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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
4907:
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3277:
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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:
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427:
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When the partial bijection is on the same set, it is sometimes called a
4820:
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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:
3510:
3282:
3104:
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
4364:
3710:
3555:
3154:
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:
3514:
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
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4893:
4786:
4638:
4331:
4254:
4148:
4052:
3941:
3868:
3803:
3718:
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3548:
3435:
3398:
3310:
3200:
3088:
3029:
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
3526:
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.
4352:
3947:
3715:
3533:
3519:
3511:
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
5252:
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:
5213:History of mathematical logic
2160:Bijections are precisely the
782:defines a bijection from the
5284:Basic concepts in set theory
5138: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
5310:
4202:Schröder–Bernstein theorem
3929:Monadic predicate calculus
3588:Foundations of mathematics
3371: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
5248:
5235:Philosophy of mathematics
5184:Automated theorem proving
4355:
4309:Von Neumann–Bernays–Gödel
3950:
3172: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
4885:Self-verifying theories
4706:Tarski's axiomatization
3657:Tarski's undefinability
3652:incompleteness theorems
2740:D'Angelo; West (2000).
2379:Ax–Grothendieck theorem
5289:Mathematical relations
5279:Functions and mappings
5259:Mathematics portal
4870:Proof of impossibility
4518:propositional variable
3828:Propositional calculus
3130: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,
5128:Kolmogorov complexity
5081:Computably enumerable
4981:Model complete theory
4773:Principia Mathematica
3833:Propositional formula
3662:Banach–Tarski paradox
3353:Principia Mathematica
3187:Transfinite induction
3046:(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:
5076:Church–Turing thesis
5063:Computability theory
4272:continuum hypothesis
3790:Square of opposition
3648:Gödel's completeness
3427:Burali-Forti paradox
3182:Set-builder notation
3135:Continuum hypothesis
3075: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:
5230:Mathematical object
5121:P versus NP problem
5086:Computable function
4880:Reverse mathematics
4806:Logical consequence
4683:primitive recursive
4678:elementary function
4451:Free/bound variable
4304:Tarski–Grothendieck
3823:Logical connectives
3753:Logical equivalence
3603:Logical consequence
3388:Tarski–Grothendieck
2659:Schumacher (1996).
2279:one-to-one partial
1581:functional notation
1478:
1004:injective functions
820:one-to-one function
5294:Types of functions
5028:Transfer principle
4991:Semantics of logic
4976:Categorical theory
4952:Non-standard model
4466:Logical connective
3593:Information theory
