7105:
31:
2337:
in mathematical logic are not necessarily forbidden. Set equality in ZFC is capable of declairing these indiscernibles as not equal, but an equality solely defined by these properties is not. Thus these properties form a strictly weaker notion of equality than set equality in ZFC. Outside of
1720:. There is no standard notation that distinguishes an equation from an identity, or other use of the equality relation: one has to guess an appropriate interpretation from the semantics of expressions and the context. Sometimes, but not always, an identity is written with a
2510:
is distinct from reflexivity in two main ways: first, the Law of
Identity applies only to cases of equality, and second, it is not restricted to elements of a set. However, many mathematicians refer to both as "Reflexivity", which is generally harmless.
327:
The truth of an equality depends on an interpretation of its members. In the above examples, the equalities are true if the members are interpreted as numbers or sets, but are false if the members are interpreted as expressions or sequences of symbols.
311:
4012:
3947:
4082:
between shapes. Similarly to isomorphisms of sets, the difference between isomorphisms and equality/congruence between such mathematical objects with properties and structure was one motivation for the development of
1805:
4340:
1695:
4522:
4263:
4192:
4451:
3843:
1170:
3054:
2727:
3791:
are distinct as fractions (as different strings of symbols) but they "represent" the same rational number (the same point on a number line). This distinction gives rise to the notion of a
2859:
2564:
2422:
2002:
403:
3886:
are not equal sets â the first consists of letters, while the second consists of numbers â but they are both sets of three elements and thus isomorphic, meaning that there is a
3616:
3257:
1222:
4843:
Deutsch, Harry and Pawel
Garbacz, "Relative Identity", The Stanford Encyclopedia of Philosophy (Fall 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.), forthcoming URL:
5484:
3370:
1860:
964:
1578:
908:
2990:
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is replaced with any number, then the two expressions take the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same
2897:
3881:
3485:
3441:
2062:
1096:
799:
755:
685:
6159:
3564:
2596:
2227:
2155:
2124:
214:
2275:
4056:
2448:
1532:
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Equality of sets is axiomatized in set theory in two different ways, depending on whether the axioms are based on a first-order language with or without equality.
3172:
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158:
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990:
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1718:
1303:
1283:
1259:
1046:
1022:
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and these sets cannot be identified without making such a choice â any statement that identifies them "depends on choice of identification". This distinction,
5191:
More generally, equality itself can be formally said to be a "reflexive relation". Just not as relation within ZFC, but as a "meta-relation", within some of
6242:
5383:
1610:
is an equality that is true for all values of its variables in a given domain. An "equation" may sometimes mean an identity, but more often than not, it
229:
4353:
is a matter of convenience; by this we save the labor of defining equality and proving all its properties; this burden is now assumed by the logic."
2311:
of equality, as they are usually sufficient for deducing most properties of equality that mathematicians care about. (See the following subsection)
1337:. Equality is a predicate, which may be true for some values of the variables (if any) and false for other values. More specifically, equality is a
2357:
However, apart from cases dealing with indiscernibles, these properties taken as axioms of equality are equivalent to equality as defined in ZFC.
2021:, generally states that if two things are equal, then any property of one must be a property of the other. It can be stated formally as: for every
3958:
3896:
6556:
2296:
Note that this says "Equality implies these two properties" not that "These properties define equality"; this is intentional. This makes it an
3491:
This is also sometimes included in the axioms of equality, but isn't necessary as it can be deduced from the other two axioms as shown above.
6714:
5351:
5325:
5244:
5222:
5068:
4855:
Forrest, Peter, "The
Identity of Indiscernibles", The Stanford Encyclopedia of Philosophy (Winter 2020 Edition), Edward N. Zalta (ed.), URL:
4680:
5502:
6569:
5892:
5274:
4626:
Pratt, Vaughan, "Algebra", The
Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL:
5025:
2251:, mathematicians don't tend to view their objects of interest as sets. For instance, many mathematicians would say that the expression "
1727:
6154:
7144:
6574:
6564:
6301:
5507:
6052:
5498:
6710:
4705:
6807:
6551:
5376:
4270:
1617:
6112:
5805:
5546:
2240:
2030:
4458:
4199:
4128:
4649:
4377:
7139:
7068:
6770:
6533:
6528:
6353:
5774:
5458:
5195:, which may be ZFC itself. So one could describe equality as a reflexive relation in some "meta-ZFC", but not "internal-ZFC"
4755:
4345:
Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by LĂ©vy.
3706:
in the sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element).
1326:
2293:
can be proved within ZFC as well as most other formal foundations), but is closer to how most mathematicians use equality.
7063:
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6284:
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when one may be moved to coincide with the other, and the equality/congruence relation is the isomorphism classes of
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2325:
2244:
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Similarly, there should be quantifiers 'â' for a, b, and đ, so more formally, this formula would be written as:
2015:
1819:
6926:
5541:
4927:
4868:
4832:
4722:
1391:, for which the specified equality is true. Each value of the unknown for which the equation holds is called a
6856:
6395:
6247:
6230:
5953:
5433:
4543:
3571:
1871:
578:
4032:
two mathematical objects that are only equivalent for the properties and structure being considered. The word
5162:
Here đ can have any (finite) arity, however, it is written as a unary formula to avoid cumbersome notation.
2307:, only what "equality" must satify. However, the two axioms as stated are still generally useful, even as an
6758:
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321:
49:
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to be equal if they contain the same elements. Then the axiom of extensionality states that two equal sets
338:
7048:
6775:
6753:
6720:
6613:
6459:
6444:
6417:
6368:
6252:
6187:
6012:
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5973:
5847:
5678:
5655:
3724:
3647:
3143:
1607:
1334:
1225:
1000:
410:
332:
4770:
4615:
6978:
6831:
6623:
6341:
6077:
5983:
5842:
5827:
5708:
5683:
4092:
4075:
3641:
3588:
3203:
2329:, which states that two distinct things cannot have all their properties in common. In mathematics, the
7104:
1175:
6951:
6913:
6790:
6594:
6434:
6358:
6336:
6164:
6122:
6021:
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4538:
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3629:
3524:
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1815:
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These properties offer a formal reinterpretation of equality from how it is defined in standard
52:, asserting that the quantities have the same value, or that the expressions represent the same
5283:
3848:
3446:
3402:
2035:
1057:
760:
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646:
17:
7044:
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6235:
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as its only solutions. The terminology is used similarly for equations with several unknowns.
1358:
1306:
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182:
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993:
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4117:
In first-order logic with equality, the axiom of extensionality states that two sets which
1505:
7039:
7029:
6983:
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6705:
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4084:
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1866:
1614:
a subset of the variable space to be the subset where the equation is true. An example is
1338:
802:
607:
134:
3766:
3738:
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3111:
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3059:
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2902:
2784:
2758:
2732:
2601:
2479:
2453:
2074:
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1450:
969:
2247:. In ZFC, equality only means that two sets have the same elements. However, outside of
7024:
7003:
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6941:
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6691:
6289:
6279:
6269:
6264:
6198:
6072:
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5411:
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4906:
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2334:
1920:
1876:
1703:
1288:
1268:
1244:
1031:
1007:
5020:
7123:
6998:
6676:
6183:
5968:
5958:
5928:
5913:
5583:
3585:
3570:
or other things, even if more precisely defined, is not transitive (since many small
3543:
2065:
1502:
An equation can be used to define a set. For example, the set of all solution pairs
6898:
6745:
6646:
6638:
6518:
6466:
6375:
6311:
6294:
6225:
6084:
5943:
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5428:
4639:
4627:
4059:
3792:
2322:
of the second statement must be true. The converse of the
Substitution property is
2162:
809:
693:
2360:
These are sometimes taken as the definition of equality, such as in some areas of
1870:: Stating that each thing is identical with itself, without restriction. That is,
427:
7008:
6888:
6067:
6057:
6004:
5688:
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5593:
5473:
5418:
5270:
4018:
3717:
3633:
3567:
3508:
1698:
1581:
1342:
1330:
306:{\displaystyle \{x\mid x\in \mathbb {Z} {\text{ and }}0<x\leq 3\}=\{1,2,3\},}
41:
34:
4881:
5938:
5793:
5764:
5570:
5343:
4079:
2248:
1721:
1229:
433:
414:
324:
that is often expressed as "two sets that have the same elements are equal".)
4993:
4928:
http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
4893:
4869:
http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
4833:
http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
4723:
http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
2354:. This is why the properties are said to not form a complete axiomatization.
7090:
6993:
6046:
5963:
5923:
5887:
5823:
5635:
5625:
5598:
5146:
4940:"Find all Unicode Characters from Hieroglyphs to Dingbats â Unicode Compart"
4939:
4807:
4007:{\displaystyle {\text{A}}\mapsto 3,{\text{B}}\mapsto 2,{\text{C}}\mapsto 1,}
3942:{\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.}
3887:
3528:
3520:
5265:(Third ed.). Providence, Rhode Island: American Mathematical Society.
