36:
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839:. Thus, the notion of an irrational number is meaningless to even the most powerful floating-point computer. The necessity for such an extension stems not from physical observation but rather from the internal requirements of mathematical coherence. The infinitesimals entered mathematical discourse at a time when such a notion was required by mathematical developments at the time, namely the emergence of what became known as the
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The correct general statement that formulates these equivalences is called the transfer principle. Note that, in many formulas in analysis, quantification is over higher-order objects such as functions and sets, which makes the transfer principle somewhat more subtle than the above examples suggest.
1307:
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed
851:"In discussing the real line we remarked that we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line."
3165:
867:, plus, times, comparison) and quantifies only over the real numbers was assumed to be true in a reinterpreted form if we presume that it quantifies over hyperreal numbers. For example, we can state that for every real number there is another number greater than it:
1457:
3373: + 3} has exactly three members by the transfer principle. Because of the infiniteness of the domain, the complements of the images of one-to-one functions from the former set to the latter come in many sizes, but most of these functions are external.
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There are several different versions of the transfer principle, depending on what model of nonstandard mathematics is being used. In terms of model theory, the transfer principle states that a map from a standard model to a nonstandard model is an
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3637:ĆoĆ, Jerzy (1955) Quelques remarques, thĂ©orĂšmes et problĂšmes sur les classes dĂ©finissables d'algĂšbres. Mathematical interpretation of formal systems, pp. 98â113. North-Holland Publishing Co., Amsterdam.
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443:". In other words, the hyperreals appear to be Archimedean to an internal observer living in the nonstandard universe, but appear to be non-Archimedean to an external observer outside the universe.
3465:
Robinson, A. The metaphysics of the calculus, in
Problems in the Philosophy of Mathematics, ed. Lakatos (Amsterdam: North Holland), pp. 28â46, 1967. Reprinted in the 1979 Collected Works. Page 29.
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and Shelah give a construction of a definable, countably saturated elementary extension of the structure consisting of the reals and all finitary relations on it.
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already constituted a departure from the realm of immediate experience to the realm of mathematical models. The further extension, the rational numbers
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includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical realisation of a project initiated by
Leibniz.
1272:. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number like
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are expected to have the "same" properties as appreciable numbers. The transfer principle can also be viewed as a rigorous formalization of the
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3160:{\displaystyle \forall n\in {^{*}\mathbb {N} }\ \exists {\text{ internal }}A\subseteq {^{*}\mathbb {N} }\ \forall x\in {^{*}\mathbb {N} }\ .}
289:
The transfer principle appears to lead to contradictions if it is not handled correctly. For example, since the hyperreal numbers form a non-
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states that all statements of some language that are true for some structure are true for another structure. One of the first examples was
236:. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to
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The importance of these concepts stems from their role in the following proposition and is illustrated by the examples that follow it.
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outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an
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entities. Thus, the well-ordering property of the natural numbers by transfer yields the fact that every internal subset of
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1452:{\displaystyle \underbrace {\left|x\right|+\cdots +\left|x\right|} _{n{\text{ terms}}}<1{\text{ for every finite ] }}n.}
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and the reals form an
Archimedean ordered field, the property of being Archimedean ("every positive real is larger than
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344:") seems at first sight not to satisfy the transfer principle. The statement "every positive hyperreal is larger than
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2915:} (which is not standard) must be internal. To prove this, first observe that the following is trivially true:
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Suppose a proposition otherwise expressible as simply as those considered above mentions some particular sets
1548:{\displaystyle \underbrace {1+\cdots +1} _{n{\text{ terms}}}<\left|y\right|{\text{ for every finite ] }}n.}
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1005:{\displaystyle \forall x\in {}^{\star }\mathbb {R} \quad \exists y\in {}^{\star }\mathbb {R} \quad x<y.}
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2633:{\displaystyle \forall A\subseteq \mathbb {R} \dots {\text{ or }}\exists A\subseteq \mathbb {R} \dots \ .}
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1258:{\displaystyle 1<\omega ,\quad 1+1<\omega ,\quad 1+1+1<\omega ,\quad 1+1+1+1<\omega ,\ldots }
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3016:{\displaystyle \forall n\in \mathbb {N} \ \exists A\subseteq \mathbb {N} \ \forall x\in \mathbb {N} \ .}
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Suppose a proposition otherwise expressible as simply as those considered above contains the quantifier
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is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals.
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between 0 and 1 that differ from those by infinitesimals. To see this, observe that the sentence
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was a major intellectual accomplishment in its time. The addition of negative integers to form
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The transfer principle concerns the logical relation between the properties of the real numbers
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392:" is false; however the correct interpretation is "every positive hyperreal is larger than
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3288:{\displaystyle \forall {\text{ internal }}f:{^{*}\!A}\rightarrow {^{*}\mathbb {R} }\dots }
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2462:{\displaystyle \forall x\in \mathbb {R} \ (x\in {\text{ if and only if }}0\leq x\leq 1)}
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Another example is the statement that if you add 1 to a number you get a bigger number:
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922:{\displaystyle \forall x\in \mathbb {R} \quad \exists y\in \mathbb {R} \quad x<y.}
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has a least element. In this section internal sets are discussed in more detail.
1928:{\displaystyle \forall x\in \mathbb {R} {\text{ and }}\exists x\in \mathbb {R} .}
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function. By a typical application of the transfer principle, every hyperreal
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after the changes specified above and the replacement of the quantifiers with
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The appropriate setting for the hyperreal transfer principle is the world of
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is simple enough for the transfer principle to apply to it) and must contain
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2006:{\displaystyle \forall x\in \mathbb {R} \ \exists y\in \mathbb {R} \ x+y=0.}
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is an infinite integer, then the complement of the image of any internal
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2293:{\displaystyle ^{\ast }=\{\,x\in \mathbb {R} :0\leq x\leq 1\,\}^{\ast }}
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as a subfield. Unlike the reals, the hyperreals do not form a standard
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3538:, Studies in Logic and the Foundations of Mathematics (3rd ed.),
2490:(since the sentence expressing the non-existence of an upper bound of
859:
development of the hyperreals turned out to be possible if every true
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3587:
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722:
446:
A freshman-level accessible formulation of the transfer principle is
1120:{\displaystyle \forall x\in {}^{\star }\mathbb {R} \quad x<x+1.}
6056:
5771:
4836:
4182:
4027:
1797:{\displaystyle {^{*}\!f}:{^{*}\!A}\rightarrow {^{*}\mathbb {R} };}
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1836:
can be expressed via functions of finitely many variables (e.g. (
1681:. The standard sets belong to a much larger class of subsets of
232:, and then point out that this language applies equally well to *
6016:
5744:
3986:
3677:
1312:, but the ultrafilter itself cannot be explicitly constructed.
2367:{\displaystyle \{\,x\in {^{*}\mathbb {R} }:0\leq x\leq 1\,\},}
29:
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elementary embedding (similar, but only for statements with
2834:. Consequently the set of all infinitesimals is external.
