}
682:{\displaystyle X}
662:{\displaystyle x}
603:identity function
477:
476:
389:Generalizations
18:Partial bijection
16:(Redirected from
5294:
5250:
5249:
5201:History of logic
5196:Category of sets
5089:Decision problem
4868:Ordinal analysis
4809:Sequent calculus
4707:Boolean algebras
4647:
4646:
4621:
4592:logical/constant
4346:
4332:
4255:Zermelo–Fraenkel
4006:Set operations:
3941:
3878:
3709:
3689:Löwenheim–Skolem
3576:Formal semantics
3528:
3521:
3514:
3505:
3483:Bertrand Russell
3473:John von Neumann
3458:Abraham Fraenkel
3453:Richard Dedekind
3415:Suslin's problem
3326:Cantor's theorem
3043:De Morgan's laws
2908:
2875:
2868:
2861:
2852:
2842:
2841:
2823:
2798:
2789:
2788:. Prentice Hall.
2780:
2771:
2759:
2745:
2744:. Prentice Hall.
2736:
2727:
2718:
2709:
2700:
2699:. Prentice Hall.
2691:
2682:
2673:
2672:. Prentice-Hall.
2668:O'Leary (2003).
2664:
2655:
2646:
2645:. Prentice-Hall.
2637:
2628:
2603:
2602:
2600:
2573:
2555:
2549:
2548:
2528:
2522:
2521:
2501:
2495:
2494:
2474:
2468:
2467:
2465:
2463:
2449:
2443:
2432:
2426:
2420:
2373:
2368:
2367:
2345:
2333:
2317:
2305:
1876:
1874:
1873:
1868:
1845:
1843:
1842:
1837:
1832:
1831:
1808:
1807:
1787:
1786:
1753:
1751:
1750:
1745:
1709:
1707:
1706:
1701:
1609:inverse function
1510:
1508:
1507:
1502:
1500:
1496:
1477:
1472:
1467:
1403:
1401:
1400:
1395:
1393:
1389:
1371:
1370:
1365:
1203:over the reals,
937:each element of
915:each element of
840:
790:, which has the
775:
773:
772:
767:
752:
750:
749:
744:
732:
730:
729:
724:
688:
686:
685:
680:
668:
666:
665:
660:
648:
646:
645:
640:
596:
588:
586:
585:
580:
553:
551:
550:
545:
469:
462:
455:
267:
266:
260:
252:
251:
245:
237:
236:
232:
222:
221:
215:
207:
206:
200:
192:
191:
187:
177:
176:
170:
162:
161:
157:
147:
146:
140:
132:
131:
125:
117:
116:
112:
79:
56:
21:
5302:
5301:
5297:
5296:
5295:
5293:
5292:
5291:
5262:
5261:
5260:
5255:
5244:
5237:
5182:Category theory
5172:Algebraic logic
5155:
5126:Lambda calculus
5064:Church encoding
5050:
5026:Truth predicate
4882:
4848:Complete theory
4771:
4640:
4636:
4632:
4627:
4619:
4339: and
4335:
4330:
4316:
4292:New Foundations
4260:axiom of choice
4243:
4205:Gödel numbering
4145: and
4137:
4041:
3926:
3876:
3857:
3806:Boolean algebra
3792:
3756:Equiconsistency
3721:Classical logic
3698:
3679:Halting problem
3667: and
3643: and
3631: and
3630:
3625:Theorems (
3620:
3537:
3532:
3502:
3497:
3424:
3403:
3387:
3352:New Foundations
3299:
3189:
3108:Cardinal number
3091:
3077:
3018:
2909:
2900:
2884:
2879:
2827:
2826:
2808:
2805:
2792:
2783:
2774:
2768:
2748:
2739:
2730:
2721:
2712:
2703:
2694:
2686:Maddox (2002).
2685:
2681:. Random House.
2676:
2667:
2658:
2649:
2640:
2631:
2619:
2612:
2607:
2606:
2578:
2570:
2557:
2556:
2552:
2545:
2530:
2529:
2525:
2518:
2503:
2502:
2498:
2491:
2476:
2475:
2471:
2461:
2459:
2451:
2450:
2446:
2433:
2429:
2421:
2417:
2412:
2399:Category theory
2394:Bijective proof
2369:
2362:
2359:
2354:
2353:
2352:
2349:
2346:
2337:
2334:
2325:
2318:
2309:
2306:
2294:
2270:is a subset of
2201:
2158:
2156:Category theory
2143:of elements of
2129:total orderings
2098:is a bijection.