4735:
7075:
6873:
6321:
6026:
5620:
4067:
3702:. It follows that equality is the finest equivalence relation on any set
1384:
30:
6671:
5463:
5063:. Oxford Logic Guides. Vol. 36. Clarendon Press. pp. 83â111.
5001:
4968:
3512:
582:
4771:
http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
4616:
http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
4058:) is frequently used for this kind of equality, and is defined as the
5361:
413:(equality of functions), or that the two expressions denote the same
4985:
3666:. The identity relation is an equivalence relation. Conversely, let
431:("equal", "like", "comparable", "similar"), which itself stems from
4786:
6215:
5561:
5406:
5150:
5005:
4845:
https://plato.stanford.edu/entries/identity-relative/#StanAccoIden
2290:
1800:{\displaystyle \left(x+1\right)\left(x+1\right)\equiv x^{2}+2x+1.}
29:
3735:
the latter being equivalence classes of fractions: the fractions
48:
is a relationship between two quantities or, more generally, two
2318:
of equality, meaning, if they were to define equality, then the
1822:, equality is often described through the following properties:
5365:
4857:
https://plato.stanford.edu/entries/identity-indiscernible/#Form
4025:
and is one motivation for the development of category theory.
3650:, equality is the archetype of the more general concept of an
2346:
has attracted much controversy and criticism, especially from
4831:
4721:
3584:
A questionable equality under test may be denoted using the
4335:{\displaystyle (\forall z,(z\in x\iff z\in y))\implies x=y}
3503:
that do not have any notion of equality. This reflects the
1690:{\displaystyle \left(x+1\right)\left(x+1\right)=x^{2}+2x+1}
1365:, and its computation from the two expressions is known as
4349:"The reason why we take up first-order predicate calculus
1387:
is the problem of finding values of some variable, called
1285:. In fact, equality is the unique equivalence relation on
4517:{\displaystyle x=y\implies \forall z,(x\in z\iff y\in z)}
4258:{\displaystyle x=y\implies \forall z,(x\in z\iff y\in z)}
4187:{\displaystyle x=y\implies \forall z,(z\in x\iff z\in y)}
3952:
However, there are other choices of isomorphism, such as
3709:
In some contexts, equality is sharply distinguished from
4446:{\displaystyle (x=y)\ :=\ \forall z,(z\in x\iff z\in y)}
4358:
Set equality based on first-order logic without equality
320:
have the same elements. (This equality results from the
219:
are two notations for the same number. Similarly, using
5153:, but is written as unary to avoid cumbersome notation.
4808:"Identity â math word definition â Math Open Reference"
27:
Relationship asserting that two quantities are the same
3624:
Relation with equivalence, congruence, and isomorphism
1178:
1104:
106:
of the equality and are distinguished by calling them
4769:
4614:
4461:
4380:
4273:
4202:
4131:
4113:
Set equality based on first-order logic with equality
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3851:
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3405:
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3114:
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2905:
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2430:
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2257:
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2134:
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2038:
1932:
1899:
1879:
1830:
1730:
1706:
1620:
1540:
1508:
1479:
1453:
1405:
1341:(i.e., a two-argument predicate) which may produce a
1291:
1271:
1247:
1060:
1034:
1010:
972:
916:
868:
825:
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719:
649:
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341:
232:
185:
137:
4362:
In first-order logic without equality, two sets are
3838:{\displaystyle \{{\text{A}},{\text{B}},{\text{C}}\}}
2303:
of equality. That is, it does not say what equality
1165:{\textstyle {\frac {d}{da}}g(a)={\frac {d}{da}}h(a)}
7017:
6912:
6744:
6637:
6489:
6182:
6105:
5999:
5903:
5792:
5719:
5654:
5569:
5560:
5482:
5399:
4926:Equality axioms. Encyclopedia of Mathematics. URL:
4867:Equality axioms. Encyclopedia of Mathematics. URL:
4967:
4516:
4445:
4334:
4257:
4186:
4050:
4006:
3941:
3875:
3837:
3783:
3755:
3610:
3558:
3479:
3435:
3391:
3364:
3251:
3166:
3126:
3100:
3074:
3049:{\displaystyle (b=a)\implies (bRc\Rightarrow aRc)}
3048:
2984:
2943:
2917:
2891:
2853:
2799:
2773:
2747:
2722:{\displaystyle (a=b)\implies (aRa\Rightarrow bRa)}
2721:
2657:
2616:
2590:
2558:
2494:
2468:
2442:
2416:
2269:
2221:
2149:
2118:
2089:
2056:
1996:
1911:
1885:
1854:
1799:
1712:
1689:
1572:
1526:
1491:
1465:
1439:
1297:
1277:
1253:
1216:
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1040:
1016:
984:
958:
902:
854:
793:
749:
679:
625:
397:
305:
208:
176:denote or represent the same object. For example,
152:
3670:be an equivalence relation, and let us denote by
4628:https://plato.stanford.edu/entries/algebra/#Laws
4598:
3574:can add up to something big). However, equality
1261:, those first three properties make equality an
124:. Two objects that are not equal are said to be
4911:First-Order Logic and Automated Theorem Proving
4882:"Identity and Individuality in Quantum Theory"
5377:
5061:Twenty Five Years of Constructive Type Theory
5059:. In Sambin, Giovanni; Smith, Jan M. (eds.).
1989:
1955:
8:
5276:When is one thing equal to some other thing?
5057:"The groupoid interpretation of type theory"
4698:Mengen – Relationen – Funktionen
3870:
3852:
3832:
3808:
2289:which can be grounded in ZFC (that is, both
297:
279:
273:
233:
6203:
5798:
5566:
5384:
5370:
5362:
5099:
4501:
4497:
4475:
4471:
4430:
4426:
4322:
4318:
4302:
4298:
4242:
4238:
4216:
4212:
4171:
4167:
4145:
4141:
3654:on a set: those binary relations that are
3286:
3282:
3018:
3014:
2691:
2687:
1952:
1948:
5239:. Mineola, New York: Dover Publications.
5217:. Mineola, New York: Dover Publications.
5123:
4675:. San Francisco, CA: Dover Publications.
4602:
4460:
4379:
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4201:
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3990:
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3962:
3960:
3928:
3914:
3900:
3898:
3850:
3827:
3819:
3811:
3806:
3773:
3768:
3745:
3740:
3600:
3595:
3593:
3592:
3590:
3551:
3527:. In other words, there cannot exist any
3448:
3404:
3378:
3265:
3205:
3150:
3113:
3087:
3061:
2997:
2956:
2930:
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1507:
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1410:
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1246:
1238:If restricted to the elements of a given
1194:
1177:
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253:
249:
248:
231:
195:
184:
136:
5320:. Mineola, New York: Dover Publication.
1399:the equation. For example, the equation
5138:
4700:(4th ed.). ZĂŒrich: Harri Deutsch.
4575:
1224:. An operation over functions (i.e. an
5127:
5091:
4652:from the original on 15 September 2020
4582:
2854:{\displaystyle xRy\Leftrightarrow x=y}
2559:{\displaystyle xRy\Leftrightarrow x=y}
2417:{\displaystyle xRy\Leftrightarrow x=y}
1997:{\displaystyle (a=b)\implies {\bigl }}
1588:; therefore, this equation is called
1395:of the given equation; also stated as
559:: Informally, this just means that if
4953:
4913:(Berlin/Heidelberg: Springer, 1990),
2314:If these properties were to define a
398:{\displaystyle (x+1)^{2}=x^{2}+2x+1,}
7:
5111:
5095:
4966:Eilenberg, S.; Mac Lane, S. (1942).
4594:
4121:the same elements are the same set.
3511:, defined by formulas involving the
1919:. It is the first of the historical
437:("equal", "level", "fair", "just").
5026:Stanford Encyclopedia of Philosophy
4886:Stanford Encyclopedia of Philosophy
4028:In some cases, one may consider as
3531:for deciding such an equality (see
1374:Relational operator § Equality
425:The word is derived from the Latin
5308:. New York: Van Nostrand Reinhold.