654:
is the natural extension of the integer part function. If
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The theorem to the effect that each proposition valid over
132:
An incipient form of a transfer principle was described by
3175:
As with internal sets, so with internal functions: Replace
3299:
when applying the transfer principle, and similarly with
3214:{\displaystyle \forall f:A\rightarrow \mathbb {R} \dots }
5740:
2911:
is an infinite integer, then the set {1, ...,
2889:{\displaystyle {^{*}\mathbb {N} }\setminus \mathbb {N} }
2554:{\displaystyle {^{*}\mathbb {N} }\setminus \mathbb {N} }
2178:{\displaystyle \scriptstyle A\,\subseteq \,\mathbb {R} }
1666:{\displaystyle \scriptstyle A\,\subseteq \,\mathbb {R} }
795:, is more familiar to a layperson than their completion
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3378:
This last example motivates an important definition: A
1058:{\displaystyle \forall x\in \mathbb {R} \quad x<x+1}
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2837:The well-ordering principle implies every nonempty
1474:of the nonzero infinitesimals, are infinite, i.e.,
1295:, but by virtue of their order they carry an order
3394:one-to-one correspondence with {1, ...,
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647:{\displaystyle {}^{*}\!\lfloor \,\cdot \,\rfloor }
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2069:{\displaystyle \forall x\in {^{*}\!\mathbb {R} }}
2059:
1852:), relations among finitely many variables (e.g.
1767:
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1612:
1350:. Like all ordered fields that properly include
1139:The transfer principle however doesn't mean that
693:
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593:
278:of all statements in a language), or sometimes a
3613:"Elementary Calculus: An Infinitesimal Approach"
3453:"Elementary Calculus: An Infinitesimal Approach"
1815:. Sets and functions that are not internal are
1319:In its most general form, transfer is a bounded
609:{\displaystyle x\geq {}^{*}\!\lfloor x\rfloor ,}
209:, and the properties of a larger field denoted *
2382:between 0 and 1 inclusive, but also members of
3418:Elementary Calculus: An Infinitesimal Approach
453:Elementary Calculus: An Infinitesimal Approach
201:Hyperreal number § The transfer principle
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3446:
3444:
932:The same will then also hold for hyperreals:
152:, who used infinitesimals to define both the
8:
3569:"A definable nonstandard model of the reals"
3555:Hardy, Michael: "Scaled Boolean algebras".
3477:"A definable nonstandard model of the reals"
2845:has a smallest member. Consequently the set
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3475:Kanovei, Vladimir; Shelah, Saharon (2004),
1719:{\displaystyle f:A\rightarrow \mathbb {R} }
1147:have identical behavior. For instance, in *
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2115:{\displaystyle \forall x\in \mathbb {R} ,}
863:statement that uses basic arithmetic (the
540:{\displaystyle \lfloor \,\cdot \,\rfloor }
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3495:
3353:from the infinite set {1, ...,
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80:Learn how and when to remove this message
2506:, but must not contain anything between
1860:), finitary logical connectives such as
717:Generalizations of the concept of number
512:{\displaystyle x\geq \lfloor x\rfloor ,}
104:, which states that any sentence in the
43:This article includes a list of general
3440:
2878:
2543:
1832:Suppose a proposition that is true of
246:the sentences of are interpreted in *
224:The idea is to express analysis over
171:proved the transfer principle for any
3808:Infinitesimal strain theory (physics)
2900:of all infinite integers is external.
1623:{\displaystyle A\subseteq {^{*}\!A},}
1068:which will also hold for hyperreals:
195:Transfer principle for the hyperreals
7:
1941:For example, one such proposition is
266:, is called the transfer principle.
175:system. Its most common use is in
5938:Analytic and synthetic propositions
5809:Formal semantics (natural language)
2477:, and apply the transfer principle.
191:is also true of hyperreal numbers.
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3306:
3234:
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3064:
3036:
2964:
2947:
2930:
2726:
2664:
2607:
2585:
2395:
2135:
2095:
2040:
1971:
1954:
1908:
1889:
1078:
1025:
966:
942:
892:
877:
148:. Similar tendencies are found in
49:it lacks sufficient corresponding
25:
3910:Transcendental law of homogeneity
3803:Constructive nonstandard analysis
3747:The Method of Mechanical Theorems
3734:Criticism of nonstandard analysis
2185:. Such a proposition is true in
6150:
5724:
3761:
3559:29 (2002), no. 2, 243–292.
2197:" replaced by the corresponding
1558:The underlying set of the field
34:
3793:Synthetic differential geometry
1689:sets. Similarly each function
1358:. It means that some members
1303:Constructions of the hyperreals
1268:but there is no such number in
1221:
1196:
1177:
1101:
1039:
989:
965:
906:
891:
3263:
3200:
3151:
3125:
3007:
2981:
2759:
2723:
2697:
2661:
2644:Such a proposition is true in
2502: + 1 if it contains
2456:
2433:
2421:
2412:
2378:including not only members of
2234:
2221:
2019:Such a proposition is true in
1772:
1708:
1536: for every finite ]
1440: for every finite ]
1:
5685:History of mathematical logic
3962:Analyse des Infiniment Petits
3798:Smooth infinitesimal analysis
3390:is one that can be placed in
2648:if and only if it is true in
2189:if and only if it is true in
2023:if and only if it is true in
1637:is finite. Sets of the form
1633:with equality if and only if
1466:is 0. Some other members of
721:Historically, the concept of
274:(an embedding preserving the
5610:Primitive recursive function
2807:{\displaystyle \mathbb {N} }
2486:must have no upper bound in
832:{\displaystyle \mathbb {Q} }
810:{\displaystyle \mathbb {R} }
788:{\displaystyle \mathbb {Q} }
766:{\displaystyle \mathbb {Z} }
744:{\displaystyle \mathbb {N} }
240:rather than to all sets. As
3627:Encyclopedia of Mathematics
3611:Keisler, H. Jerome (2000).
2830:has a least upper bound in
2826:that has an upper bound in
2514: + 1. Members of
1135:Differences between R and R
6224:
4674:SchröderâBernstein theorem
4401:Monadic predicate calculus
4060:Foundations of mathematics
3650:Princeton University Press
3357:} into {1, ...,
2438: if and only if
1462:The only infinitesimal in
372:for some positive integer
324:for some positive integer
228:in a suitable language of
198:
118:algebraically closed field
6145:
6022:Necessity and sufficiency
5778:
5720:
5707:Philosophy of mathematics
5656:Automated theorem proving
4827:
4781:Von NeumannâBernaysâGödel
4422:
3926:Gottfried Wilhelm Leibniz
3759:
3620:Kuhlmann, F.-V. (2001) ,
3574:Journal of Symbolic Logic
3484:Journal of Symbolic Logic
2565:are "infinite integers".)
2201:. Here are two examples:
571:satisfies the inequality
484:satisfies the inequality
3332:{\displaystyle \forall }
3312:{\displaystyle \exists }
2141:{\displaystyle \exists }
1340:nonstandard real numbers
1151:there exists an element
6198:Mathematical principles
5357:Self-verifying theories
5178:Tarski's axiomatization
4129:Tarski's undefinability
4124:incompleteness theorems
3424:Principle of Permanence
729:to the natural numbers
154:continuity of functions
146:principle of permanence
136:under the name of "the
102:the Lefschetz principle
64:more precise citations.