3542:Mathematical logic
2977: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:
18:Bijective relation
5266:
5265:
5198:Abstract category
5001:Theories of truth
4811:Rule of inference
4801:Natural deduction
4782:
4781:
4327:
4326:
4032:Cartesian product
3937:
3936:
3843:Many-valued logic
3818:Boolean functions
3701:Russell's paradox
3676:diagonal argument
3573:First-order logic
3508:
3507:
3417:Russell's paradox
3366:Zermelo–Fraenkel
3267:Dedekind-infinite
3140:Diagonal argument
3039: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
16:(Redirected from
5301:
5257:
5256:
5208:History of logic
5203:Category of sets
5096:Decision problem
4875:Ordinal analysis
4816:Sequent calculus
4714:Boolean algebras
4654:
4653:
4628:
4599:logical/constant
4353:
4339:
4262:Zermelo–Fraenkel
4013:Set operations:
3948:
3885:
3716:
3696:Löwenheim–Skolem
3583:Formal semantics
3535:
3528:
3521:
3512:
3490:Bertrand Russell
3480:John von Neumann
3465:Abraham Fraenkel
3460:Richard Dedekind
3422:Suslin's problem
3333:Cantor's theorem
3050: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:
5309:
5308:
5304:
5303:
5302:
5300:
5299:
5298:
5269:
5268:
5267:
5262:
5251:
5244:
5189:Category theory
5179:Algebraic logic
5162:
5133:Lambda calculus
5071:Church encoding
5057:
5033:Truth predicate
4889:
4855:Complete theory
4778:
4647:
4643:
4639:
4634:
4626:
4346: and
4342:
4337:
4323:
4299:New Foundations
4267:axiom of choice
4250:
4212:Gödel numbering
4152: and
4144:
4048:
3933:
3883:
3864:
3813:Boolean algebra
3799:
3763:Equiconsistency
3728:Classical logic
3705:
3686:Halting problem
3674: and
3650: and
3638: and
3637:
3632:Theorems (
3627:
3544:
3539:
3509:
3504:
3431:
3410:
3394:
3359:New Foundations
3306:
3196:
3115:Cardinal number
3098:
3084:
3025:
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:
5307:
5305:
5297:
5296:
5291:
5286:
5281:
5271:
5270:
5264:
5263:
5249:
5246:
5245:
5243:
5242:
5237:
5232:
5227:
5222:
5221:
5220:
5210:
5205:
5200:
5191:
5186:
5181:
5176:
5174:Abstract logic
5170:
5168:
5164:
5163:
5161:
5160:
5155:
5153:Turing machine
5150:
5145:
5140:
5135:
5130:
5125:
5124:
5123:
5118:
5113:
5108:
5103:
5093:
5091:Computable set
5088:
5083:
5078:
5073:
5067:
5065:
5059:
5058:
5056:
5055:
5050:
5045:
5040:
5035:
5030:
5025:
5020:
5019:
5018:
5013:
5008:
4998:
4993:
4988:
4986:Satisfiability
4983:
4978:
4973:
4972:
4971:
4961:
4960:
4959:
4949:
4948:
4947:
4942:
4937:
4932:
4927:
4917:
4916:
4915:
4910:
4903:Interpretation
4899:
4897:
4891:
4890:
4888:
4887:
4882:
4877:
4872:
4867:
4857:
4852:
4851:
4850:
4849:
4848:
4838:
4833:
4823:
4818:
4813:
4808:
4803:
4798:
4792:
4790:
4784:
4783:
4780:
4779:
4777:
4776:
4768:
4767:
4766:
4765:
4760:
4759:
4758:
4753:
4748:
4728:
4727:
4726:
4724:minimal axioms
4721:
4710:
4709:
4708:
4697:
4696:
4695:
4690:
4685:
4680:
4675:
4670:
4657:
4655:
4636:
4635:
4633:
4632:
4631:
4630:
4618:
4613:
4612:
4611:
4606:
4601:
4596:
4586:
4581:
4576:
4571:
4570:
4569:
4564:
4554:
4553:
4552:
4547:
4542:
4537:
4527:
4522:
4521:
4520:
4515:
4510:
4500:
4499:
4498:
4493:
4488:
4483:
4478:
4473:
4463:
4458:
4453:
4448:
4447:
4446:
4441:
4436:
4431:
4421:
4416:
4414:Formation rule
4411:
4406:
4405:
4404:
4399:
4389:
4388:
4387:
4377:
4372:
4367:
4362:
4356:
4350:
4333:Formal systems
4329:
4328:
4325:
4324:
4322:
4321:
4316:
4311:
4306:
4301:
4296:
4291:
4286:
4281:
4276:
4275:
4274:
4269:
4258:
4256:
4252:
4251:
4249:
4248:
4247:
4246:
4236:
4231:
4230:
4229:
4222:Large cardinal
4219:
4214:
4209:
4204:
4199:
4185:
4184:
4183:
4178:
4173:
4158:
4156:
4146:
4145:
4143:
4142:
4141:
4140:
4135:
4130:
4120:
4115:
4110:
4105:
4100:
4095:
4090:
4085:
4080:
4075:
4070:
4065:
4059:
4057:
4050:
4049:
4047:
4046:
4045:
4044:
4039:
4034:
4029:
4024:
4019:
4011:
4010:
4009:
4004:
3994:
3989:
3987:Extensionality
3984:
3982:Ordinal number
3979:
3969:
3964:
3963:
3962:
3951:
3945:
3939:
3938:
3935:
3934:
3932:
3931:
3926:
3921:
3916:
3911:
3906:
3901:
3900:
3899:
3889:
3888:
3887:
3874:
3872:
3866:
3865:
3863:
3862:
3861:
3860:
3855:
3850:
3840:
3835:
3830:
3825:
3820:
3815:
3809:
3807:
3801:
3800:
3798:
3797:
3792:
3787:
3782:
3777:
3772:
3767:
3766:
3765:
3755:
3750:
3745:
3740:
3735:
3730:
3724:
3722:
3713:
3707:
3706:
3704:
3703:
3698:
3693:
3688:
3683:
3678:
3666:Cantor's
3664:
3659:
3654:
3644:
3642:
3629:
3628:
3626:
3625:
3620:
3615:
3610:
3605:
3600:
3595:
3590:
3585:
3580:
3575:
3570:
3565:
3564:
3563:
3552:
3550:
3546:
3545:
3540:
3538:
3537:
3530:
3523:
3515:
3506:
3505:
3503:
3502:
3497:
3495:Thoralf Skolem
3492:
3487:
3482:
3477:
3472:
3467:
3462:
3457:
3452:
3447:
3441:
3439:
3433:
3432:
3430:
3429:
3424:
3419:
3413:
3411:
3409:
3408:
3405:
3399:
3396:
3395:
3393:
3392:
3391:
3390:
3385:
3380:
3379:
3378:
3363:
3362:
3361:
3349:
3348:
3347:
3336:
3335:
3330:
3325:
3320:
3314:
3312:
3308:
3307:
3305:
3304:
3299:
3294:
3289:
3280:
3275:
3270:
3260:
3255:
3254:
3253:
3248:
3243:
3233:
3223:
3218:
3213:
3207:
3205:
3198:
3197:
3195:
3194:
3189:
3184:
3179:
3177:Ordinal number
3174:
3169:
3164:
3159:
3158:
3157:
3152:
3142:
3137:
3132:
3127:
3122:
3112:
3107:
3101:
3099:
3097:
3096:
3093:
3089:
3086:
3085:
3083:
3082:
3077:
3072:
3067:
3062:
3057:
3055:Disjoint union
3052:
3047:
3041:
3035:
3033:
3027:
3026:
3024:
3023:
3022:
3021:
3016:
3005:
3004:
3002:Martin's axiom
2999:
2994:
2989:
2984:
2979:
2974:
2969:
2967:Extensionality
2964:
2963:
2962:
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:
5306:
5295:
5292:
5290:
5287:
5285:
5282:
5280:
5277:
5276:
5274:
5261:
5260:
5255:
5247:
5241:
5238:
5236:
5233:
5231:
5228:
5226:
5223:
5219:
5216:
5215:
5214:
5211:
5209:
5206:
5204:
5201:
5199:
5195:
5192:
5190:
5187:
5185:
5182:
5180:
5177:
5175:
5172:
5171:
5169:
5165:
5159:
5156:
5154:
5151:
5149:
5148:Recursive set
5146:
5144:
5141:
5139:
5136:
5134:
5131:
5129:
5126:
5122:
5119:
5117:
5114:
5112:
5109:
5107:
5104:
5102:
5099:
5098:
5097:
5094:
5092:
5089:
5087:
5084:
5082:
5079:
5077:
5074:
5072:
5069:
5068:
5066:
5064:
5060:
5054:
5051:
5049:
5046:
5044:
5041:
5039:
5036:
5034:
5031:
5029:
5026:
5024:
5021:
5017:
5014:
5012:
5009:
5007:
5004:
5003:
5002:
4999:
4997:
4994:
4992:
4989:
4987:
4984:
4982:
4979:
4977:
4974:
4970:
4967:
4966:
4965:
4962:
4958:
4957:of arithmetic
4955:
4954:
4953:
4950:
4946:
4943:
4941:
4938:
4936:
4933:
4931:
4928:
4926:
4923:
4922:
4921:
4918:
4914:
4911:
4909:
4906:
4905:
4904:
4901:
4900:
4898:
4896:
4892:
4886:
4883:
4881:
4878:
4876:
4873:
4871:
4868:
4865:
4864:from ZFC
4861:
4858:
4856:
4853:
4847:
4844:
4843:
4842:
4839:
4837:
4834:
4832:
4829:
4828:
4827:
4824:
4822:
4819:
4817:
4814:
4812:
4809:
4807:
4804:
4802:
4799:
4797:
4794:
4793:
4791:
4789:
4785:
4775:
4774:
4770:
4769:
4764:
4763:non-Euclidean