5306:Introduction to Mathematical Logic
4476:
4405:
4277:
4217:
4146:
4021:, is of fundamental importance in
3611:{\displaystyle {\stackrel {?}{=}}}
3252:{\displaystyle \phi (x):f(a)=f(x)}
1831:
1217:{\textstyle f(x)={\frac {dx}{da}}}
25:
3722:For example, one may distinguish
7103:
4019:between equality and isomorphism
3694:is equivalent with the equality
168:are any expressions, means that
4969:"Group Extensions and Homology"
2368:Derivations of basic properties
2285:or meaningless. This is a more
1590:the equation of the unit circle
18:Transitive property of equality
4511:
4498:
4485:
4472:
4440:
4427:
4414:
4393:
4381:
4319:
4315:
4312:
4299:
4286:
4274:
4252:
4239:
4226:
4213:
4181:
4168:
4155:
4142:
3995:
3981:
3967:
3933:
3919:
3905:
3474:
3468:
3459:
3453:
3430:
3424:
3415:
3409:
3365:{\displaystyle (a=b)\implies }
3359:
3356:
3353:
3347:
3338:
3332:
3326:
3323:
3320:
3317:
3311:
3302:
3296:
3290:
3287:
3283:
3279:
3267:
3246:
3240:
3231:
3225:
3216:
3210:
3161:
3155:
3043:
3031:
3019:
3015:
3011:
2999:
2967:
2961:
2839:
2716:
2704:
2692:
2688:
2684:
2672:
2640:
2634:
2544:
2402:
2216:
2210:
2144:
2138:
2113:
2107:
2048:
2042:
1984:
1978:
1972:
1969:
1963:
1949:
1945:
1933:
1855:{\displaystyle \forall a(a=a)}
1849:
1837:
1521:
1509:
1188:
1182:
1159:
1153:
1129:
1123:
1085:
1079:
1070:
1064:
959:{\displaystyle f(x,y)=x/y^{2}}
932:
920:
773:
767:
674:
668:
659:
653:
620:
614:
355:
342:
1:
7064:History of mathematical logic
5019:Marquis, Jean-Pierre (2019).
4733:Sobolev, S.K. (originator). "
4564:Proportionality (mathematics)
3682:, consisting of all elements
3443:by reflexivity, we have that
2861:), assume there are elements
2781:by Reflexivity, we have that
2566:), assume there are elements
2476:by the Law of identity, thus
1573:{\displaystyle x^{2}+y^{2}=1}
903:{\displaystyle a^{2}/b^{2}=2}
585:without changing its meaning.
6989:Primitive recursive function
5029:. Department of Philosophy,
4599:Mac Lane & Birkhoff 1999
4549:List of mathematical symbols
3174:, assume there are elements
3108:by assumption, we have that
2985:{\displaystyle \phi (x):xRc}
2658:{\displaystyle \phi (x):xRa}
1440:{\displaystyle x^{2}-6x+5=0}
855:{\displaystyle a^{2}=2b^{2}}
4743:Encyclopedia of Mathematics
4038:(and the associated symbol
2241:ZermeloâFraenkel set theory
2012:: Sometimes referred to as
417:(equality of polynomials).
7161:
6053:SchröderâBernstein theorem
5780:Monadic predicate calculus
5439:Foundations of mathematics
4102:
3890:between them. For example
3627:
2892:{\displaystyle a,b,c\in S}
2344:identity of indiscernibles
2333:is usually rejected since
2331:identity of indiscernibles
2326:identity of indiscernibles
7145:Equivalence (mathematics)
7099:
7086:Philosophy of mathematics
7035:Automated theorem proving
6206:
6160:Von NeumannâBernaysâGödel
5801:
5336:Shoenfield, Joseph Robert
5193:metatheory in mathematics
3876:{\displaystyle \{1,2,3\}}
3546:" (denoted by the symbol
3480:{\displaystyle f(a)=f(b)}
3436:{\displaystyle f(a)=f(a)}
2624:. Then, take the formula
2309:incomplete axiomatization
2057:{\displaystyle \phi (x),}
1353:) from its arguments. In
1091:{\displaystyle g(a)=h(a)}
794:{\displaystyle f(x)=2x-5}
750:{\displaystyle 2a-5=2b-5}
680:{\displaystyle f(a)=f(b)}
5318:Logic for mathematicians
4374:Set theory definition:
4066:between the objects. In
3559:{\displaystyle \approx }
3200:, then take the formula
2951:. Then take the formula
2812:Transitivity of Equality
2591:{\displaystyle a,b\in S}
2222:{\displaystyle \phi (x)}
2150:{\displaystyle \phi (b)}
2119:{\displaystyle \phi (a)}
209:{\displaystyle 1.5=3/2,}
50:mathematical expressions
6736:Self-verifying theories
6557:Tarski's axiomatization
5508:Tarski's undefinability
5503:incompleteness theorems
4880:French, Steven (2019).
4105:Axiom of extensionality
3507:of the equality of two
2375:Reflexivity of Equality
2316:complete axiomatization
2270:{\displaystyle 1\cup 2}
1820:mathematical philosophy
1357:, equality is called a
322:axiom of extensionality
7110:Mathematics portal
6721:Proof of impossibility
6369:propositional variable
5679:Propositional calculus
4787:"What is an Equation?"
4712:Here: sect.3.5, p.103.
4518:
4447:
4336:
4259:
4188:
4099:Equality in set theory
4052:
4051:{\displaystyle \cong }
4008:
3943:
3877:
3839:
3785:
3757:
3612:
3560:
3544:is approximately equal
3481:
3437:
3393:
3366:
3253:
3168:
3128:
3102:
3076:
3050:
2986:
2945:
2919:
2893:
2855:
2801:
2775:
2749:
2723:
2659:
2618:
2592:
2560:
2496:
2470:
2444:
2443:{\displaystyle a\in S}
2418:
2348:corpuscular philosophy
2271:
2243:(ZFC) or other formal
2223:
2151:
2120:
2091:
2058:
1998:
1913:
1887:
1856:
1801:
1714:
1691:
1574:
1528:
1493:
1467:
1441:
1299:
1279:
1255:
1218:
1166:
1092:
1042:
1018:
986:
960:
904:
856:
795:
751:
681:
627:
399:
307:
210:
154:
37:
7140:Elementary arithmetic
6979:Kolmogorov complexity
6932:Computably enumerable
6832:Model complete theory
6624:Principia Mathematica
5684:Propositional formula
5513:BanachâTarski paradox
4974:Annals of Mathematics
4640:"Definition of EQUAL"
4519:
4448:
4337:
4260:
4189:
4093:univalent foundations
4053:
4009:
3944:
3878:
3840:
3786:
3758:
3642:Congruence (geometry)
3613:
3561:
3517:arithmetic operations
3482:
3438:
3394:
3367:
3254:
3169:
3129:
3103:
3077:
3051:
2987:
2946:
2920:
2894:
2856:
2823:induced by equality (
2802:
2776:
2750:
2724:
2660:
2619:
2593:
2561:
2528:induced by equality (
2497:
2471:
2445:
2419:
2386:induced by equality (
2272:
2224:
2161:For example: For all
2152:
2121:
2092:
2059:
2009:Substitution property
1999:
1921:three laws of thought
1914:
1888:
1857:
1802:
1715:
1692:
1575:
1529:
1527:{\displaystyle (x,y)}
1494:
1468:
1442:
1317:Equality as predicate
1300:
1280:
1256:
1219:
1167:
1093:
1043:
1019:
987:
961:
905:
857:
796:
752:
682:
628:
590:Operation application
400:
308:
211:
155:
90:". In this equality,
33:
6927:ChurchâTuring thesis
6914:Computability theory
6123:continuum hypothesis
5641:Square of opposition
5499:Gödel's completeness
5314:Rosser, John Barkley
5211:Kleene, Stephen Cole
5126:, pp. 159â161.
4696:Lilly Görke (1974).
4673:Set Theory and Logic
4597:, pp. 13, 358.
4539:Homotopy type theory
4459:
4378:
4271:
4200:
4129:
4089:homotopy type theory
4042:
3959:
3897:
3849:
3805:
3798:Similarly, the sets
3767:
3739:
3690:. Then the relation
3652:equivalence relation
3630:Equivalence relation
3589:
3550:
3533:Richardson's theorem
3525:exponential function
3495:Approximate equality
3447:
3403:
3377:
3264:
3204:
3167:{\displaystyle f(x)}
3149:
3139:Function application
3112:
3086:
3060:
2996:
2955:
2929:
2903:
2865:
2827:
2785:
2759:
2733:
2669:
2628:
2602:
2570:
2532:
2517:Symmetry of Equality
2480:
2454:
2428:
2390:
2287:abstracted framework
2255:
2204:
2132:
2101:
2075:
2036:
1930:
1897:
1877:
1828:
1728:
1704:
1618:
1538:
1506:
1477:
1451:
1403:
1355:computer programming
1333:which may have some
1289:
1269:
1263:equivalence relation
1245:
1176:
1102:
1058:
1032:
1008:
970:
914:
866:
823:
761:
717:
647:
626:{\displaystyle f(x)}
608:
577:in any mathematical
339:
230:
221:set builder notation
183:
153:{\displaystyle x=y,}
135:
7081:Mathematical object
6972:P versus NP problem
6937:Computable function
6731:Reverse mathematics
6657:Logical consequence
6534:primitive recursive
6529:elementary function
6302:Free/bound variable
6155:TarskiâGrothendieck
5674:Logical connectives
5604:Logical equivalence
5454:Logical consequence
5031:Stanford University
4812:www.mathopenref.com
4785:; Watt, Stephen M.