27:Concept in model theory
5731:Mathematics portal
5342:Proof of impossibility
4990:propositional variable
4300:Propositional calculus
3855:Standard part function
3597:10.2178/jsl/1080938834
3506:10.2178/jsl/1080938834
3333:
3313:
3289:
3215:
3161:
3017:
2890:
2808:
2772:
2704:
2634:
2555:
2463:
2368:
2294:
2179:
2142:
2116:
2070:
2007:
1929:
1798:
1729:extends to a function
1720:
1667:
1624:
1549:
1453:
1342:properly includes the
1259:
1121:
1059:
1006:
923:
841:infinitesimal calculus
833:
811:
789:
767:
745:
713:is infinite, as well.
707:
674:is infinite, then the
668:
648:
610:
565:
541:
513:
478:
437:
414:
386:
366:
338:
318:
262:, is also valid over *
6157:Philosophy portal
5600:Kolmogorov complexity
5553:Computably enumerable
5453:Model complete theory
5245:Principia Mathematica
4305:Propositional formula
4134:BanachâTarski paradox
3941:Augustin-Louis Cauchy
3753:Cavalieri's principle
3646:Non-standard analysis
3429:Generality of algebra
3369: + 2,
3365: + 1,
3334:
3314:
3290:
3216:
3162:
3018:
2891:
2809:
2773:
2705:
2635:
2556:
2464:
2369:
2295:
2180:
2143:
2117:
2071:
2008:
1930:
1876:, and the quantifiers
1799:
1721:
1668:
1625:
1550:
1454:
1287:containing the reals
1260:
1122:
1060:
1007:
924:
834:
812:
790:
768:
746:
708:
669:
649:
611:
566:
542:
514:
479:
438:
415:
387:
367:
339:
319:
116:is also true for any
112:that is true for the
6208:Nonstandard analysis
5548:ChurchâTuring thesis
5535:Computability theory
4744:continuum hypothesis
4262:Square of opposition
4120:Gödel's completeness
3783:Nonstandard calculus
3778:Nonstandard analysis
3622:"Transfer principle"
3451:Keisler, H. Jerome.
3323:
3303:
3239: internal
3231:
3185:
3069: internal
3033:
2927:
2859:
2796:
2731: internal
2720:
2669: internal
2658:
2582:
2524:
2392:
2310:
2218:
2157:
2132:
2092:
2037:
1951:
1886:
1736:
1696:
1645:
1593:
1481:
1377:
1323:between structures.
1321:elementary embedding
1162:
1075:
1022:
939:
874:
821:
799:
777:
755:
733:
681:
658:
620:
575:
555:
523:
488:
468:
427:
396:
376:
348:
328:
300:
272:elementary embedding
181:nonstandard analysis
162:Dirac delta function
160:) and a form of the
106:first-order language
5819:Philosophy of logic
5702:Mathematical object
5593:P versus NP problem
5558:Computable function
5352:Reverse mathematics
5278:Logical consequence
5155:primitive recursive
5150:elementary function
4923:Free/bound variable
4776:TarskiâGrothendieck
4295:Logical connectives
4225:Logical equivalence
4075:Logical consequence
3967:Elementary Calculus
3848:Individual concepts
3788:Internal set theory
3563:Kanovei, Vladimir;
3557:Adv. in Appl. Math.
3530:Chang, Chen Chung;
3402: ∈
3348:one-to-one function
2771:{\displaystyle \ .}
2027:when the quantifier
1827:transfer principle:
413:{\displaystyle 1/n}
365:{\displaystyle 1/n}
317:{\displaystyle 1/n}
284:bounded quantifiers
18:Transfer principles
6118:Rules of inference
6087:Mathematical logic
5829:Semantics of logic
5500:Transfer principle
5463:Semantics of logic
5448:Categorical theory
5424:Non-standard model
4938:Logical connective
4065:Information theory
4014:Mathematical logic
3860:Transfer principle
3724:Leibniz's notation
3532:Keisler, H. Jerome
3329:
3309:
3285:
3211:
3157:
3013:
2886:
2804:
2768:
2700:
2630:
2551:
2459:
2364:
2290:
2175:
2174:
2138:
2128:and similarly for
2112:
2066:
2003:
1925:
1813:internal functions
1809:standard functions
1794:
1716:
1663:
1662:
1620:
1545:
1519:
1507:
1470:, the reciprocals
1449:
1431:
1419:
1362: â 0 of
1255:
1117:
1055:
1002:
919:
829:
807:
785:
763:
741:
703:
664:
644:
606:
561:
537:
509:
474:
433:
420:for some positive
410:
382:
362:
334:
314:
230:mathematical logic
98:transfer principle
6185:
6184:
6141:
6140:
5975:Deductive closure
5921:
5920:
5860:Critical thinking
5738:
5737:
5670:Abstract category
5473:Theories of truth
5283:Rule of inference
5273:Natural deduction
5254:
5253:
4799:
4798:
4504:Cartesian product
4409:
4408:
4315:Many-valued logic
4290:Boolean functions
4173:Russell's paradox
4148:diagonal argument
4045:First-order logic
3980:
3979:
3895:Law of continuity
3885:Levi-Civita field
3870:Increment theorem
3829:Hyperreal numbers
3659:978-0-691-04490-3
3642:Robinson, Abraham
3549:978-0-444-88054-3
3240:
3140:
3124:
3096:
3070:
3063:
2996:
2980:
2963:
2946:
2764:
2732:
2670:
2626:
2605:
2439:
2411:
1987:
1970:
1906:
1807:these are called
1586:. In every case
1537:
1516:
1486:
1484:
1441:
1428:
1382:
1380:
861:first-order logic
667:{\displaystyle x}
564:{\displaystyle x}
477:{\displaystyle x}
436:{\displaystyle n}
385:{\displaystyle n}
337:{\displaystyle n}