2004:
1989:symmetric group
1953:
1945:cardinal number
1899:
1851:
1850:
1820:
1796:
1775:
1756:
1755:
1728:
1727:
1684:
1683:
1669:
1561:
1486:
1482:
1457:
1456:
1379:
1375:
1360:
1355:
1354:
1201:linear function
1139:
1122:
1076:natural numbers
1068:
1042:
1021:batting line-up
1017:
1012:
901:binary relation
897:
873:diffeomorphisms
854:symmetric group
843:cardinal number
838:
836:natural numbers
792:division by two
755:
754:
735:
734:
691:
690:
671:
670:
651:
650:
607:
606:
594:
556:
555:
524:
523:
473:
437:
398:Binary relation
384:
351:
271:
265:
257:
250:
242:
231:
227:
220:
212:
205:
197:
186:
182:
175:
167:
156:
152:
145:
137:
130:
122:
111:
107:
66:
28:
23:
22:
15:
12:
11:
5:
5300:
5298:
5290:
5289:
5284:
5279:
5274:
5264:
5263:
5257:
5256:
5242:
5239:
5238:
5236:
5235:
5230:
5225:
5220:
5215:
5214:
5213:
5203:
5198:
5193:
5184:
5179:
5174:
5169:
5167:Abstract logic
5163:
5161:
5157:
5156:
5154:
5153:
5148:
5146:Turing machine
5143:
5138:
5133:
5128:
5123:
5118:
5117:
5116:
5111:
5106:
5101:
5096:
5086:
5084:Computable set
5081:
5076:
5071:
5066:
5060:
5058:
5052:
5051:
5049:
5048:
5043:
5038:
5033:
5028:
5023:
5018:
5013:
5012:
5011:
5006:
5001:
4991:
4986:
4981:
4979:Satisfiability
4976:
4971:
4966:
4965:
4964:
4954:
4953:
4952:
4942:
4941:
4940:
4935:
4930:
4925:
4920:
4910:
4909:
4908:
4903:
4896:Interpretation
4892:
4890:
4884:
4883:
4881:
4880:
4875:
4870:
4865:
4860:
4850:
4845:
4844:
4843:
4842:
4841:
4831:
4826:
4816:
4811:
4806:
4801:
4796:
4791:
4785:
4783:
4777:
4776:
4773:
4772:
4770:
4769:
4761:
4760:
4759:
4758:
4753:
4752:
4751:
4746:
4741:
4721:
4720:
4719:
4717:minimal axioms
4714:
4703:
4702:
4701:
4690:
4689:
4688:
4683:
4678:
4673:
4668:
4663:
4650:
4648:
4629:
4628:
4626:
4625:
4624:
4623:
4611:
4606:
4605:
4604:
4599:
4594:
4589:
4579:
4574:
4569:
4564:
4563:
4562:
4557:
4547:
4546:
4545:
4540:
4535:
4530:
4520:
4515:
4514:
4513:
4508:
4503:
4493:
4492:
4491:
4486:
4481:
4476:
4471:
4466:
4456:
4451:
4446:
4441:
4440:
4439:
4434:
4429:
4424:
4414:
4409:
4407:Formation rule
4404:
4399:
4398:
4397:
4392:
4382:
4381:
4380:
4370:
4365:
4360:
4355:
4349:
4343:
4326:Formal systems
4322:
4321:
4318:
4317:
4315:
4314:
4309:
4304:
4299:
4294:
4289:
4284:
4279:
4274:
4269:
4268:
4267:
4262:
4251:
4249:
4245:
4244:
4242:
4241:
4240:
4239:
4229:
4224:
4223:
4222:
4215:Large cardinal
4212:
4207:
4202:
4197:
4192:
4178:
4177:
4176:
4171:
4166:
4151:
4149:
4139:
4138:
4136:
4135:
4134:
4133:
4128:
4123:
4113:
4108:
4103:
4098:
4093:
4088:
4083:
4078:
4073:
4068:
4063:
4058:
4052:
4050:
4043:
4042:
4040:
4039:
4038:
4037:
4032:
4027:
4022:
4017:
4012:
4004:
4003:
4002:
3997:
3987:
3982:
3980:Extensionality
3977:
3975:Ordinal number
3972:
3962:
3957:
3956:
3955:
3944:
3938:
3932:
3931:
3928:
3927:
3925:
3924:
3919:
3914:
3909:
3904:
3899:
3894:
3893:
3892:
3882:
3881:
3880:
3867:
3865:
3859:
3858:
3856:
3855:
3854:
3853:
3848:
3843:
3833:
3828:
3823:
3818:
3813:
3808:
3802:
3800:
3794:
3793:
3791:
3790:
3785:
3780:
3775:
3770:
3765:
3760:
3759:
3758:
3748:
3743:
3738:
3733:
3728:
3723:
3717:
3715:
3706:
3700:
3699:
3697:
3696:
3691:
3686:
3681:
3676:
3671:
3659:Cantor's
3657:
3652:
3647:
3637:
3635:
3622:
3621:
3619:
3618:
3613:
3608:
3603:
3598:
3593:
3588:
3583:
3578:
3573:
3568:
3563:
3558:
3557:
3556:
3545:
3543:
3539:
3538:
3533:
3531:
3530:
3523:
3516:
3508:
3499:
3498:
3496:
3495:
3490:
3488:Thoralf Skolem
3485:
3480:
3475:
3470:
3465:
3460:
3455:
3450:
3445:
3440:
3434:
3432:
3426:
3425:
3423:
3422:
3417:
3412:
3406:
3404:
3402:
3401:
3398:
3392:
3389:
3388:
3386:
3385:
3384:
3383:
3378:
3373:
3372:
3371:
3356:
3355:
3354:
3342:
3341:
3340:
3329:
3328:
3323:
3318:
3313:
3307:
3305:
3301:
3300:
3298:
3297:
3292:
3287:
3282:
3273:
3268:
3263:
3253:
3248:
3247:
3246:
3241:
3236:
3226:
3216:
3211:
3206:
3200:
3198:
3191:
3190:
3188:
3187:
3182:
3177:
3172:
3170:Ordinal number
3167:
3162:
3157:
3152:
3151:
3150:
3145:
3135:
3130:
3125:
3120:
3115:
3105:
3100:
3094:
3092:
3090:
3089:
3086:
3082:
3079:
3078:
3076:
3075:
3070:
3065:
3060:
3055:
3050:
3048:Disjoint union
3045:
3040:
3034:
3028:
3026:
3020:
3019:
3017:
3016:
3015:
3014:
3009:
2998:
2997:
2995:Martin's axiom
2992:
2987:
2982:
2977:
2972:
2967:
2962:
2960:Extensionality
2957:
2952:
2947:
2946:
2945:
2940:
2935:
2925:
2919:
2917:
2911:
2910:
2903:
2901:
2899:
2898:
2892:
2890:
2886:
2885:
2880:
2878:
2877:
2870:
2863:
2855:
2849:
2848:
2843:
2824:
2804:
2803:External links
2801:
2800:
2799:
2790:
2781:
2779:. Brooks/Cole.