4559:Logical equivalence
4064:isomorphism classes
3784:{\displaystyle 2/4}
3756:{\displaystyle 1/2}
3638:Congruence relation
3399:by assumption, and
3392:{\displaystyle a=b}
3127:{\displaystyle aRc}
3101:{\displaystyle bRc}
3075:{\displaystyle b=a}
2944:{\displaystyle bRc}
2918:{\displaystyle aRb}
2800:{\displaystyle bRa}
2774:{\displaystyle aRa}
2755:by assumption, and
2748:{\displaystyle a=b}
2617:{\displaystyle aRb}
2495:{\displaystyle aRa}
2469:{\displaystyle a=a}
2090:{\displaystyle a=b}
1912:{\displaystyle a=a}
1492:{\displaystyle x=5}
1466:{\displaystyle x=1}
1307:equivalence classes
1050:over some variable
985:{\displaystyle y=b}
56:. Equality between
54:mathematical object
7130:Mathematical logic
6879:Transfer principle
6842:Semantics of logic
6827:Categorical theory
6803:Non-standard model
6317:Logical connective
5444:Information theory
5393:Mathematical logic
5340:Mathematical Logic
5302:Mendelson, Elliott
5289:on 24 October 2019
5255:Mac Lane, Saunders
5215:Mathematical Logic
5130:, pp. 211â213
4514:
4455:Set theory axiom:
4443:
4332:
4267:Set theory axiom:
4255:
4184:
4076:equal or congruent
4070:for instance, two
4048:
4004:
3939:
3873:
3835:
3781:
3753:
3608:
3556:
3477:
3433:
3389:
3362:
3249:
3164:
3124:
3098:
3072:
3046:
2982:
2941:
2915:
2889:
2851:
2797:
2771:
2745:
2719:
2655:
2614:
2588:
2556:
2492:
2466:
2440:
2414:
2267:
2219:
2147:
2116:
2087:
2054:
1994:
1909:
1883:
1852:
1816:mathematical logic
1797:
1710:
1687:
1570:
1524:
1489:
1463:
1437:
1295:
1275:
1251:
1214:
1162:
1088:
1038:
1014:
999:Given real-valued
982:
956:
900:
852:
791:
747:
677:
623:
395:
303:
206:
150:
131:A formula such as
78:, and pronounced "
38:
7117:
7116:
7049:Abstract category
6852:Theories of truth
6662:Rule of inference
6652:Natural deduction
6633:
6632:
6178:
6177:
5883:Cartesian product
5788:
5787:
5694:Many-valued logic
5669:Boolean functions
5552:Russell's paradox
5527:diagonal argument
5424:First-order logic
5353:978-1-56881-135-2
5327:978-0-486-46898-3
5259:Birkhoff, Garrett
5246:978-0-486-42079-0
5224:978-0-486-42533-7
5145:đ can have any (
5070:978-0-19-158903-4
5053:Streicher, Thomas
5051:Hofmann, Martin;
5021:"Category Theory"
4682:978-0-486-63829-4
4671:Stoll, Robert R.
4404:
4398:
4087:, as well as for
3993:
3979:
3965:
3931:
3917:
3903:
3830:
3822:
3814:
3676:equivalence class
3605:
3576:almost everywhere
3082:by symmetry, and
2815:: Given some set
2520:: Given some set
2378:: Given some set
2362:first-order logic
2352:quantum mechanics
2283:abuse of notation
1886:{\displaystyle a}
1713:{\displaystyle x}
1586:analytic geometry
1298:{\displaystyle S}
1278:{\displaystyle S}
1254:{\displaystyle S}
1212:
1148:
1118:
1041:{\displaystyle h}
1017:{\displaystyle g}
256:
16:(Redirected from
7152:
7135:Binary relations
7108:
7107:
7059:History of logic
7054:Category of sets
6947:Decision problem
6726:Ordinal analysis
6667:Sequent calculus
6565:Boolean algebras
6505:
6504:
6479:
6450:logical/constant
6204:
6190:
6113:ZermeloâFraenkel
5864:Set operations:
5799:
5736:
5567:
5547:LöwenheimâSkolem
5434:Formal semantics
5386:
5379:
5372:
5363:
5357:
5342:(2nd ed.).
5331:
5309:
5297:
5296:
5294:
5288:
5282:, archived from
5281:
5273:(12 June 2007),
5266:
5250:
5237:Basic set theory
5228:
5196:
5189:
5183:
5160:
5154:
5143:
5131:
5121:
5115:
5109:
5103:
5089:
5083:
5082:
5048:
5042:
5041:
5039:
5037:
5016:
5010:
5009:
4971:
4963:
4957:
4950:
4944:
4943:
4936:
4930:
4924:
4918:
4904:
4898:
4897:
4877:
4871:
4865:
4859:
4853:
4847:
4841:
4835:
4829:
4823:
4822:
4820:
4818:
4804:
4798:
4797:
4795:
4793:
4779:
4773:
4767:
4761:
4731:
4725:
4719:
4713:
4711:
4693:
4687:
4686:
4668:
4662:
4661:
4659:
4657:
4636:
4630:
4624:
4618:
4612:
4606:
4592:
4586:
4580:
4554:Logical equality
4523:
4521:
4520:
4515:
4452:
4450:
4449:
4444:
4402:
4396:
4368:are contained in
4341:
4339:
4338:
4333:
4264:
4262:
4261:
4256:
4193:
4191:
4190:
4185:
4072:geometric shapes
4057:
4055:
4054:
4049:
4013:
4011:
4010:
4005:
3994:
3991:
3980:
3977:
3966:
3963:
3948:
3946:
3945:
3940:
3932:
3929:
3918:
3915:
3904:
3901:
3882:
3880:
3879:
3874:
3844:
3842:
3841:
3836:
3831:
3828:
3823:
3820:
3815:
3812:
3790:
3788:
3787:
3782:
3777:
3762:
3760:
3759:
3754:
3749:
3731:rational numbers
3617:
3615:
3614:
3609:
3607:
3606:
3604:
3599:
3594:
3565:
3563:
3562:
3557:
3486:
3484:
3483:
3478:
3442:
3440:
3439:
3434:
3398:
3396:
3395:
3390:
3371:
3369:
3368:
3363:
3258:
3256:
3255:
3250:
3199:
3185:
3179:
3173:
3171:
3170:
3165:
3133:
3131:
3130:
3125:
3107:
3105:
3104:
3099:
3081:
3079:
3078:
3073:
3055:
3053:
3052:
3047:
2991:
2989:
2988:
2983:
2950:
2948:
2947:
2942:
2924:
2922:
2921:
2916:
2898:
2896:
2895:
2890:
2860:
2858:
2857:
2852:
2822:
2819:with a relation
2818:
2806:
2804:
2803:
2798:
2780:
2778:
2777:
2772:
2754:
2752:
2751:
2746:
2728:
2726:
2725:
2720:
2664:
2662:
2661:
2656:
2623:
2621:
2620:
2615:
2597:
2595:
2594:
2589:
2565:
2563:
2562:
2557:
2527:
2524:with a relation
2523:
2501:
2499:
2498:
2493:
2475:
2473:
2472:
2467:
2449:
2447:
2446:
2441:
2423:
2421:
2420:
2415:
2385:
2382:with a relation
2381:
2276:
2274:
2273:
2268:
2235:
2228:
2226:
2225:
2220:
2199:
2192:
2185:
2175:
2169:
2156:
2154:
2153:
2148:
2125:
2123:
2122:
2117:
2096:
2094:
2093:
2088:
2070:
2063:
2061:
2060:
2055:
2028:
2024:
2003:
2001:
2000:
1995:
1993:
1992:
1959:
1958:
1918:
1916:
1915:
1910:
1892:
1890:
1889:
1884:
1861:
1859:
1858:
1853:
1806:
1804:
1803:
1798:
1781:
1780:
1768:
1764:
1749:
1745:
1719:
1717:
1716:
1711:
1697:is true for all
1696:
1694:
1693:
1688:
1671:
1670:
1658:
1654:
1639:
1635:
1597:Equation solving
1579:
1577:
1576:
1571:
1563:
1562:
1550:
1549:
1534:of the equation
1533:
1531:
1530:
1525:
1498:
1496:
1495:
1490:
1472:
1470:
1469:
1464:
1446:
1444:
1443:
1438:
1415:
1414:
1304:
1302:
1301:
1296:
1284:
1282:
1281:
1276:
1260:
1258:
1257:
1252:
1223:
1221:
1220:
1215:
1213:
1211:
1203:
1195:
1171:
1169:
1168:
1163:
1149:
1147:
1136:
1119:
1117:
1106:
1097:
1095:
1094:
1089:
1053:
1049:
1047:
1045:
1044:
1039:
1025:
1023:
1021:
1020:
1015:
994:binary operation
991:
989:
988:
983:
965:
963:
962:
957:
955:
954:
945:
909:
907:
906:
901:
893:
892:
883:
878:
877:
861:
859:
858:
853:
851:
850:
835:
834:
818:
814:
800:
798:
797:
792:
756:
754:
753:
748:
712:
702:
698:
686:
684:
683:
678:
642:
632:
630:
629:
624:
600:
596:
576:
572:
568:
550:
540:
530:
520:
516:
512:
500:
490:
480:
476:
464:
454:
441:Basic properties
408:
404:
402:
401:
396:
376:
375:
363:
362:
312:
310:
309:
304:
257:
254:
252:
215:
213:
212:
207:
199:
175:
171:
167:
163:
159:
157:
156:
151:
101:
95:
89:
83:
77:
67:
61:
21:
7160:
7159:
7155:
7154:
7153:
7151:
7150:
7149:
7120:
7119:
7118:
7113:
7102:
7095:
7040:Category theory
7030:Algebraic logic
7013:
6984:Lambda calculus
6922:Church encoding
6908:
6884:Truth predicate
6740:
6706:Complete theory
6629:
6498:
6494:
6490:
6485:
6477:
6197: and
6193:
6188:
6174:
6150:New Foundations
6118:axiom of choice
6101:
6063:Gödel numbering
6003: and
5995:
5899:
5784:
5734:
5715:
5664:Boolean algebra
5650:
5614:Equiconsistency
5579:Classical logic
5556:
5537:Halting problem
5525: and
5501: and
5489: and
5488:
5483:Theorems (
5478:
5395:
5390:
5360:
5354:
5334:
5328:
5312:
5300:
5292:
5290:
5286:
5279:
5269:
5253:
5247:
5231:
5225:
5209:
5205:
5200:
5199:
5190:
5186:
5165:
5163:
5161:
5157:
5144:
5140:
5135:
5134:
5122:
5118:
5110:
5106:
5100:Shoenfield 2001
5094:, p. 189.