215:hyperreal numbers
185:hyperreal numbers
138:Law of Continuity
90:
89:
82:
16:(Redirected from
6215:
6155:
6154:
6153:
6075:
5840:
5804:Computer science
5765:
5758:
5751:
5742:
5729:
5728:
5680:History of logic
5675:Category of sets
5568:Decision problem
5347:Ordinal analysis
5288:Sequent calculus
5186:Boolean algebras
5126:
5125:
5100:
5071:logical/constant
4825:
4811:
4734:ZermeloâFraenkel
4485:Set operations:
4420:
4357:
4188:
4168:LöwenheimâSkolem
4055:Formal semantics
4007:
4000:
3993:
3984:
3936:Pierre de Fermat
3931:Abraham Robinson
3771:Related branches
3765:
3698:
3691:
3684:
3675:
3670:
3634:
3616:
3607:
3590:
3552:
3517:
3516:
3499:
3481:
3472:
3466:
3463:
3457:
3456:
3448:
3342:For example: If
3338:
3336:
3335:
3330:
3318:
3316:
3315:
3310:
3294:
3292:
3291:
3286:
3281:
3280:
3275:
3274:
3262:
3257:
3256:
3241:
3238:
3220:
3218:
3217:
3212:
3207:
3166:
3164:
3163:
3158:
3141:
3138:
3122:
3121:
3120:
3115:
3114:
3094:
3093:
3092:
3087:
3086:
3071:
3068:
3061:
3060:
3059:
3054:
3053:
3022:
3020:
3019:
3014:
2997:
2994:
2978:
2977:
2961:
2960:
2944:
2943:
2895:
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2892:
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2871:
2870:
2813:
2811:
2810:
2805:
2803:
2777:
2775:
2774:
2769:
2762:
2755:
2754:
2749:
2748:
2733:
2730:
2709:
2707:
2706:
2703:{\displaystyle }
2701:
2693:
2692:
2687:
2686:
2671:
2668:
2639:
2637:
2636:
2631:
2624:
2620:
2606:
2603:
2598:
2560:
2558:
2557:
2552:
2550:
2542:
2541:
2536:
2535:
2468:
2466:
2465:
2460:
2440:
2437:
2409:
2408:
2373:
2371:
2370:
2365:
2338:
2337:
2332:
2331:
2299:
2297:
2296:
2291:
2289:
2288:
2260:
2242:
2241:
2193:with each such "
2184:
2182:
2181:
2176:
2173:
2147:
2145:
2144:
2139:
2121:
2119:
2118:
2113:
2108:
2075:
2073:
2072:
2067:
2065:
2064:
2058:
2057:
2012:
2010:
2009:
2004:
1985:
1984:
1968:
1967:
1934:
1932:
1931:
1926:
1921:
1907:
1904:
1902:
1803:
1801:
1800:
1795:
1790:
1789:
1784:
1783:
1771:
1766:
1765:
1753:
1748:
1747:
1725:
1723:
1722:
1717:
1715:
1672:
1670:
1669:
1664:
1661:
1629:
1627:
1626:
1621:
1616:
1611:
1610:
1566:under a mapping
1562:is the image of
1554:
1552:
1551:
1546:
1538:
1535:
1533:
1518:
1517:
1514:
1508:
1503:
1458:
1456:
1455:
1450:
1442:
1439:
1430:
1429:
1426:
1420:
1415:
1414:
1394:
1354:, this field is
1314:Vladimir Kanovei
1279:The hyperreals *
1264:
1262:
1261:
1256:
1126:
1124:
1123:
1118:
1100:
1095:
1094:
1089:
1064:
1062:
1061:
1056:
1038:
1011:
1009:
1008:
1003:
988:
983:
982:
977:
964:
959:
958:
953:
928:
926:
925:
920:
905:
890:
838:
836:
835:
830:
828:
816:
814:
813:
808:
806:
794:
792:
791:
786:
784:
772:
770:
769:
764:
762:
750:
748:
747:
742:
740:
712:
710:
709:
704:
692:
691:
686:
673:
671:
670:
665:
653:
651:
650:
645:
631:
630:
625:
615:
613:
612:
607:
592:
591:
586:
570:
568:
567:
562:
546:
544:
543:
538:
518:
516:
515:
510:
483:
481:
480:
475:
442:
440:
439:
434:
419:
417:
416:
411:
406:
391:
389:
388:
383:
371:
369:
368:
363:
358:
343:
341:
340:
335:
323:
321:
320:
315:
310:
177:Abraham Robinson
173:hyperreal number
122:characteristic 0
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
6223:
6222:
6218:
6217:
6216:
6214:
6213:
6212:
6188:
6187:
6186:
6181:
6151:
6149:
6137:
6101:
6092:Boolean algebra
6066:
5917:
5908:Metamathematics
5886:
5838:
5792:
5774:
5769:
5739:
5734:
5723:
5716:
5661:Category theory
5651:Algebraic logic
5634:
5605:Lambda calculus
5543:Church encoding
5529:
5505:Truth predicate
5361:
5327:Complete theory
5250:
5119:
5115:
5111:
5106:
5098:
4818: and
4814:
4809:
4795:
4771:New Foundations
4739:axiom of choice
4722:
4684:Gödel numbering
4624: and
4616:
4520:
4405:
4355:
4336:
4285:Boolean algebra
4271:
4235:Equiconsistency
4200:Classical logic
4177:
4158:Halting problem
4146: and
4122: and
4110: and
4109:
4104:Theorems (
4099:
4016:
4011:
3981:
3976:
3972:Cours d'Analyse
3950:
3914:
3905:Microcontinuity
3890:Hyperfinite set
3843:
3839:Surreal numbers
3812:
3766:
3757:
3729:Integral symbol
3707:
3702:
3660:
3640:
3619:
3610:
3565:Shelah, Saharon
3562:
3550:
3529:
3526:
3521:
3520:
3479:
3474:
3473:
3469:
3464:
3460:
3450:
3449:
3442:
3437:
3413:
3321:
3320:
3301:
3300:
3267:
3249:
3229:
3228:
3183:
3182:
3139: iff
3107:
3079:
3046:
3031:
3030:
2995: iff
2925:
2924:
2863:
2857:
2856:
2818:Every nonempty
2794:
2793:
2786:
2741:
2718:
2717:
2679:
2656:
2655:
2580:
2579:
2528:
2522:
2521:
2390:
2389:
2324:
2308:
2307:
2280:
2233:
2216:
2215:
2155:
2154:
2130:
2129:
2090:
2089:
2050:
2035:
2034:
1949:
1948:
1905: and
1884:
1883:
1776:
1758:
1740:
1734:
1733:
1694:
1693:
1643:
1642:
1603:
1591:
1590:
1523:
1487:
1479:
1478:
1404:
1384:
1383:
1375:
1374:
1356:non-Archimedean
1329:
1305:
1160:
1159:
1137:
1087:
1073:
1072:
1020:
1019:
975:
951:
937:
936:
872:
871:
865:natural numbers
857:self-consistent
819:
818:
797:
796:
775:
774:
753:
752:
731:
730:
719:
684:
679:
678:
656:
655:
623:
618:
617:
584:
573:
572:
553:
552:
521:
520:
486:
485:
466:
465:
462:
425:
424:
394:
393:
374:
373:
346:
345:
326:
325:
298:
297:
203:
197:
158:Cours d'Analyse
130:
114:complex numbers
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
6221:
6219:
6211:
6210:
6205:
6200:
6190:
6189:
6183:
6182:
6180:
6179:
6174:
6164:
6159:
6146:
6143:
6142:
6139:
6138:
6136:
6135:
6130:
6125:
6120:
6115:
6109:
6107:
6103:
6102:
6100:
6099:
6094:
6089:
6083:
6081:
6072:
6068:
6067:
6065:
6064:
6059:
6054:
6049:
6044:
6039:
6034:
6029:
6024:
6019:
6014:
6009:
6004:
5999:
5998:
5997:
5987:
5982:
5977:
5972:
5967:
5966:
5965:
5960:
5950:
5945:
5940:
5935:
5929:
5927:
5923:
5922:
5919:
5918:
5916:
5915:
5910:
5905:
5900:
5894:
5892:
5888:
5887:
5885:
5884:
5879:
5874:
5869:
5868:
5867:
5862:
5852:
5846:
5844:
5837:
5836:
5831:
5826:
5821:
5816:
5811:
5806:
5800:
5798:
5794:
5793:
5791:
5790:
5785:
5779:
5776:
5775:
5770:
5768:
5767:
5760:
5753:
5745:
5736:
5735:
5721:
5718:
5717:
5715:
5714:
5709:
5704:
5699:
5694:
5693:
5692:
5682:
5677:
5672:
5663:
5658:
5653:
5648:
5646:Abstract logic
5642:
5640:
5636:
5635:
5633:
5632:
5627:
5625:Turing machine
5622:
5617:
5612:
5607:
5602:
5597:
5596:
5595:
5590:
5585:
5580:
5575:
5565:
5563:Computable set
5560:
5555:
5550:
5545:
5539:
5537:
5531:
5530:
5528:
5527:
5522:
5517:
5512:
5507:
5502:
5497:
5492:
5491:
5490:
5485:
5480:
5470:
5465:
5460:
5458:Satisfiability
5455:
5450:
5445:
5444:
5443:
5433:
5432:
5431:
5421:
5420:
5419:
5414:
5409:
5404:
5399:
5389:
5388:
5387:
5382:
5375:Interpretation
5371:
5369:
5363:
5362:
5360:
5359:
5354:
5349:
5344:
5339:
5329:
5324:
5323:
5322:
5321:
5320:
5310:
5305:
5295:
5290:
5285:
5280:
5275:
5270:
5264:
5262:
5256:
5255:
5252:
5251:
5249:
5248:
5240:
5239:
5238:
5237:
5232:
5231:
5230:
5225:
5220:
5200:
5199:
5198:
5196:minimal axioms
5193:
5182:
5181:
5180:
5169:
5168:
5167:
5162:
5157:
5152:
5147:
5142:
5129:
5127:
5108:
5107:
5105:
5104:
5103:
5102:
5090:
5085:
5084:
5083:
5078:
5073:
5068:
5058:
5053:
5048:
5043:
5042:
5041:
5036:
5026:
5025:
5024:
5019:
5014:
5009:
4999:
4994:
4993:
4992:
4987:
4982:
4972:
4971:
4970:
4965:
4960:
4955:
4950:
4945:
4935:
4930:
4925:
4920:
4919:
4918:
4913:
4908:
4903:
4893:
4888:
4886:Formation rule
4883:
4878:
4877:
4876:
4871:
4861:
4860:
4859:
4849:
4844:
4839:
4834:
4828:
4822:
4805:Formal systems
4801:
4800:
4797:
4796:
4794:
4793:
4788:
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4748:
4747:
4746:
4741:
4730:
4728:
4724:
4723:
4721:
4720:
4719:
4718:
4708:
4703:
4702:
4701:
4694:Large cardinal
4691:
4686:
4681:
4676:
4671:
4657:
4656:
4655:
4650:
4645:
4630:
4628:
4618:
4617:
4615:
4614:
4613:
4612:
4607:
4602:
4592:
4587:
4582:
4577:
4572:
4567:
4562:
4557:
4552:
4547:
4542:
4537:
4531:
4529:
4522:
4521:
4519:
4518:
4517:
4516:
4511:
4506:
4501:
4496:
4491:
4483:
4482:
4481:
4476:
4466:
4461:
4459:Extensionality
4456:
4454:Ordinal number
4451:
4441:
4436:
4435:
4434:
4423:
4417:
4411:
4410:
4407:
4406:
4404:
4403:
4398:
4393:
4388:
4383:
4378:
4373:
4372:
4371:
4361:
4360:
4359:
4346:
4344:
4338:
4337:
4335:
4334:
4333:
4332:
4327:
4322:
4312:
4307:
4302:
4297:
4292:
4287:
4281:
4279:
4273:
4272:
4270:
4269:
4264:
4259:
4254:
4249:
4244:
4239:
4238:
4237:
4227:
4222:
4217:
4212:
4207:
4202:
4196:
4194:
4185:
4179:
4178:
4176:
4175:
4170:
4165:
4160:
4155:
4150:
4138:Cantor's
4136:
4131:
4126:
4116:
4114:
4101:
4100:
4098:
4097:
4092:
4087:
4082:
4077:
4072:
4067:
4062:
4057:
4052:
4047:
4042:
4037:
4036:
4035:
4024:
4022:
4018:
4017:
4012:
4010:
4009:
4002:
3995:
3987:
3978:
3977:
3975:
3974:
3969:
3964:
3958:
3956:
3952:
3951:
3949:
3948:
3946:Leonhard Euler
3943:
3938:
3933:
3928:
3922:
3920:
3919:Mathematicians
3916:
3915:
3913:
3912:
3907:
3902:
3897:
3892:
3887:
3882:
3877:
3872:
3867:
3862:
3857:
3851:
3849:
3845:
3844:
3842:
3841:
3836:
3831:
3826:
3820:
3818:
3817:Formalizations
3814:
3813:
3811:
3810:
3805:
3800:
3795:
3790:
3785:
3780:
3774:
3772:
3768:
3767:
3760:
3758:
3756:
3755:
3750:
3743:
3736:
3731:
3726:
3721:
3715:
3713:
3709:
3708:
3705:Infinitesimals
3703:
3701:
3700:
3693:
3686:
3678:
3672:
3671:
3658:
3638:
3635:
3617:
3608:
3560:
3553:
3548:
3525:
3522:
3519:
3518:
3467:
3458:
3455:. p. 902.