2772:
2766:
2746:
2737:
2728:
2719:
2710:
2701:
2692:
2683:
2674:
2665:
2656:
2647:
2638:
2629:
2611:
2608:
2605:
2604:
2591:(2): 428–438.
2568:
2550:
2543:
2523:
2516:
2496:
2489:
2469:
2444:
2436:total relation
2427:
2414:
2413:
2411:
2408:
2407:
2406:
2401:
2396:
2391:
2386:
2381:
2375:
2374:
2358:
2355:
2351:
2350:
2347:
2340:
2338:
2335:
2328:
2326:
2319:
2312:
2310:
2307:
2300:
2297:
2296:
2295:
2293:
2290:
2281:transformation
2213:total function
2200:
2197:
2157:
2154:
2153:
2152:
2121:
2120:
2119:
2109:
2099:
2070:
2069:
2068:
2018:
2002:
1973:
1952:
1949:
1937:equinumerosity
1922:if and only if
1898:
1895:
1866:
1862:
1858:
1835:
1830:
1827:
1823:
1819:
1815:
1811:
1806:
1803:
1799:
1795:
1791:
1785:
1782:
1778:
1774:
1770:
1766:
1763:
1743:
1739:
1735:
1699:
1695:
1691:
1668:
1665:
1660:
1659:
1571:(indicated by
1560:
1557:
1556:
1555:
1516:
1499:
1495:
1492:
1489:
1485:
1481:
1476:
1471:
1466:
1413:
1392:
1388:
1385:
1382:
1378:
1374:
1369:
1364:
1307:
1256:
1149:
1135:
1118:
1082:, which maps 2
1067:
1064:
1060:
1059:
1056:
1053:
1050:
1041:
1038:
1016:
1013:
1011:
1008:
958:
957:
948:no element of
946:
935:
926:no element of
924:
896:
893:
869:homeomorphisms
839:(1, 2, 3, ...)
765:
762:
742:
722:
719:
716:
713:
710:
707:
704:
701:
698:
678:
658:
638:
635:
632:
629:
626:
623:
620:
617:
614:
578:
575:
572:
569:
566:
563:
543:
540:
537:
534:
531:
516:if and only if
475:
474:
472:
471:
464:
457:
449:
446:
445:
439:
438:
436:
435:
430:
425:
420:
415:
410:
405:
400:
394:
391:
390:
386:
385:
383:
382:
377:
372:
367:
361:
358:
357:
353:
352:
350:
349:
344:
339:
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
283:
280:
279:
273:
272:
270:
269:
263:
254:
248:
239:
229:
224:
218:
209:
203:
194:
184:
179:
173:
164:
154:
149:
143:
134:
128:
119:
109:
103:
100:
99:
88:
87:
81:
80:
63:
62:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5299:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5269:
5267:
5254:
5253:
5248:
5240:
5234:
5231:
5229:
5226:
5224:
5221:
5219:
5216:
5212:
5209:
5208:
5207:
5204:
5202:
5199:
5197:
5194:
5192:
5188:
5185:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5165:
5164:
5162:
5158:
5152:
5149:
5147:
5144:
5142:
5141:Recursive set
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5115:
5112:
5110:
5107:
5105:
5102:
5100:
5097:
5095:
5092:
5091:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5067:
5065:
5062:
5061:
5059:
5057:
5053:
5047:
5044:
5042:
5039:
5037:
5034:
5032:
5029:
5027:
5024:
5022:
5019:
5017:
5014:
5010:
5007:
5005:
5002:
5000:
4997:
4996:
4995:
4992:
4990:
4987:
4985:
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4963:
4960:
4959:
4958:
4955:
4951:
4950:of arithmetic
4948:
4947:
4946:
4943:
4939:
4936:
4934:
4931:
4929:
4926:
4924:
4921:
4919:
4916:
4915:
4914:
4911:
4907:
4904:
4902:
4899:
4898:
4897:
4894:
4893:
4891:
4889:
4885:
4879:
4876:
4874:
4871:
4869:
4866:
4864:
4861:
4858:
4857:from ZFC
4854:
4851:
4849:
4846:
4840:
4837:
4836:
4835:
4832:
4830:
4827:
4825:
4822:
4821:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4795:
4792:
4790:
4787:
4786:
4784:
4782:
4778:
4768:
4767:
4763:
4762:
4757:
4756:non-Euclidean
4754:
4750:
4747:
4745:
4742:
4740:
4739:
4735:
4734:
4732:
4729:
4728:
4726:
4722:
4718:
4715:
4713:
4710:
4709:
4708:
4704:
4700:
4697:
4696:
4695:
4691:
4687:
4684:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4662:
4659:
4658:
4656:
4652:
4651:
4649:
4644:
4638:
4633:Example
4630:
4622:
4617:
4616:
4615:
4612:
4610:
4607:
4603:
4600:
4598:
4595:
4593:
4590:
4588:
4585:
4584:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4561:
4558:
4556:
4553:
4552:
4551:
4548:
4544:
4541:
4539:
4536:
4534:
4531:
4529:
4526:
4525:
4524:
4521:
4519:
4516:
4512:
4509:
4507:
4504:
4502:
4499:
4498:
4497:
4494:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4461:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4438:
4435:
4433:
4430:
4428:
4425:
4423:
4420:
4419:
4418:
4415:
4413:
4410:
4408:
4405:
4403:
4400:
4396:
4393:
4391:
4390:by definition
4388:
4387:
4386:
4383:
4379:
4376:
4375:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4350:
4347:
4344:
4342:
4338:
4333:
4327:
4323:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4277:Kripke–Platek
4275:
4273:
4270:
4266:
4263:
4261:
4258:
4257:
4256:
4253:
4252:
4250:
4246:
4238:
4235:
4234:
4233:
4230:
4228:
4225:
4221:
4218:
4217:
4216:
4213:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
4190:
4186:
4182:
4179:
4175:
4172:
4170:
4167:
4165:
4162:
4161:
4160:
4156:
4153:
4152:
4150:
4148:
4144:
4140:
4132:
4129:
4127:
4124:
4122:
4121:constructible
4119:
4118:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4087:
4084:
4082:
4079:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4054:
4053:
4051:
4049:
4044:
4036:
4033:
4031:
4028:
4026:
4023:
4021:
4018:
4016:
4013:
4011:
4008:
4007:
4005:
4001:
3998:
3996:
3993:
3992:
3991:
3988:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3967:
3963:
3961:
3958:
3954:
3951:
3950:
3949:
3946:
3945:
3942:
3939:
3937:
3933:
3923:
3920:
3918:
3915:
3913:
3910:
3908:
3905:
3903:
3900:
3898:
3895:
3891:
3888:
3887:
3886:
3883:
3879:
3874:
3873:
3872:
3869:
3868:
3866:
3864:
3860:
3852:
3849:
3847:
3844:
3842:
3839:
3838:
3837:
3834:
3832:
3829:
3827:
3824:
3822:
3819:
3817:
3814:
3812:
3809:
3807:
3804:
3803:
3801:
3799:
3798:Propositional
3795:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3761:
3757:
3754:
3753:
3752:
3749:
3747:
3744:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3726:Logical truth
3724:
3722:
3719:
3718:
3716:
3714:
3710:
3707:
3705:
3701:
3695:
3692:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3670:
3666:
3662:
3658:
3656:
3653:
3651:
3648:
3646:
3642:
3639:
3638:
3636:
3634:
3628:
3623:
3617:
3614:
3612:
3609:
3607:
3604:
3602:
3599:
3597:
3594:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3572:
3569:
3567:
3564:
3562:
3559:
3555:
3552:
3551:
3550:
3547:
3546:
3544:
3540:
3536:
3529:
3524:
3522:
3517:
3515:
3510:
3509:
3506:
3494:
3493:Ernst Zermelo
3491:
3489:
3486:
3484:
3481:
3479:
3478:Willard Quine
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3435:
3433:
3431:
3430:Set theorists
3427:
3421:
3418:
3416:
3413:
3411:
3408:
3407:
3405:
3399:
3397:
3394:
3393:
3390:
3382:
3379:
3377:
3376:Kripke–Platek
3374:
3370:
3367:
3366:
3365:
3362:
3361:
3360:
3357:
3353:
3350:
3349:
3348:
3347:
3343:
3339:
3336:
3335:
3334:
3331:
3330:
3327:
3324:
3322:
3319:
3317:
3314:
3312:
3309:
3308:
3306:
3302:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3279:
3274:
3272:
3269:
3267:
3264:
3261:
3257:
3254:
3252:
3249:
3245:
3242:
3240:
3237:
3235:
3232:
3231:
3230:
3227:
3224:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3201:
3199:
3196:
3192:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3166:
3163:
3161:
3158:
3156:
3153:
3149:
3146:
3144:
3141:
3140:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3113:
3109:
3106:
3104:
3101:
3099:
3096:
3095:
3093:
3087:
3084:
3083:
3080:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3038:
3035:
3033:
3030:
3029:
3027:
3025:
3021:
3013:
3012:specification
3010:
3008:
3005:
3004:
3003:
3000:
2999:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2944:
2941:
2939:
2936:
2934:
2931:
2930:
2929:
2926:
2924:
2921:
2920:
2918:
2916:
2912:
2907:
2897:
2894:
2893:
2891:
2887:
2883:
2876:
2871:
2869:
2864:
2862:
2857:
2856:
2853:
2847:
2844:
2839:
2838:
2833:
2830:
2825:
2821:
2817:
2816:
2811:
2807:
2806:
2802:
2796:
2791:
2787:
2782:
2778:
2773:
2769:
2767:9780534103200
2763:
2760:. Wadsworth.