5090:
5086:
5071:
5050:
5049:
5045:
5035:
5033:
5018:
5017:
5013:
4986:10.2307/1968966
4965:
4964:
4960:
4951:
4947:
4938:
4937:
4933:
4925:
4921:
4905:
4901:
4879:
4878:
4874:
4866:
4862:
4854:
4850:
4842:
4838:
4830:
4826:
4816:
4814:
4806:
4805:
4801:
4791:
4789:
4783:Marcus, Solomon
4781:
4780:
4776:
4768:
4764:
4758:
4732:
4728:
4720:
4716:
4708:
4695:
4694:
4690:
4683:
4670:
4669:
4665:
4655:
4653:
4645:Merriam-Webster
4638:
4637:
4633:
4625:
4621:
4613:
4609:
4593:
4589:
4581:
4577:
4572:
4530:
4457:
4456:
4376:
4375:
4370:the same sets.
4360:
4269:
4268:
4198:
4197:
4127:
4126:
4115:
4107:
4101:
4085:category theory
4074:are said to be
4040:
4039:
4023:category theory
3957:
3956:
3895:
3894:
3847:
3846:
3803:
3802:
3765:
3764:
3737:
3736:
3644:
3628:Main articles:
3626:
3587:
3586:
3548:
3547:
3540:binary relation
3499:There are some
3497:
3445:
3444:
3401:
3400:
3375:
3374:
3372:
3262:
3261:
3260:
3202:
3201:
3191:
3181:
3175:
3147:
3146:
3110:
3109:
3084:
3083:
3058:
3057:
2994:
2993:
2953:
2952:
2927:
2926:
2901:
2900:
2863:
2862:
2825:
2824:
2820:
2816:
2783:
2782:
2757:
2756:
2731:
2730:
2667:
2666:
2626:
2625:
2600:
2599:
2568:
2567:
2530:
2529:
2525:
2521:
2508:Law of identity
2478:
2477:
2452:
2451:
2426:
2425:
2388:
2387:
2383:
2379:
2370:
2253:
2252:
2230:
2202:
2201:
2194:
2187:
2177:
2171:
2165:
2130:
2129:
2099:
2098:
2073:
2072:
2068:
2034:
2033:
2026:
2022:
1928:
1927:
1895:
1894:
1875:
1874:
1867:Law of identity
1826:
1825:
1812:
1772:
1754:
1750:
1735:
1731:
1726:
1725:
1702:
1701:
1662:
1644:
1640:
1625:
1621:
1616:
1615:
1604:
1554:
1541:
1536:
1535:
1504:
1503:
1475:
1474:
1449:
1448:
1447:has the values
1406:
1401:
1400:
1381:
1339:binary relation
1319:
1287:
1286:
1267:
1266:
1243:
1242:
1204:
1196:
1174:
1173:
1140:
1110:
1100:
1099:
1056:
1055:
1051:
1030:
1029:
1027:
1006:
1005:
1003:
968:
967:
946:
912:
911:
884:
869:
864:
863:
842:
826:
821:
820:
816:
812:
803:unary operation
759:
758:
715:
714:
704:
700:
696:
688:
645:
644:
634:
606:
605:
598:
594:
574:
570:
560:
542:
532:
522:
518:
514:
510:
492:
482:
478:
474:
456:
452:
443:
423:
406:
367:
354:
337:
336:
255: and
228:
227:
181:
180:
173:
169:
165:
161:
133:
132:
118:right-hand side
97:
91:
85:
79:
69:
63:
57:
28:
23:
22:
15:
12:
11:
5:
7158:
7156:
7148:
7147:
7142:
7137:
7132:
7122:
7121:
7115:
7114:
7100:
7097:
7096:
7094:
7093:
7088:
7083:
7078:
7073:
7072:
7071:
7061:
7056:
7051:
7042:
7037:
7032:
7027:
7025:Abstract logic
7021:
7019:
7015:
7014:
7012:
7011:
7006:
7004:Turing machine
7001:
6996:
6991:
6986:
6981:
6976:
6975:
6974:
6969:
6964:
6959:
6954:
6944:
6942:Computable set
6939:
6934:
6929:
6924:
6918:
6916:
6910:
6909:
6907:
6906:
6901:
6896:
6891:
6886:
6881:
6876:
6871:
6870:
6869:
6864:
6859:
6849:
6844:
6839:
6837:Satisfiability
6834:
6829:
6824:
6823:
6822:
6812:
6811:
6810:
6800:
6799:
6798:
6793:
6788:
6783:
6778:
6768:
6767:
6766:
6761:
6754:Interpretation
6750:
6748:
6742:
6741:
6739:
6738:
6733:
6728:
6723:
6718:
6708:
6703:
6702:
6701:
6700:
6699:
6689:
6684:
6674:
6669:
6664:
6659:
6654:
6649:
6643:
6641:
6635:
6634:
6631:
6630:
6628:
6627:
6619:
6618:
6617:
6616:
6611:
6610:
6609:
6604:
6599:
6579:
6578:
6577:
6575:minimal axioms
6572:
6561:
6560:
6559:
6548:
6547:
6546:
6541:
6536:
6531:
6526:
6521:
6508:
6506:
6487:
6486:
6484:
6483:
6482:
6481:
6469:
6464:
6463:
6462:
6457:
6452:
6447:
6437:
6432:
6427:
6422:
6421:
6420:
6415:
6405:
6404:
6403:
6398:
6393:
6388:
6378:
6373:
6372:
6371:
6366:
6361:
6351:
6350:
6349:
6344:
6339:
6334:
6329:
6324:
6314:
6309:
6304:
6299:
6298:
6297:
6292:
6287:
6282:
6272:
6267:
6265:Formation rule
6262:
6257:
6256:
6255:
6250:
6240:
6239:
6238:
6228:
6223:
6218:
6213:
6207:
6201:
6184:Formal systems
6180:
6179:
6176:
6175:
6173:
6172:
6167:
6162:
6157:
6152:
6147:
6142:
6137:
6132:
6127:
6126:
6125:
6120:
6109:
6107:
6103:
6102:
6100:
6099:
6098:
6097:
6087:
6082:
6081:
6080:
6073:Large cardinal
6070:
6065:
6060:
6055:
6050:
6036:
6035:
6034:
6029:
6024:
6009:
6007:
5997:
5996:
5994:
5993:
5992:
5991:
5986:
5981:
5971:
5966:
5961:
5956:
5951:
5946:
5941:
5936:
5931:
5926:
5921:
5916:
5910:
5908:
5901:
5900:
5898:
5897:
5896:
5895:
5890:
5885:
5880:
5875:
5870:
5862:
5861:
5860:
5855:
5845:
5840:
5838:Extensionality
5835:
5833:Ordinal number
5830:
5820:
5815:
5814:
5813:
5802:
5796:
5790:
5789:
5786:
5785:
5783:
5782:
5777:
5772:
5767:
5762:
5757:
5752:
5751:
5750:
5740:
5739:
5738:
5725:
5723:
5717:
5716:
5714:
5713:
5712:
5711:
5706:
5701:
5691:
5686:
5681:
5676:
5671:
5666:
5660:
5658:
5652:
5651:
5649:
5648:
5643:
5638:
5633:
5628:
5623:
5618:
5617:
5616:
5606:
5601:
5596:
5591:
5586:
5581:
5575:
5573:
5564:
5558:
5557:
5555:
5554:
5549:
5544:
5539:
5534:
5529:
5517:Cantor's
5515:
5510:
5505:
5495:
5493:
5480:
5479:
5477:
5476:
5471:
5466:
5461:
5456:
5451:
5446:
5441:
5436:
5431:
5426:
5421:
5416:
5415:
5414:
5403:
5401:
5397:
5396:
5391:
5389:
5388:
5381:
5374:
5366:
5359:
5358:
5352:
5332:
5326:
5310:
5298:
5267:
5251:
5245:
5229:
5223:
5206:
5204:
5201:
5198:
5197:
5184:
5155:
5137:
5136:
5133:
5132:
5124:Mendelson 1964
5116:
5104:
5102:, p. 239.