4761:
4757:
4754:
4752:
4749:
4747:
4746:
4742:
4741:
4739:
4736:
4735:
4733:
4729:
4725:
4722:
4720:
4717:
4716:
4715:
4711:
4707:
4704:
4703:
4702:
4698:
4694:
4691:
4689:
4686:
4684:
4681:
4679:
4676:
4674:
4671:
4669:
4666:
4665:
4663:
4659:
4658:
4656:
4651:
4645:
4640:Example
4637:
4629:
4624:
4623:
4622:
4619:
4617:
4614:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4592:
4591:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4568:
4565:
4563:
4560:
4559:
4558:
4555:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4532:
4531:
4528:
4526:
4523:
4519:
4516:
4514:
4511:
4509:
4506:
4505:
4504:
4501:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4468:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4426:
4425:
4422:
4420:
4417:
4415:
4412:
4410:
4407:
4403:
4400:
4398:
4397:by definition
4395:
4394:
4393:
4390:
4386:
4383:
4382:
4381:
4378:
4376:
4373:
4371:
4368:
4366:
4363:
4361:
4358:
4357:
4354:
4351:
4349:
4345:
4340:
4334:
4330:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4287:
4285:
4284:Kripke–Platek
4282:
4280:
4277:
4273:
4270:
4268:
4265:
4264:
4263:
4260:
4259:
4257:
4253:
4245:
4242:
4241:
4240:
4237:
4235:
4232:
4228:
4225:
4224:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4197:
4193:
4189:
4186:
4182:
4179:
4177:
4174:
4172:
4169:
4168:
4167:
4163:
4160:
4159:
4157:
4155:
4151:
4147:
4139:
4136:
4134:
4131:
4129:
4128:constructible
4126:
4125:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4060:
4058:
4056:
4051:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4014:
4012:
4008:
4005:
4003:
4000:
3999:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3974:
3970:
3968:
3965:
3961:
3958:
3957:
3956:
3953:
3952:
3949:
3946:
3944:
3940:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3905:
3902:
3898:
3895:
3894:
3893:
3890:
3886:
3881:
3880:
3879:
3876:
3875:
3873:
3871:
3867:
3859:
3856:
3854:
3851:
3849:
3846:
3845:
3844:
3841:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3810:
3808:
3806:
3805:Propositional
3802:
3796:
3793:
3791:
3788:
3786:
3783:
3781:
3778:
3776:
3773:
3771:
3768:
3764:
3761:
3760:
3759:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3739:
3736:
3734:
3733:Logical truth
3731:
3729:
3726:
3725:
3723:
3721:
3717:
3714:
3712:
3708:
3702:
3699:
3697:
3694:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3673:
3669:
3665:
3663:
3660:
3658:
3655:
3653:
3649:
3646:
3645:
3643:
3641:
3635:
3630:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3562:
3559:
3558:
3557:
3554:
3553:
3551:
3547:
3543:
3536:
3531:
3529:
3524:
3522:
3517:
3516:
3513:
3501:
3500:Ernst Zermelo
3498:
3496:
3493:
3491:
3488:
3486:
3485:Willard Quine
3483:
3481:
3478:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3446:
3443:
3442:
3440:
3438:
3437:Set theorists
3434:
3428:
3425:
3423:
3420:
3418:
3415:
3414:
3412:
3406:
3404:
3401:
3400:
3397:
3389:
3386:
3384:
3383:Kripke–Platek
3381:
3377:
3374:
3373:
3372:
3369:
3368:
3367:
3364:
3360:
3357:
3356:
3355:
3354:
3350:
3346:
3343:
3342:
3341:
3338:
3337:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3315:
3313:
3309:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3286:
3281:
3279:
3276:
3274:
3271:
3268:
3264:
3261:
3259:
3256:
3252:
3249:
3247:
3244:
3242:
3239:
3238:
3237:
3234:
3231:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3208:
3206:
3203:
3199:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3163:
3160:
3156:
3153:
3151:
3148:
3147:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3120:
3116:
3113:
3111:
3108:
3106:
3103:
3102:
3100:
3094:
3091:
3090:
3087:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3045:
3042:
3040:
3037:
3036:
3034:
3032:
3028:
3020:
3019:specification
3017:
3015:
3012:
3011:
3010:
3007:
3006:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2968:
2965:
2961:
2958:
2957:
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:
5250:
5048:Ultraproduct
4895:Model theory
4860:Independence
4796:Formal proof
4788:Proof theory
4771:
4744:
4701:real numbers
4673:second-order
4584:Substitution
4461:Metalanguage
4402:conservative
4375:Axiom schema
4319:Constructive
4289:Morse–Kelley
4255:Set theories
4234:Aleph number
4227:inaccessible
4195:
4133:Grothendieck
4017:intersection
3904:Higher-order
3892:Second-order
3838:Truth tables
3795:Venn diagram
3578:Formal proof
3450:Georg Cantor
3445:Paul Bernays
3376:Morse–Kelley
3351:
3284:
3283:Subset
3230:hereditarily
3192:Venn diagram
3171:
3150:ordered pair
3065:Intersection
3009: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:
5158:Type theory
5106:undecidable
5038:Truth value
4925:equivalence
4604:non-logical
4217:Enumeration
4207:Isomorphism
4154:cardinality
4138:Von Neumann
4103:Ultrafilter
4068:Uncountable
4002:equivalence
3919:Quantifiers
3909:Fixed-point
3878:First-order
3758:Consistency
3743:Proposition
3720:Traditional
3691:Lindström's
3681:Compactness
3623:Type theory
3568:Cardinality
3475:Thomas Jech
3318:Alternative
3297:Uncountable
3251:Ultrafilter
3110:Cardinality
3014: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
5273:Categories
4969:elementary
4662:arithmetic
4530:Quantifier
4508:functional
4380:Expression
4098:Transitive
4042:identities
4027:complement
3960:hereditary
3943:Set theory
3470:Kurt Gödel
3455:Paul Cohen
3292:Transitive
3060:Identities
3044:Complement
3031:Operations
2992:Regularity
2960:projective
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
5240:Supertask
5143:Recursion
5101:decidable
4935:saturated
4913:of models
4836:deductive
4831:axiomatic
4751:Hilbert's
4738:Euclidean
4719:canonical
4642:axiomatic
4574:Signature
4503:Predicate
4392:Extension
4314:Ackermann
4239:Operation
4118:Universal
4108:Recursive
4083:Singleton
4078:Inhabited
4063:Countable
4053:Types of
4037:power set
4007:partition
3924:Predicate
3870:Predicate
3785:Syllogism
3775:Soundness
3748:Inference
3738:Tautology
3640:paradoxes
3403:Paradoxes
3323:Axiomatic
3302:Universal
3278:Singleton
3273:Recursive
3216:Countable
3211:Amorphous
3070:Power set
2987: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
5225:Logicism
5218:timeline
5194:Concrete
5053:Validity
5023:T-schema
5016:Kripke's
5011:Tarski's
5006:semantic
4996:Strength
4945:submodel
4940:spectrum
4908:function
4756:Tarski's
4745:Elements
4732:geometry
4688:Robinson
4609:variable
4594:function
4567:spectrum
4557:Sentence
4513:variable
4456:Language
4409:Relation
4370:Automata
4360:Alphabet
4344:language
4198:-jection
4176:codomain
4162:Function
4123:Universe
4093:Infinite
3997:Relation
3780:Validity
3770:Argument
3668:theorem,
3407:Problems
3311:Theories
3287:Superset
3263:Infinite
3092:Concepts
2972: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.