2758:
2757:
2751:
2747:
2743:
2738:
2734:
2729:
2725:
2720:
2716:
2711:
2707:
2702:
2698:
2693:
2689:
2684:
2680:
2675:
2671:
2666:
2662:
2657:
2653:
2648:
2644:
2639:
2635:
2632:Wolf (1998).
2630:
2626:
2622:
2618:
2617:
2616:
2609:
2599:
2594:
2590:
2586:
2582:
2576:
2571:
2565:
2561:
2554:
2551:
2546:
2540:
2536:
2535:
2527:
2524:
2519:
2513:
2509:
2508:
2500:
2497:
2492:
2486:
2482:
2481:
2473:
2470:
2458:
2457:brilliant.org
2454:
2448:
2445:
2441:
2437:
2431:
2428:
2424:
2419:
2416:
2409:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2376:
2372:
2366:
2361:
2356:
2344:
2339:
2332:
2327:
2323:
2316:
2311:
2304:
2299:
2291:
2289:
2287:
2283:
2282:
2275:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2233:
2229:
2225:
2220:
2218:
2214:
2210:
2206:
2198:
2196:
2194:
2190:
2189:homomorphisms
2186:
2182:
2181:
2176:
2172:
2171:
2167:
2163:
2155:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2122:
2117:
2113:
2110:
2107:
2103:
2100:
2097:
2094:
2093:
2091:
2087:
2083:
2079:
2075:
2071:
2066:
2062:
2058:
2054:
2050:
2046:
2042:
2041:
2039:
2035:
2032:| and subset
2031:
2027:
2023:
2022:cardinalities
2019:
2016:
2013:
2009:
2005:
1998:
1994:
1990:
1986:
1982:
1978:
1974:
1971:
1967:
1963:
1959:
1955:
1954:
1950:
1948:
1946:
1942:
1941:infinite sets
1938:
1934:
1930:
1926:
1923:
1920:
1916:
1912:
1908:
1904:
1896:
1894:
1892:
1888:
1884:
1880:
1864:
1860:
1856:
1847:
1828:
1825:
1821:
1813:
1804:
1801:
1797:
1789:
1783:
1780:
1772:
1768:
1764:
1741:
1737:
1733:
1725:
1721:
1717:
1713:
1697:
1693:
1689:
1682:
1673:
1666:
1664:
1657:
1653:
1649:
1645:
1641:
1637:
1633:
1629:
1628:
1627:
1625:
1621:
1616:
1614:
1610:
1606:
1602:
1598:
1594:
1591:and going to
1590:
1586:
1582:
1578:
1574:
1570:
1566:
1558:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1517:
1514:
1497:
1490:
1487:
1483:
1479:
1474:
1469:
1454:
1450:
1446:
1442:
1438:
1434:
1430:
1426:
1422:
1418:
1415:The function
1414:
1411:
1407:
1390:
1383:
1380:
1376:
1372:
1367:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1261:
1258:The function
1257:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1151:The function
1150:
1148:is bijective.