5098:, p. 13.
5084:
5069:
5043:
5011:
4980:(4): 757â831.
4958:
4945:
4931:
4919:
4899:
4872:
4860:
4848:
4836:
4824:
4799:
4774:
4762:
4756:
4726:
4714:
4706:
4688:
4681:
4663:
4631:
4619:
4607:
4603:Mendelson 1964
4587:
4585:, p. 163.
4574:
4573:
4571:
4568:
4567:
4566:
4561:
4556:
4551:
4546:
4541:
4536:
4534:Extensionality
4529:
4526:
4525:
4524:
4513:
4510:
4507:
4504:
4500:
4496:
4493:
4490:
4487:
4484:
4481:
4478:
4474:
4470:
4467:
4464:
4453:
4442:
4439:
4436:
4433:
4429:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4401:
4395:
4392:
4389:
4386:
4383:
4359:
4356:
4355:
4354:
4343:
4342:
4331:
4328:
4325:
4321:
4317:
4314:
4311:
4308:
4305:
4301:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4265:
4254:
4251:
4248:
4245:
4241:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4215:
4211:
4208:
4205:
4194:
4183:
4180:
4177:
4174:
4170:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4144:
4140:
4137:
4134:
4114:
4111:
4103:Main article:
4100:
4097:
4047:
4015:
4014:
4003:
4000:
3997:
3989:
3986:
3983:
3975:
3972:
3969:
3950:
3949:
3938:
3935:
3927:
3924:
3921:
3913:
3910:
3907:
3884:
3883:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3834:
3826:
3818:
3810:
3780:
3776:
3772:
3752:
3748:
3744:
3625:
3622:
3603:
3598:
3555:
3505:undecidability
3496:
3493:
3489:
3488:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3388:
3385:
3382:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3285:
3281:
3278:
3275:
3272:
3269:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3163:
3160:
3157:
3154:
3135:
3123:
3120:
3117:
3097:
3094:
3091:
3071:
3068:
3065:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3017:
3013:
3010:
3007:
3004:
3001:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2940:
2937:
2934:
2914:
2911:
2908:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2808:
2796:
2793:
2790:
2770:
2767:
2764:
2744:
2741:
2738:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2690:
2686:
2683:
2680:
2677:
2674:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2613:
2610:
2607:
2587:
2584:
2581:
2578:
2575:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2504:
2503:
2491:
2488:
2485:
2465:
2462:
2459:
2439:
2436:
2433:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2369:
2366:
2335:indiscernibles
2300:axiomatization
2266:
2263:
2260:
2218:
2215:
2212:
2209:
2159:
2158:
2146:
2143:
2140:
2137:
2115:
2112:
2109:
2106:
2086:
2083:
2080:
2053:
2050:
2047:
2044:
2041:
1991:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1957:
1951:
1947:
1944:
1941:
1938:
1935:
1925:
1924:
1908:
1905:
1902:
1882:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1811:
1808:
1796:
1793:
1790:
1787:
1784:
1779:
1775:
1771:
1767:
1763:
1760:
1757:
1753:
1748:
1744:
1741:
1738:
1734:
1709:
1686:
1683:
1680:
1677:
1674:
1669:
1665:
1661:
1657:
1653:
1650:
1647:
1643:
1638:
1634:
1631:
1628:
1624:
1613:
1603:
1600:
1591:
1569:
1566:
1561:
1557:
1553:
1548:
1544:
1523:
1520:
1517:
1514:
1511:
1488:
1485:
1482:
1462:
1459:
1456:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1413:
1409:
1398:
1394:
1390:
1380:
1377:
1335:free variables
1318:
1315:
1294:
1274:
1250:
1236:
1235:
1234:
1233:
1228:), called the
1210:
1207:
1202:
1199:
1193:
1190:
1187:
1184:
1181:
1161:
1158:
1155:
1152:
1146:
1143:
1139:
1134:
1131:
1128:
1125:
1122:
1116:
1113:
1109:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1037:
1013:
997:
981:
978:
975:
953:
949:
944:
940:
937:
934:
931:
928:
925:
922:
919:
899:
896:
891:
887:
882:
876:
872:
849:
845:
841:
838:
833:
829:
806:
790:
787:
784:
781:
778:
775:
772:
769:
766:
746:
743:
740:
737:
734:
731:
728:
725:
722:
676:
673:
670:
667:
664:
661:
658:
655:
652:
622:
619:
616:
613:
586:
552:
502:
466:
442:
439:
422:
419:
405:means that if
394:
391:
388:
385:
382:
379:
374:
370:
366:
361:
357:
353:
350:
347:
344:
316:since the two
314:
313:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
251:
247:
244:
241:
238:
235:
217:
216:
205:
202:
198:
194:
191:
188:
149:
146:
143:
140:
109:left-hand side
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7157:
7146:
7143:
7141:
7138:
7136:
7133:
7131:
7128:
7127:
7125:
7112:
7111:
7106:
7098:
7092:
7089:
7087:
7084:
7082:
7079:
7077:
7074:
7070:
7067:
7066:
7065:
7062:
7060:
7057:
7055:
7052:
7050:
7046:
7043:
7041:
7038:
7036:
7033:
7031:
7028:
7026:
7023:
7022:
7020:
7016:
7010:
7007:
7005:
7002:
7000:
6999:Recursive set
6997:
6995:
6992:
6990:
6987:
6985:
6982:
6980:
6977:
6973:
6970:
6968:
6965:
6963:
6960:
6958:
6955:
6953:
6950:
6949:
6948:
6945:
6943:
6940:
6938:
6935:
6933:
6930:
6928:
6925:
6923:
6920:
6919:
6917:
6915:
6911:
6905:
6902:
6900:
6897:
6895:
6892:
6890:
6887:
6885:
6882:
6880:
6877:
6875:
6872:
6868:
6865:
6863:
6860:
6858:
6855:
6854:
6853:
6850:
6848:
6845:
6843:
6840:
6838:
6835:
6833:
6830:
6828:
6825:
6821:
6818:
6817:
6816:
6813:
6809:
6808:of arithmetic
6806:
6805:
6804:
6801:
6797:
6794:
6792:
6789:
6787:
6784:
6782:
6779:
6777:
6774:
6773:
6772:
6769:
6765:
6762:
6760:
6757:
6756:
6755:
6752:
6751:
6749:
6747:
6743:
6737:
6734:
6732:
6729:
6727:
6724:
6722:
6719:
6716:
6715:from ZFC
6712:
6709:
6707:
6704:
6698:
6695:
6694:
6693:
6690:
6688:
6685:
6683:
6680:
6679:
6678:
6675:
6673:
6670:
6668:
6665:
6663:
6660:
6658:
6655:
6653:
6650:
6648:
6645:
6644:
6642:
6640:
6636:
6626:
6625:
6621:
6620:
6615:
6614:non-Euclidean
6612:
6608:
6605:
6603:
6600:
6598:
6597:
6593:
6592:
6590:
6587:
6586:
6584:
6580:
6576:
6573:
6571:
6568:
6567:
6566:
6562:
6558:
6555:
6554:
6553:
6549:
6545:
6542:
6540:
6537:
6535:
6532:
6530:
6527:
6525:
6522:
6520:
6517:
6516:
6514:
6510:
6509:
6507:
6502:
6496:
6491:Example
6488:
6480:
6475:
6474:
6473:
6470:
6468:
6465:
6461:
6458:
6456:
6453:
6451:
6448:
6446:
6443:
6442:
6441:
6438:
6436:
6433:
6431:
6428:
6426:
6423:
6419:
6416:
6414:
6411:
6410:
6409:
6406:
6402:
6399:
6397:
6394:
6392:
6389:
6387:
6384:
6383:
6382:
6379:
6377:
6374:
6370:
6367:
6365:
6362:
6360:
6357:
6356:
6355:
6352:
6348:
6345:
6343:
6340:
6338:
6335:
6333:
6330:
6328:
6325:
6323:
6320:
6319:
6318:
6315:
6313:
6310:
6308:
6305:
6303:
6300:
6296:
6293:
6291:
6288:
6286:
6283:
6281:
6278:
6277:
6276:
6273:
6271:
6268:
6266:
6263:
6261:
6258:
6254:
6251:
6249:
6248:by definition
6246:
6245:
6244:
6241:
6237:
6234:
6233:
6232:
6229:
6227:
6224:
6222:
6219:
6217:
6214:
6212:
6209:
6208:
6205:
6202:
6200:
6196:
6191:
6185:
6181:
6171:
6168:
6166:
6163:
6161:
6158:
6156:
6153:
6151:
6148:
6146:
6143:
6141:
6138:
6136:
6135:KripkeâPlatek
6133:
6131:
6128:
6124:
6121:
6119:
6116:
6115:
6114:
6111:
6110:
6108:
6104:
6096:
6093:
6092:
6091:
6088:
6086:
6083:
6079:
6076:
6075:
6074:
6071:
6069:
6066:
6064:
6061:
6059:
6056:
6054:
6051:
6048:
6044:
6040:
6037:
6033:
6030:
6028:
6025:
6023:
6020:
6019:
6018:
6014:
6011:
6010:
6008:
6006:
6002:
5998:
5990:
5987:
5985:
5982:
5980:
5979:constructible
5977:
5976:
5975:
5972:
5970:
5967:
5965:
5962:
5960:
5957:
5955:
5952:
5950:
5947:
5945:
5942:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5922:
5920:
5917:
5915:
5912:
5911:
5909:
5907:
5902:
5894:
5891:
5889:
5886:
5884:
5881:
5879:
5876:
5874:
5871:
5869:
5866:
5865:
5863:
5859:
5856:
5854:
5851:
5850:
5849:
5846:
5844:
5841:
5839:
5836:
5834:
5831:
5829:
5825:
5821:
5819:
5816:
5812:
5809:
5808:
5807:
5804:
5803:
5800:
5797:
5795:
5791:
5781:
5778:
5776:
5773:
5771:
5768:
5766:
5763:
5761:
5758:
5756:
5753:
5749:
5746:
5745:
5744:
5741:
5737:
5732:
5731:
5730:
5727:
5726:
5724:
5722:
5718:
5710:
5707:
5705:
5702:
5700:
5697:
5696:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5661:
5659:
5657:
5656:Propositional
5653:
5647:
5644:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5622:
5619:
5615:
5612:
5611:
5610:
5607:
5605:
5602:
5600:
5597:
5595:
5592:
5590:
5587:
5585:
5584:Logical truth
5582:
5580:
5577:
5576:
5574:
5572:
5568:
5565:
5563:
5559:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5533:
5530:
5528:
5524:
5520:
5516:
5514:
5511:
5509:
5506:
5504:
5500:
5497:
5496:
5494:
5492:
5486:
5481:
5475:
5472:
5470:
5467:
5465:
5462:
5460:
5457:
5455:
5452:
5450:
5447:
5445:
5442:
5440:
5437:
5435:
5432:
5430:
5427:
5425:
5422:
5420:
5417:
5413:
5410:
5409:
5408:
5405:
5404:
5402:
5398:
5394:
5387:
5382:
5380:
5375:
5373:
5368:
5367:
5364:
5355:
5349:
5345:
5341:
5337:
5333:
5329:
5323:
5319:
5315:
5311:
5307:
5303:
5299:
5285:
5278:
5277:
5272:
5268:
5264:
5260:
5256:
5252:
5248:
5242:
5238:
5234:
5230:
5226:
5220:
5216:
5212:
5208:
5207:
5202:
5194:
5188:
5185:
5181:
5177:
5173:
5169:
5159:
5156:
5152:
5148:
5142:
5139:
5129:
5125:
5120:
5117:
5113:
5108:
5105:
5101:
5097:
5093:
5088:
5085:
5080:
5076:
5072:
5066:
5062:
5058:
5054:
5047:
5044:
5032:
5028:
5027:
5022:
5015:
5012:
5007:
5003:
4999:
4995:
4991:
4987:
4983:
4979:
4975:
4970:
4962:
4959:
4955:
4949:
4946:
4941:
4935:
4932:
4929:
4923:
4920:
4916:
4912:
4908:
4903:
4900:
4895:
4891:
4887:
4883:
4876:
4873:
4870:
4864:
4861:
4858:
4852:
4849:
4846:
4840:
4837:
4834:
4828:
4825:
4813:
4809:
4803:
4800:
4788:
4784:
4778:
4775:
4772:
4766:
4763:
4759:
4753:
4749:
4745:
4744:
4739:
4737:
4730:
4727:
4724:
4718:
4715:
4709:
4707:3-87144-118-X
4703:
4699:
4692:
4689:
4684:
4678:
4674:
4667:
4664:
4651:
4647:
4646:
4641:
4635:
4632:
4629:
4623:
4620:
4617:
4611:
4608:
4604:
4601:, p. 2.
4600:
4596:
4591:
4588:
4584:
4579:
4576:
4569:
4565:
4562:
4560:
4557:
4555:
4552:
4550:
4547:
4545:
4542:
4540:
4537:
4535:
4532:
4531:
4527:
4508:
4505:
4502:
4494:
4491:
4488:
4482:
4479:
4468:
4465:
4462:
4454:
4437:
4434:
4431:
4423:
4420:
4417:
4411:
4408:
4399:
4390:
4387:
4384:
4373:
4372:
4371:
4369:
4365:
4357:
4352:
4351:with equality
4348:
4347:
4346:
4329:
4326:
4323:
4309:
4306:
4303:
4295:
4292:
4289:
4283:
4280:
4266:
4249:
4246:
4243:
4235:
4232:
4229:
4223:
4220:
4209:
4206:
4203:
4196:Logic axiom:
4195:
4178:
4175:
4172:
4164:
4161:
4158:
4152:
4149:
4138:
4135:
4132:
4125:Logic axiom:
4124:
4123:
4122:
4120:
4112:
4110:
4106:
4098:
4096:
4094:
4090:
4086:
4081:
4077:
4073:
4069:
4065:
4061:
4045:
4037:
4036:
4031:
4026:
4024:
4020:
4001:
3998:
3987:
3984:
3973:
3970:
3955:
3954:
3953:
3936:
3925:
3922:
3911:
3908:
3893:
3892:
3891:
3889:
3867:
3864:
3861:
3858:
3855:
3824:
3816:
3801:
3800:
3799:
3796:
3794:
3778:
3774:
3770:
3750:
3746:
3742:
3734:
3732:
3727:
3726:
3721:
3719:
3714:
3713:
3707:
3705:
3701:
3698: =
3697:
3693:
3689:
3685:
3681:
3677:
3673:
3669:
3665:
3661:
3657:
3653:
3649:
3643:
3639:
3635:
3631:
3623:
3621:
3619:
3601:
3596:
3582:
3580:
3577:
3573:
3569:
3553:
3545:
3541:
3536:
3534:
3530:
3526:
3522:
3518:
3514:
3510:
3506:
3502:
3501:logic systems
3494:
3492:
3471:
3465:
3462:
3456:
3450:
3427:
3421:
3418:
3412:
3406:
3386:
3383:
3380:
3350:
3344:
3341:
3335:
3329:
3314:
3308:
3305:
3299:
3293:
3276:
3273:
3270:
3259:. So we have
3243:
3237:
3234:
3228:
3222:
3219:
3213:
3207:
3198:
3194:
3189:
3184:
3178:
3158:
3152:
3145:
3142:: Given some
3141:
3140:
3136:
3121:
3118:
3115:
3095:
3092:
3089:
3069:
3066:
3063:
3040:
3037:
3034:
3028:
3025:
3022:
3008:
3005:
3002:
2992:. So we have
2979:
2976:
2973:
2970:
2964:
2958:
2938:
2935:
2932:
2912:
2909:
2906:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2848:
2845:
2842:
2836:
2833:
2830:
2814:
2813:
2809:
2794:
2791:
2788:
2768:
2765:
2762:
2742:
2739:
2736:
2713:
2710:
2707:
2701:
2698:
2695:
2681:
2678:
2675:
2665:. So we have
2652:
2649:
2646:
2643:
2637:
2631:
2611:
2608:
2605:
2585:
2582:
2579:
2576:
2573:
2553:
2550:
2547:
2541:
2538:
2535:
2519:
2518:
2514:
2513:
2512:
2509:
2489:
2486:
2483:
2463:
2460:
2457:
2437:
2434:
2431:
2411:
2408:
2405:
2399:
2396:
2393:
2377:
2376:
2372:
2371:
2367:
2365:
2363:
2358:
2355:
2353:
2349:
2345:
2341:
2336:
2332:
2328:
2327:
2321:
2317:
2312:
2310:
2306:
2302:
2301:
2294:
2292:
2288:
2284:
2280:
2264:
2261:
2258:
2250:
2246:
2242:
2237:
2233:
2213:
2207:
2197:
2190:
2184:
2180:
2174:
2168:
2164:
2141:
2135:
2128:
2110:
2104:
2084:
2081:
2078:
2067:
2066:free variable
2051:
2045:
2039:
2032:
2020:
2019:
2017:
2011:
2010:
2006:
2005:
2004:
1981:
1975:
1966:
1960:
1942:
1939:
1936:
1922:
1906:
1903:
1900:
1880:
1873:
1869:
1868:
1864:
1863:
1862:
1846:
1843:
1840:
1834:
1823:
1821:
1817:
1809:
1807:
1794:
1791:
1788:
1785:
1782:
1777:
1773:
1769:
1765:
1761:
1758:
1755:
1751:
1746:
1742:
1739:
1736:
1732:
1723:
1707:
1700:
1684:
1681:
1678:
1675:
1672:
1667:
1663:
1659:
1655:
1651:
1648:
1645:
1641:
1636:
1632:
1629:
1626:
1622:
1611:
1609:
1601:
1599:
1598:
1593:
1589:
1587:
1583:
1567:
1564:
1559:
1555:
1551:
1546:
1542:
1518:
1515:
1512:
1500:
1486:
1483:
1480:
1460:
1457:
1454:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1411:
1407:
1396:
1392:
1388:
1386:
1378:
1376:
1375:
1370:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1316:
1314:
1312:
1308:
1292:
1272:
1264:
1248:
1241:
1231:
1227:
1208:
1205:
1200:
1197:
1191:
1185:
1179:
1156:
1150:
1144:
1141:
1137:
1132:
1126:
1120:
1114:
1111:
1107:
1082:
1076:
1073:
1067:
1061:
1035:
1011:
1002:
998:
995:
979:
976:
973:
951:
947:
942:
938:
935:
929:
926:
923:
917:
897:
894:
889:
885:
880:
874:
870:
847:
843:
839:
836:
831:
827:
811:
807:
804:
788:
785:
782:
779:
776:
770:
764:
744:
741:
738:
735:
732:
729:
726:
723:
720:
711:
707:
695:
691:
690:
689:For example:
671:
665:
662:
656:
650:
641:
637:
617:
611:
604:
592:
591:
587:
584:
580:
567:
563:
558:
557:
553:
549:
545:
539:
535:
529:
525:
508:
507:
503:
499:
495:
489:
485:
472:
471:
467:
463:
459:
450:
449:
445:
444:
440:
438:
436:
435:
430:
429:
420:
418:
416:
412:
392:
389:
386:
383:
380:
377:
372:
368:
364:
359:
351:
348:
345:
334:
329:
325:
323:
319:
300:
294:
291:
288:
285:
282:
276:
270:
267:
264:
261:
258:
245:
242:
239:
236:
226:
225:
224:
222:
203:
200:
196:
192:
189:
186:
179:
178:
177:
147:
144:
141:
138:
129:
127:
123:
119:
115:
111:
110:
105:
100:
94:
88:
82:
76:
73: =
72:
66:
60:
55:
51:
47:
43:
36:
32:
19:
7101:
6899:Ultraproduct
6746:Model theory
6711:Independence
6647:Formal proof
6639:Proof theory
6622:
6595:
6552:real numbers
6524:second-order
6435:Substitution
6312:Metalanguage
6253:conservative
6226:Axiom schema
6170:Constructive
6140:MorseâKelley
6106:Set theories
6085:Aleph number
6078:inaccessible
5984:Grothendieck
5868:intersection
5755:Higher-order
5743:Second-order
5689:Truth tables
5646:Venn diagram
5429:Formal proof
5339:
5317:
5305:
5291:, retrieved
5284:the original
5275:
5271:Mazur, Barry
5262:
5236:
5233:LĂ©vy, Azriel
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5119:
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5107:
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5036:26 September
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4977:
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810:real numbers
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694:real numbers
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601:, with some
593:: for every
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506:Transitivity
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7009:Type theory
6957:undecidable
6889:Truth value
6776:equivalence
6455:non-logical
6068:Enumeration
6058:Isomorphism
6005:cardinality
5989:Von Neumann
5954:Ultrafilter
5919:Uncountable
5853:equivalence
5770:Quantifiers
5760:Fixed-point
5729:First-order
5609:Consistency
5594:Proposition
5571:Traditional
5542:Lindström's
5532:Compactness
5474:Type theory
5419:Cardinality
5293:13 December
5128:Rosser 2008
5092:Kleene 2002
4915:pp. 198â200
4907:Fitting, M.
4792:27 February
4583:Rosser 2008
3718:isomorphism
3712:equivalence
3634:Isomorphism
3572:differences
2298:incomplete
2245:foundations
1582:unit circle
1343:truth value
1331:proposition
448:Reflexivity
114:left member
68:is written
42:mathematics
35:Equals sign
7124:Categories
6820:elementary
6513:arithmetic
6381:Quantifier
6359:functional
6231:Expression
5949:Transitive
5893:identities
5878:complement
5811:hereditary
5794:Set theory
5344:A K Peters
5203:References
4954:Mazur 2007
4817:1 December
4757:1402006098
4544:Inequality
4080:isometries
4035:congruence
3686:such that
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3190:such that
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2598:such that
2424:), assume
2249:set theory
2029:, and any
1722:triple bar
1602:Identities
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1580:forms the
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1372:See also:
1367:comparison
1363:expression
1311:singletons
1230:derivative
579:expression
455:, one has
415:polynomial
335:, such as
7091:Supertask
6994:Recursion
6952:decidable
6786:saturated
6764:of models
6687:deductive
6682:axiomatic
6602:Hilbert's
6589:Euclidean
6570:canonical
6493:axiomatic
6425:Signature
6354:Predicate
6243:Extension
6165:Ackermann
6090:Operation
5969:Universal
5959:Recursive
5934:Singleton
5929:Inhabited
5914:Countable
5904:Types of
5888:power set
5858:partition
5775:Predicate
5721:Predicate
5636:Syllogism
5626:Soundness
5599:Inference
5589:Tautology
5491:paradoxes
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5316:(2008) .
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3888:bijection
3725:fractions
3660:symmetric
3656:reflexive
3554:≈
3529:algorithm
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2340:pure math
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2208:ϕ
2136:ϕ
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1976:ϕ
1973:⇒
1961:ϕ
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1327:predicate
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1001:functions
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268:≤
246:∈
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7069:timeline
7045:Concrete
6904:Validity
6874:T-schema
6867:Kripke's
6862:Tarski's
6857:semantic
6847:Strength
6796:submodel
6791:spectrum
6759:function
6607:Tarski's
6596:Elements
6583:geometry
6539:Robinson
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6408:Sentence
6364:variable
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6260:Relation
6221:Automata
6211:Alphabet
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5519:theorem,
5304:(1964).
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4656:9 August
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428:aequÄlis
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6401:rank
6295:open
6189:list
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5906:sets
5765:Free
5735:list
5485:list
5412:list
5348:ISBN
5322:ISBN
5295:2009
5241:ISBN
5219:ISBN
5065:ISBN
5038:2022
4990:ISSN
4890:ISSN
4819:2019
4794:2019
4752:ISBN
4702:ISBN
4677:ISBN
4658:2020
4091:and
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170:x
166:y
162:x
148:,
145:y
142:=
139:x
99:B
93:A
87:B
81:A
75:B
71:A
65:B
59:A
20:)
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