5167:Related
4964:Diagram
4862: (
4841:Hilbert
4826:Systems
4821:Theorem
4699:of the
4644:systems
4424:Formula
4419:Grammar
4335: (
4279:General
3992:Forcing
3977:Element
3897:Monadic
3672:paradox
3613:Theorem
3549:General
3345:General
3340:Zermelo
3246:subbase
3228: (
3167:Forcing
3145:Element
3117: (
3095:Methods
2982: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
4930:finite
4693:Skolem
4646:
4621:Theory
4589:Symbol
4579:String
4562:atomic
4439:ground
4434:closed
4429:atomic
4385:ground
4348:syntax
4244:binary
4171:domain
4088:Finite
3853:finite
3711:Logics
3670:
3618:Theory
3236:Filter
3226:Finite
3162:Family
3105: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
4920:Model
4668:Peano
4525:Proof
4365:Arity
4294:Naive
4181:image
4113:Fuzzy
4073:Empty
4022:union
3967:Class
3608:Model
3598:Lemma
3556:Axiom
3328:Naive
3258:Fuzzy
3221:Empty
3204:types
3155:tuple
3125:Class
3119:large
3080:Union
2997: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
5043:Type
4846:list
4650:list
4627:list
4616:Term
4550:rank
4444:open
4338:list
4150:Maps
4055:sets
3914:Free
3884:list
3634:list
3561:list
3241: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
4730:of
4712:of
4660:of
4192:Sur
4166:Map
3973:Ur-
3955:Set
3202: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
5275::
5116:NP
4740::
4734::
4664::
4341:),
4196:Bi
4188:In
2834:.
2818:,
2812:,
2587:.
2583:.
2455:.
2274:.
2268:B′
2256:A′
2252:B′
2248:A′
2219:.
2151:!.
2088::
2067:|.
2017:).
1964:→
1960::
1893:.
1846:.
1722::
1714::
1658:).
1650:=
1622::
1575::
1550::
1542::
1534::
1427:,
1423:→
1419::
1325:,
1321:→
1317::
1313:,
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::
5196:/
5111:P
4866:)
4652:)
4648:(
4545:∀
4540:!
4535:∃
4496:=
4491:↔
4486:→
4481:∧
4476:∨
4471:¬
4194:/
4190:/
4164:/
3975:)
3971:(
3858:∞
3848:3
3636:)
3534:e
3527:t
3520:v
3285:·
3269:)
3265:(
3232:)
3121:)
2874:e
2867:t
2860:v
2840:.
2770:.
2601:.
2595::
2572:.
2547:.
2520:.
2493:.
2466:.
2442:.
2324:)
2272:B
2264:A
2250:→
2246::
2244:f
2236:R
2232:R
2228:B
2224:A
2149:n
2145:S
2137:S
2133:S
2125:S
2118:.
2112:f
2108:.
2102:f
2096:f
2086:f
2078:Y
2074:X
2065:B
2061:B
2059:(
2057:f
2053:A
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
1834:)
1829:1
1822:g
1818:(
1810:)
1805:1
1798:f
1794:(
1790:=
1784:1
1777:)
1773:f
1765:g
1762:(
1742:f
1734:g
1720:g
1712:f
1698:f
1690:g
1656:x
1654:(
1652:f
1648:y
1644:X
1640:x
1636:Y
1632:y
1620:f
1601:Y
1593:X
1589:Y
1573:f
1569:X
1565:f
1554:.
1548:h
1540:g
1532:f
1528:Y
1524:X
1513:h
1498:)
1491:,
1488:0
1484:[
1475:+
1470:0
1465:R
1449:h
1445:h
1441:h
1437:x
1433:x
1431:(
1429:h
1425:R
1421:R
1417:h
1406:g
1391:)
1384:,
1381:0
1377:(
1368:+
1363:R
1351:g
1347:x
1345:(
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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