1147:
1143:
1138:
1134:
1130:
1126:
1121:
1117:
1114:
1110:
1106:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1072:
1065:
1063:
1057:
1054:
1051:
1048:
1047:
1046:
1039:
1037:
1034:
1030:
1026:
1022:
1019:Consider the
1014:
1009:
1007:
1005:
1001:
997:
993:
989:
985:
982:
978:
974:
970:
967:
963:
955:
951:
947:
944:
940:
936:
933:
929:
925:
922:
918:
914:
913:
912:
910:
906:
902:
894:
892:
890:
886:
882:
878:
874:
870:
866:
862:
861:automorphisms
857:
855:
851:
846:
844:
837:
834:to the first
833:
829:
824:
822:
821:
816:
812:
808:
804:
800:
795:
793:
789:
785:
781:
776:
763:
760:
740:
720:
717:
708:
702:
696:
676:
656:
636:
633:
624:
618:
612:
604:
600:
592:
576:
573:
567:
564:
561:
541:
535:
532:
529:
521:
517:
512:
510:
506:
502:
498:
494:
490:
486:
482:
470:
465:
463:
458:
456:
451:
450:
448:
447:
444:
440:
434:
431:
429:
426:
424:
421:
419:
416:
414:
411:
409:
406:
404:
401:
399:
396:
395:
393:
392:
387:
381:
378:
376:
373:
371:
368:
366:
363:
362:
360:
359:
356:Constructions
354:
348:
345:
343:
340:
338:
335:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
282:
281:
278:
274:
268:
255:
253:
240:
238:
225:
223:
210:
208:
195:
193:
180:
178:
165:
163:
150:
148:
135:
133:
120:
118:
105:
104:
102:
101:
98:
94:
89:
86:
82:
77:
73:
69:
65:
64:
61:
57:
51:
47:
43:
39:
34:
30:
19:
5243:
5041:Ultraproduct
4888:Model theory
4853:Independence
4789:Formal proof
4781:Proof theory
4764:
4737:
4694:real numbers
4666:second-order
4577:Substitution
4454:Metalanguage
4395:conservative
4368:Axiom schema
4312:Constructive
4282:Morse–Kelley
4248:Set theories
4227:Aleph number
4220:inaccessible
4188:
4126:Grothendieck
4010:intersection
3897:Higher-order
3885:Second-order
3831:Truth tables
3788:Venn diagram
3571:Formal proof
3443:Georg Cantor
3438:Paul Bernays
3369:Morse–Kelley
3344:
3277:
3276:Subset
3223:hereditarily
3185:Venn diagram
3164:
3143:ordered pair
3058:Intersection
3002:Axiom schema
2835:
2813:
2794:
2785:
2776:
2755:
2741:
2732:
2726:. MacMillan.
2723:
2714:
2705:
2696:
2695:Lay (2001).
2687:
2678:
2669:
2660:
2651:
2642:
2633:
2627:. MacMillan.
2624:
2613:
2588:
2584:
2559:
2553:
2533:
2526:
2506:
2499:
2479:
2472:
2460:. Retrieved
2456:
2447:
2439:
2435:
2430:
2418:
2321:
2278:
2276:
2271:
2267:
2263:
2255:
2251:
2247:
2243:
2242:a bijection
2235:
2231:
2227:
2223:
2221:
2208:
2202:
2192:
2178:
2168:
2162:isomorphisms
2159:
2148:
2144:
2141:permutations
2136:
2132:
2124:
2111:
2101:
2095:
2089:
2085:
2077:
2073:
2064:
2060:
2056:
2052:
2048:
2044:
2037:
2033:
2029:
2025:
2011:
2007:
2000:
1996:
1992:
1980:
1976:
1965:
1961:
1957:
1928:
1924:
1918:
1914:
1906:
1902:
1900:
1886:
1878:
1848:
1723:
1719:
1715:
1711:
1678:
1661:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1623:
1619:
1617:
1604:
1600:
1596:
1592:
1588:
1587:starting in
1576:
1572:
1568:
1567:with domain
1564:
1563:A bijection
1562:
1551:
1547:
1543:
1539:
1535:
1531:
1527:
1523:
1512:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1416:
1412:function ln.
1405:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1318:
1314:
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1259:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1204:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1145:
1141:
1136:
1132:
1128:
1124:
1119:
1115:
1108:
1107:For any set
1099:
1095:
1091:
1087:
1083:
1061:
1043:
1032:
1028:
1018:
1003:
991:
983:
976:
972:
968:
959:
953:
949:
942:
938:
931:
927:
920:
916:
908:
904:
898:
865:isomorphisms
858:
847:
825:
818:
814:
810:
802:
796:
791:
788:even numbers
779:
777:
590:
513:
488:
484:
480:
478:
423:Higher-order
346:
75:
71:
67:
49:
45:
41:
37:
29:
5151:Type theory
5099:undecidable
5031:Truth value
4918:equivalence
4597:non-logical
4210:Enumeration
4200:Isomorphism
4147:cardinality
4131:Von Neumann
4096:Ultrafilter
4061:Uncountable
3995:equivalence
3912:Quantifiers
3902:Fixed-point
3871:First-order
3751:Consistency
3736:Proposition
3713:Traditional
3684:Lindström's
3674:Compactness
3616:Type theory
3561:Cardinality
3468:Thomas Jech
3311:Alternative
3290:Uncountable
3244:Ultrafilter
3103:Cardinality
3007:replacement
2955:Determinacy
2832:"Bijection"
2810:"Bijection"
2717:. PWS-Kent.
2425:, p. 3
2082:finite sets
1956:A function
1911:finite sets
1897:Cardinality
1681:composition
1667:Composition
1302:)). If the
1274:) = arctan(
988:surjections
879:, and most
850:permutation
408:Multivalued
370:Composition
365:Restriction
5266:Categories
4962:elementary
4655:arithmetic
4523:Quantifier
4501:functional
4373:Expression
4091:Transitive
4035:identities
4020:complement
3953:hereditary
3936:Set theory
3463:Kurt Gödel
3448:Paul Cohen
3285:Transitive
3053:Identities
3037:Complement
3024:Operations
2985:Regularity
2923:Adjunction
2882:Set theory
2636:. Freeman.
2610:References
2462:7 December
2106:surjection
1951:Properties
1891:surjective
1630:for every
1613:invertible
1597:in general
1341:such that
1294:(that is,
1000:injections
895:Definition
832:finite set
807:surjective
803:one-to-one
520:invertible
342:Surjective
332:Measurable
327:Continuous
302:Polynomial
5233:Supertask
5136:Recursion
5094:decidable
4928:saturated
4906:of models
4829:deductive
4824:axiomatic
4744:Hilbert's
4731:Euclidean
4712:canonical
4635:axiomatic
4567:Signature
4496:Predicate
4385:Extension
4307:Ackermann
4232:Operation
4111:Universal
4101:Recursive
4076:Singleton
4071:Inhabited
4056:Countable
4046:Types of
4030:power set
4000:partition
3917:Predicate
3863:Predicate
3778:Syllogism
3768:Soundness
3741:Inference
3731:Tautology
3633:paradoxes
3396:Paradoxes
3316:Axiomatic
3295:Universal
3271:Singleton
3266:Recursive
3209:Countable
3204:Amorphous
3063:Power set
2980:Power set
2938:dependent
2933:countable
2837:MathWorld
2820:EMS Press
2750:Cupillari
2423:Hall 1959
2322:bijection
2116:injection
2015:factorial
1883:injective
1861:∘
1826:−
1814:∘
1802:−
1781:−
1769:∘
1738:∘
1694:∘
1494:∞
1480:≡
1387:∞
1373:≡
1298:= arctan(
799:injective
733:for each
649:for each
599:composing
571:→
539:→
481:bijection
347:Bijective
337:Injective
312:Algebraic
91:Types by
5218:Logicism
5211:timeline
5187:Concrete
5046:Validity
5016:T-schema
5009:Kripke's
5004:Tarski's
4999:semantic
4989:Strength
4938:submodel
4933:spectrum
4901:function
4749:Tarski's
4738:Elements
4725:geometry
4681:Robinson
4602:variable
4587:function
4560:spectrum
4550:Sentence
4506:variable
4449:Language
4402:Relation
4363:Automata
4353:Alphabet
4337:language
4191:-jection
4169:codomain
4155:Function
4116:Universe
4086:Infinite
3990:Relation
3773:Validity
3763:Argument
3661:theorem,
3400:Problems
3304:Theories
3280:Superset
3256:Infinite
3085:Concepts
2965:Infinity
2889:Overview
2752:(1989).
2677:Morash.
2623:(1959).
2575:preprint
2357:See also
2254:, where
2240:graph of
2166:category
1605:function
1559:Inverses
1304:codomain
1080:integers
1010:Examples
962:function
828:counting
784:integers
509:relation
501:codomain
497:function
428:Morphism
413:Implicit
317:Analytic
307:Rational
292:Identity
287:Constant
97:codomain
74: (
60:Function
52:(1) = D.
5160:Related
4957:Diagram
4855: (
4834:Hilbert
4819:Systems
4814:Theorem
4692:of the
4637:systems
4417:Formula
4412:Grammar
4328: (
4272:General
3985:Forcing
3970:Element
3890:Monadic
3665:paradox
3606:Theorem
3542:General
3338:General
3333:Zermelo
3239:subbase
3221: (
3160:Forcing
3138:Element
3110: (
3088:Methods
2975:Pairing
2822:, 2001
2577:citing
2292:Gallery
2238:is the
2164:in the
2055:| and |
1511:, then
1443:(−1) =
1404:, then
1231:(where
1094:− 1 to
1078:to the
1025:cricket
786:to the
591:inverse
433:Functor
403:Partial
380:Inverse
4923:finite
4686:Skolem
4639:
4614:Theory
4582:Symbol
4572:String
4555:atomic
4432:ground
4427:closed
4422:atomic
4378:ground
4341:syntax
4237:binary
4164:domain
4081:Finite
3846:finite
3704:Logics
3663:
3611:Theory
3229:Filter
3219:Finite
3155:Family
3098:Almost
2943:global
2928:Choice
2915:Axioms
2797:. MAA.
2775:Bond.
2764:
2566:
2541:
2514:
2487:
2260:subset
2185:groups
2114:is an
2063:)| = |
2051:)| = |
1987:, the
1453:domain
1111:, the
1098:, for
966:domain
899:For a
518:it is
505:domain
322:Smooth
297:Linear
93:domain
4913:Model
4661:Peano
4518:Proof
4358:Arity
4287:Naive
4174:image
4106:Fuzzy
4066:Empty
4015:union
3960:Class
3601:Model
3591:Lemma
3549:Axiom
3321:Naive
3251:Fuzzy
3214:Empty
3197:types
3148:tuple
3118:Class
3112:large
3073:Union
2990:Union
2793:Ash.
2410:Notes
2258:is a
2104:is a
2090:X → Y
2006:, or
1985:group
1970:graph
1724:Y → Z
1716:X → Y
1646:with
1624:X → Y
1577:X → Y
1552:X → Y
1544:Y → X
1536:X → Y
1171:) = 2
1090:and 2
964:with
945:, and
495:is a
487:, or
418:Space
5036:Type
4839:list
4643:list
4620:list
4609:Term
4543:rank
4437:open
4331:list
4143:Maps
4048:sets
3907:Free
3877:list
3627:list
3554:list
3234:base
2762:ISBN
2564:ISBN
2539:ISBN
2512:ISBN
2485:ISBN
2464:2019
2266:and
2175:sets
2080:are
2076:and
1927:and
1917:and
1909:are
1905:and
1885:and
1718:and
1679:The
1538:and
1526:and
1435:) =
1309:The
1290:) =
1223:) =
1195:) =
1144:) =
1102:≥ 0.
1086:to −
1002:(or
990:(or
981:onto
811:onto
809:(or
801:(or
689:and
589:the
493:sets
95:and
4723:of
4705:of
4653:of
4185:Sur
4159:Map
3966:Ur-
3948:Set
3195:Set
2593:doi
2589:200
2262:of
2226:to
2183:of
2180:Grp
2173:of
2170:Set
2135:to
2072:If
2010:! (
1999:),
1991:of
1975:If
1901:If
1889:is
1881:is
1754:is
1642:in
1634:in
1579:in
1518:By
1337:in
1243:= (
1183:= (
753:in
669:in
593:of
5268::
5109:NP
4733::
4727::
4657::
4334:),
4189:Bi
4181:In
2834:.
2818:,
2812:,
2587:.
2583:.
2455:.
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2268:B′
2256:A′
2252:B′
2248:A′
2219:.
2151:!.
2088::
2067:|.
2017:).
1964:→
1960::
1893:.
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1722::
1714::
1658:).
1650:=
1622::
1575::
1550::
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1427:,
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1325:,
1321:→
1317::
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1262::
1251:)/
1247:−
1227:+
1225:ax
1215:,
1211:→
1207::
1163:,
1159:→
1155::
1131:,
1127:→
1123::
883:.
875:,
871:,
867:,
863:,
856:.
605::
483:,
479:A
261:→
246:→
233:→
216:→
201:→
188:→
171:→
158:→
141:→
139:𝔹
126:→
124:𝔹
115:𝔹
113:→
70:↦
44:→
40::
5189:/
5104:P
4859:)
4645:)
4641:(
4538:∀
4533:!
4528:∃
4489:=
4484:↔
4479:→
4474:∧
4469:∨
4464:¬
4187:/
4183:/
4157:/
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3964:(
3851:∞
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2272:B
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2246::
2244:f
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2228:B
2224:A
2149:n
2145:S
2137:S
2133:S
2125:S
2118:.
2112:f
2108:.
2102:f
2096:f
2086:f
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2049:A
2047:(
2045:f
2043:|
2038:B
2034:B
2030:A
2026:A
2012:X
2008:X
2003:X
2001:S
1997:X
1993:X
1981:X
1977:X
1966:R
1962:R
1958:f
1929:Y
1925:X
1919:Y
1915:X
1907:Y
1903:X
1887:g
1879:f
1865:f
1857:g
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1829:1
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1818:(
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1488:0
1484:[
1475:+
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1449:h
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1433:x
1431:(
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1425:R
1421:R
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1406:g
1391:)
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1381:0
1377:(
1368:+
1363:R
1351:g
1347:x
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1343:g
1339:R
1335:x
1331:x
1329:(
1327:g
1323:R
1319:R
1315:g
1300:x
1296:y
1292:x
1288:y
1284:y
1280:x
1276:x
1272:x
1270:(
1268:f
1264:R
1260:f
1255:.
1253:a
1249:b
1245:y
1241:x
1237:y
1233:a
1229:b
1221:x
1219:(
1217:f
1213:R
1209:R
1205:f
1197:y
1193:x
1191:(
1189:f
1185:y
1181:x
1177:y
1173:x
1169:x
1167:(
1165:f
1161:R
1157:R
1153:f
1146:x
1142:x
1140:(
1137:X
1133:1
1129:X
1125:X
1120:X
1116:1
1109:X
1100:n
1096:n
1092:n
1088:n
1084:n
1033:Y
1029:X
984:Y
977:Y
973:X
969:X
956:.
954:X
950:Y
943:X
939:Y
934:,
932:Y
928:X
923:,
921:Y
917:X
909:Y
905:X
764:.
761:Y
741:y
721:y
718:=
715:)
712:)
709:y
706:(
703:g
700:(
697:f
677:X
657:x
637:x
634:=
631:)
628:)
625:x
622:(
619:f
616:(
613:g
595:f
577:,
574:X
568:Y
565::
562:g
542:Y
536:X
533::
530:f
468:e
461:t
454:v
375:λ
264:X
259:ℂ
249:X
244:ℂ
235:ℂ
230:X
219:X
214:ℝ
204:X
199:ℝ
190:ℝ
185:X
174:X
169:ℤ
160:ℤ
155:X
144:X
129:X
110:X
78:)
76:x
72:f
68:x
50:f
46:Y
42:X
38:f
20:)
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