3439:
3438:
3436:
3433:
3432:
3431:
3426:
3421:
3412:
3409:
3408:
3407:
3375:
3374:
3340:
3328:
3308:
3297:
3296:
3295:
3284:
3279:
3273:
3269:
3265:
3261:
3255:
3251:
3247:
3244:
3236:
3223:
3222:
3221:
3210:
3206:
3202:
3199:
3196:
3193:
3190:
3177:
3176:
3172:
3171:
3170:
3169:
3168:
3167:
3156:
3153:
3150:
3147:
3144:
3136:
3133:
3130:
3127:
3119:
3113:
3109:
3105:
3102:
3099:
3091:
3085:
3081:
3077:
3074:
3066:
3058:
3052:
3048:
3044:
3041:
3038:
3025:
3024:
3023:
3012:
3009:
3006:
3003:
3000:
2992:
2989:
2986:
2983:
2976:
2972:
2969:
2966:
2959:
2955:
2952:
2949:
2942:
2938:
2935:
2932:
2917:
2916:
2904:
2903:
2902:
2901:
2898:
2897:
2896:
2884:
2880:
2875:
2869:
2865:
2849:
2848:
2847:
2846:
2802:
2785:
2784:Three examples
2782:
2781:
2780:
2779:
2778:
2767:
2761:
2758:
2753:
2747:
2743:
2739:
2736:
2728:
2725:
2712:
2711:
2710:
2699:
2696:
2691:
2685:
2681:
2677:
2674:
2666:
2663:
2642:
2641:
2640:
2629:
2623:
2619:
2615:
2612:
2609:
2604: or
2601:
2597:
2593:
2590:
2587:
2574:
2573:
2569:
2568:
2567:
2566:
2563:
2562:
2561:
2549:
2545:
2540:
2534:
2530:
2516:
2515:
2479:
2478:
2471:
2470:
2469:
2458:
2455:
2452:
2449:
2446:
2443:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2407:
2403:
2400:
2397:
2376:
2375:
2374:
2363:
2360:
2356:
2353:
2350:
2347:
2344:
2341:
2336:
2330:
2326:
2322:
2319:
2315:
2302:
2301:
2300:
2287:
2283:
2278:
2275:
2272:
2269:
2266:
2263:
2259:
2255:
2252:
2248:
2245:
2240:
2236:
2232:
2229:
2226:
2223:
2210:
2209:
2203:
2202:
2172:
2167:
2163:
2150:
2149:
2137:
2125:
2124:
2123:
2122:
2111:
2107:
2103:
2100:
2097:
2084:
2083:
2079:
2078:
2077:
2076:
2063:
2056:
2052:
2048:
2045:
2042:
2029:
2028:
2016:
2015:
2014:
2013:
2002:
1999:
1996:
1993:
1990:
1983:
1979:
1976:
1973:
1966:
1962:
1959:
1956:
1943:
1942:
1938:
1937:
1936:
1935:
1924:
1920:
1916:
1913:
1910:
1901:
1897:
1894:
1891:
1878:
1877:
1844:) âŠ
1805:
1804:
1793:
1788:
1782:
1778:
1774:
1770:
1764:
1760:
1756:
1752:
1746:
1742:
1727:
1726:
1714:
1710:
1707:
1704:
1701:
1660:
1655:
1651:
1631:
1630:
1619:
1615:
1609:
1605:
1601:
1598:
1582:to subsets of
1556:
1555:
1544:
1541:
1532:
1529:
1526:
1522:
1512:
1506:
1502:
1499:
1496:
1493:
1490:
1460:
1459:
1448:
1445:
1437:
1434:
1424:
1418:
1413:
1410:
1407:
1403:
1400:
1397:
1393:
1390:
1387:
1328:
1325:
1304:
1301:
1266:
1265:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1195:
1192:
1189:
1186:
1183:
1180:
1176:
1173:
1170:
1167:
1136:
1133:
1128:
1127:
1116:
1113:
1110:
1107:
1104:
1099:
1093:
1086:
1083:
1080:
1066:
1065:
1054:
1051:
1048:
1045:
1042:
1037:
1033:
1030:
1027:
1013:
1012:
1001:
998:
995:
992:
987:
981:
974:
971:
968:
963:
957:
950:
947:
944:
930:
929:
918:
915:
912:
909:
904:
900:
897:
894:
889:
885:
882:
879:
853:
852:
827:
805:
783:
761:
739:
718:
715:
702:
699:
696:
690:
663:
643:
639:
635:
629:
605:
602:
599:
596:
590:
583:
580:
560:
536:
532:
528:
508:
505:
502:
499:
496:
493:
473:
461:
458:
432:
409:
405:
401:
381:
361:
357:
353:
333:
313:
309:
305:
217:. The field *
196:
193:
142:infinitesimals
129:
126:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6220:
6209:
6206:
6204:
6201:
6199:
6196:
6195:
6193:
6178:
6175:
6172:
6168:
6165:
6163:
6160:
6158:
6148:
6147:
6144:
6134:
6133:Logic symbols
6131:
6129:
6126:
6124:
6121:
6119:
6116:
6114:
6111:
6110:
6108:
6104:
6098:
6095:
6093:
6090:
6088:
6085:
6084:
6082:
6080:
6076:
6073:
6069:
6063:
6060:
6058:
6055:
6053:
6050:
6048:
6045:
6043:
6040:
6038:
6035:
6033:
6030:
6028:
6025:
6023:
6020:
6018:
6015:
6013:
6012:Logical truth
6010:
6008:
6005:
6003:
6000:
5996:
5993:
5992:
5991:
5988:
5986:
5983:
5981:
5978:
5976:
5973:
5971:
5968:
5964:
5961:
5959:
5956:
5955:
5954:
5953:Contradiction
5951:
5949:
5946:
5944:
5941:
5939:
5936:
5934:
5931:
5930:
5928:
5924:
5914:
5911:
5909:
5906:
5904:
5901:
5899:
5898:Argumentation
5896:
5895:
5893:
5889:
5883:
5882:Philosophical
5880:
5878:
5877:Non-classical
5875:
5873:
5870:
5866:
5863:
5861:
5858:
5857:
5856:
5853:
5851:
5848:
5847:
5845:
5841:
5835:
5832:
5830:
5827:
5825:
5822:
5820:
5817:
5815:
5812:
5810:
5807:
5805:
5802:
5801:
5799:
5795:
5789:
5786:
5784:
5781:
5780:
5777:
5773:
5766:
5761:
5759:
5754:
5752:
5747:
5746:
5743:
5733:
5732:
5727:
5719:
5713:
5710:
5708:
5705:
5703:
5700:
5698:
5695:
5691:
5688:
5687:
5686:
5683:
5681:
5678:
5676:
5673:
5671:
5667:
5664:
5662:
5659:
5657:
5654:
5652:
5649:
5647:
5644:
5643:
5641:
5637:
5631:
5628:
5626:
5623:
5621:
5620:Recursive set
5618:
5616:
5613:
5611:
5608:
5606:
5603:
5601:
5598:
5594:
5591:
5589:
5586:
5584:
5581:
5579:
5576:
5574:
5571:
5570:
5569:
5566:
5564:
5561:
5559:
5556:
5554:
5551:
5549:
5546:
5544:
5541:
5540:
5538:
5536:
5532:
5526:
5523:
5521:
5518:
5516:
5513:
5511:
5508:
5506:
5503:
5501:
5498:
5496:
5493:
5489:
5486:
5484:
5481:
5479:
5476:
5475:
5474:
5471:
5469:
5466:
5464:
5461:
5459:
5456:
5454:
5451:
5449:
5446:
5442:
5439:
5438:
5437:
5434:
5430:
5429:of arithmetic
5427:
5426:
5425:
5422:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5398:
5395:
5394:
5393:
5390:
5386:
5383:
5381:
5378:
5377:
5376:
5373:
5372:
5370:
5368:
5364:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5337:
5336:from ZFC
5333:
5330:
5328:
5325:
5319:
5316:
5315:
5314:
5311:
5309:
5306:
5304:
5301:
5300:
5299:
5296:
5294:
5291:
5289:
5286:
5284:
5281:
5279:
5276:
5274:
5271:
5269:
5266:
5265:
5263:
5261:
5257:
5247:
5246:
5242:
5241:
5236:
5235:non-Euclidean
5233:
5229:
5226:
5224:
5221:
5219:
5218:
5214:
5213:
5211:
5208:
5207:
5205:
5201:
5197:
5194:
5192:
5189:
5188:
5187:
5183:
5179:
5176:
5175:
5174:
5170:
5166:
5163:
5161:
5158:
5156:
5153:
5151:
5148:
5146:
5143:
5141:
5138:
5137:
5135:
5131:
5130:
5128:
5123:
5117:
5112:Example
5109:
5101:
5096:
5095:
5094:
5091:
5089:
5086:
5082:
5079:
5077:
5074:
5072:
5069:
5067:
5064:
5063:
5062:
5059:
5057:
5054:
5052:
5049:
5047:
5044:
5040:
5037:
5035:
5032:
5031:
5030:
5027:
5023:
5020:
5018:
5015:
5013:
5010:
5008:
5005:
5004:
5003:
5000:
4998:
4995:
4991:
4988:
4986:
4983:
4981:
4978:
4977:
4976:
4973:
4969:
4966:
4964:
4961:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4940:
4939:
4936:
4934:
4931:
4929:
4926:
4924:
4921:
4917:
4914:
4912:
4909:
4907:
4904:
4902:
4899:
4898:
4897:
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4875:
4872:
4870:
4869:by definition
4867:
4866:
4865:
4862:
4858:
4855:
4854:
4853:
4850:
4848:
4845:
4843:
4840:
4838:
4835:
4833:
4830:
4829:
4826:
4823:
4821:
4817:
4812:
4806:
4802:
4792:
4789:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4756:KripkeâPlatek
4754:
4752:
4749:
4745:
4742:
4740:
4737:
4736:
4735:
4732:
4731:
4729:
4725:
4717:
4714:
4713:
4712:
4709:
4707:
4704:
4700:
4697:
4696:
4695:
4692:
4690:
4687:
4685:
4682:
4680:
4677:
4675:
4672:
4669:
4665:
4661:
4658:
4654:
4651:
4649:
4646:
4644:
4641:
4640:
4639:
4635:
4632:
4631:
4629:
4627:
4623:
4619:
4611:
4608:
4606:
4603:
4601:
4600:constructible
4598:
4597:
4596:
4593:
4591:
4588:
4586:
4583:
4581:
4578:
4576:
4573:
4571:
4568:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4532:
4530:
4528:
4523:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4486:
4484:
4480:
4477:
4475:
4472:
4471:
4470:
4467:
4465:
4462:
4460:
4457:
4455:
4452:
4450:
4446:
4442:
4440:
4437:
4433:
4430:
4429:
4428:
4425:
4424:
4421:
4418:
4416:
4412:
4402:
4399:
4397:
4394:
4392:
4389:
4387:
4384:
4382:
4379:
4377:
4374:
4370:
4367:
4366:
4365:
4362:
4358:
4353:
4352:
4351:
4348:
4347:
4345:
4343:
4339:
4331:
4328:
4326:
4323:
4321:
4318:
4317:
4316:
4313:
4311:
4308:
4306:
4303:
4301:
4298:
4296:
4293:
4291:
4288:
4286:
4283:
4282:
4280:
4278:
4277:Propositional
4274:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4236:
4233:
4232:
4231:
4228:
4226:
4223:
4221:
4218:
4216:
4213:
4211:
4208:
4206:
4205:Logical truth
4203:
4201:
4198:
4197:
4195:
4193:
4189:
4186:
4184:
4180:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4145:
4141:
4137:
4135:
4132:
4130:
4127:
4125:
4121:
4118:
4117:
4115:
4113:
4107:
4102:
4096:
4093:
4091:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4043:
4041:
4038:
4034:
4031:
4030:
4029:
4026:
4025:
4023:
4019:
4015:
4008:
4003:
4001:
3996:
3994:
3989:
3988:
3985:
3973:
3970:
3968:
3965:
3963:
3960:
3959:
3957:
3953:
3947:
3944:
3942:
3939:
3937:
3934:
3932:
3929:
3927:
3924:
3923:
3921:
3917:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3852:
3850:
3846:
3840:
3837:
3835:
3832:
3830:
3827:
3825:
3824:Differentials
3822:
3821:
3819:
3815:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3786:
3784:
3781:
3779:
3776:
3775:
3773:
3769:
3764:
3754:
3751:
3749:
3748:
3744:
3742:
3741:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3716:
3714:
3710:
3706:
3699:
3694:
3692:
3687:
3685:
3680:
3679:
3676:
3669:
3665:
3661:
3655:
3651:
3647:
3643:
3639:
3636:
3633:
3629:
3628:
3623:
3618:
3614:
3609:
3606:
3602:
3598:
3594:
3589:
3584:
3580:
3576:
3575:
3570:
3566:
3561:
3558:
3554:
3551:
3545:
3541:
3537:
3533:
3528:
3527:
3523:
3515:
3511:
3507:
3503:
3498:
3493:
3489:
3485:
3478:
3471:
3468:
3462:
3459:
3454:
3447:
3445:
3441:
3434:
3430:
3427:
3425:
3422:
3420:
3419:
3415:
3414:
3410:
3405:
3401:
3397:
3393:
3389:
3385:
3381:
3377:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3349:
3345:
3341:
3298:
3282:
3271:
3268:
3259:
3253:
3250:
3245:
3242:
3227:
3226:
3224:
3208:
3197:
3194:
3191:
3181:
3180:
3179:
3178:
3174:
3173:
3154:
3148:
3145:
3142:
3134:
3131:
3128:
3111:
3108:
3103:
3100:
3083:
3080:
3075:
3072:
3050:
3047:
3042:
3039:
3029:
3028:
3027:Consequently
3026:
3010:
3004:
3001:
2998:
2990:
2987:
2984:
2970:
2967:
2953:
2950:
2936:
2933:
2923:
2922:
2921:
2920:
2919:
2918:
2914:
2910:
2906:
2905:
2899:
2867:
2864:
2855:
2854:
2853:
2852:
2851:
2850:
2844:
2840:
2836:
2835:
2833:
2829:
2825:
2821:
2817:
2816:
2815:
2791:
2783:
2765:
2756:
2745:
2742:
2737:
2734:
2716:
2715:
2713:
2694:
2683:
2680:
2675:
2672:
2654:
2653:
2651:
2647:
2643:
2627:
2621:
2613:
2610:
2599:
2591:
2588:
2578:
2577:
2576:
2575:
2571:
2570:
2564:
2532:
2529:
2520:
2519:
2518:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2489:
2485:
2481:
2480:
2476:
2472:
2453:
2450:
2447:
2444:
2441:
2430:
2427:
2424:
2418:
2415:
2401:
2398:
2388:
2387:
2385:
2381:
2377:
2361:
2354:
2351:
2348:
2345:
2342:
2339:
2328:
2325:
2320:
2317:
2306:
2305:
2303:
2285:
2276:
2273:
2270:
2267:
2264:
2261:
2253:
2250:
2243:
2238:
2230:
2227:
2224:
2214:
2213:
2212:
2211:
2207:
2206:
2205:
2204:
2200:
2196:
2192:
2188:
2165:
2161:
2152:
2151:
2127:
2126:
2109:
2101:
2098:
2088:
2087:
2086:
2085:
2081:
2080:
2054:
2051:
2046:
2043:
2033:
2032:
2031:
2030:
2026:
2022:
2018:
2017:
2000:
1997:
1994:
1991:
1988:
1977:
1974:
1960:
1957:
1947:
1946:
1945:
1944:
1940:
1939:
1922:
1914:
1911:
1895:
1892:
1882:
1881:
1880:
1879:
1875:
1871:
1867:
1863:
1859:
1856: â€
1855:
1851:
1848: +
1847:
1843:
1839:
1835:
1831:
1830:
1829:
1828:
1823:
1820:
1818:
1814:
1810:
1791:
1780:
1777:
1768:
1762:
1759:
1754:
1750:
1744:
1741:
1732:
1731:
1730:
1705:
1702:
1699:
1692:
1691:
1690:
1688:
1684:
1680:
1676:
1653:
1649:
1640:
1636:
1617:
1613:
1607:
1604:
1599:
1596:
1589:
1588:
1587:
1585:
1581:
1577:
1574:from subsets
1573:
1570: âŠ
1569:
1565:
1561:
1542:
1539:
1530:
1527:
1524:
1520:
1510:
1504:
1500:
1497:
1494:
1491:
1488:
1477:
1476:
1475:
1473:
1469:
1465:
1446:
1443:
1435:
1432:
1422:
1416:
1411:
1408:
1405:
1401:
1398:
1395:
1391:
1388:
1385:
1373:
1372:
1371:
1369:
1368:infinitesimal
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1334:
1333:ordered field
1326:
1324:
1322:
1317:
1315:
1311:
1302:
1300:
1298:
1294:
1290:
1286:
1285:ordered field
1282:
1277:
1275:
1271:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1193:
1190:
1187:
1184:
1181:
1178:
1174:
1171:
1168:
1165:
1158:
1157:
1156:
1154:
1150:
1146:
1142:
1134:
1132:
1114:
1111:
1108:
1105:
1102:
1091:
1084:
1081:
1071:
1070:
1069:
1052:
1049:
1046:
1043:
1040:
1031:
1028:
1018:
1017:
1016:
999:
996:
993:
990:
979:
972:
969:
955:
948:
945:
935:
934:
933:
916:
913:
910:
907:
898:
895:
883:
880:
870:
869:
868:
866:
862:
858:
850:
849:
848:
846:
842:
728:
724:
716:
714:
697:
688:
677:
661:
637:
627:
603:
597:
588:
581:
578:
558:
550:
530:
506:
500:
494:
491:
471:
459:
457:
455:
454:
449:
444:
430:
423:
407:
403:
399:
379:
359:
355:
351:
331:
311:
307:
303:
295:
294:ordered field
292:
287:
285:
281:
277:
273:
267:
265:
261:
256:
255:
253:
249:
243:
239:
238:internal sets
235:
231:
227:
222:
220:
216:
212:
208:
202:
194:
192:
190:
186:
182:
178:
174:
170:
165:
163:
159:
155:
151:
147:
143:
139:
135:
127:
125:
123:
119:
115:
111:
107:
103:
99:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
6203:Model theory
6052:Substitution
5872:Mathematical
5797:Major fields
5722:
5520:Ultraproduct
5499:
5367:Model theory
5332:Independence
5268:Formal proof
5260:Proof theory
5243:
5216:
5173:real numbers
5145:second-order
5056:Substitution
4933:Metalanguage
4874:conservative
4847:Axiom schema
4791:Constructive
4761:MorseâKelley
4727:Set theories
4706:Aleph number
4699:inaccessible
4605:Grothendieck
4489:intersection
4376:Higher-order
4364:Second-order
4310:Truth tables
4267:Venn diagram
4050:Formal proof
3880:Internal set
3865:Hyperinteger
3859:
3834:Dual numbers
3745:
3738:
3645:
3625:
3588:math/0311165
3578:
3572:
3556:
3536:Model Theory
3535:
3497:math/0311165
3487:
3483:
3470:
3461:
3416:
3403:
3399:
3395:
3391:
3387:
3386:) subset of
3383:
3382:(pronounced
3379:
3370:
3366:
3362:
3358:
3354:
3350:
3343:
3319:in place of
2912:
2908:
2842:
2838:
2831:
2827:
2823:
2819:
2789:
2787:
2649:
2645:
2511:
2507:
2503:
2499:
2495:
2491:
2487:
2483:
2474:
2383:
2379:
2198:
2194:
2190:
2186:
2024:
2020:
1874:if...then...
1873:
1869:
1865:
1861:
1857:
1853:
1849:
1845:
1841:
1837:
1833:
1826:
1824:
1821:
1816:
1812:
1808:
1806:
1728:
1686:
1682:
1678:
1674:
1638:
1634:
1632:
1583:
1579:
1575:
1571:
1567:
1563:
1559:
1557:
1471:
1467:
1463:
1461:
1363:
1359:
1351:
1347:
1335:
1330:
1318:
1306:
1293:metric space
1288:
1280:
1278:
1273:
1269:
1267:
1152:
1148:
1144:
1140:
1138:
1129:
1067:
1014:
931:
854:
720:
676:hyperinteger
549:integer part
463:
451:
445:
422:hyperinteger
288:
279:
276:truth values
268:
263:
259:
257:
247:
245:
233:
225:
223:
218:
210:
206:
204:
189:real numbers
166:
131:
97:
94:model theory
91:
76:
70:January 2012
67:
48:
6167:WikiProject
6037:Proposition
6032:Probability
5985:Description
5926:Foundations
5630:Type theory
5578:undecidable
5510:Truth value
5397:equivalence
5076:non-logical
4689:Enumeration
4679:Isomorphism
4626:cardinality
4610:Von Neumann
4575:Ultrafilter
4540:Uncountable
4474:equivalence
4391:Quantifiers
4381:Fixed-point
4350:First-order
4230:Consistency
4215:Proposition
4192:Traditional
4163:Lindström's
4153:Compactness
4095:Type theory
4040:Cardinality
3740:The Analyst
3581:: 159â164,
3490:: 159â164,
3398:} for some
3384:star-finite
2473:is true in
1677:subsets of
1673:are called
1515: terms
1427: terms
1310:ultrafilter
464:Every real
291:Archimedean
213:called the
62:introducing
6192:Categories
6097:Set theory
5995:Linguistic
5990:Entailment
5980:Definition
5948:Consequent
5943:Antecedent
5441:elementary
5134:arithmetic
5002:Quantifier
4980:functional
4852:Expression
4570:Transitive
4514:identities
4499:complement
4432:hereditary
4415:Set theory
3719:Adequality
3524:References
2841:subset of
2822:subset of
1155:such that
199:See also:
45:references
6128:Fallacies
6123:Paradoxes
6113:Logicians
6047:Statement
6042:Reference
6007:Induction
5970:Deduction
5933:Abduction
5903:Metalogic
5850:Classical
5814:Inference
5712:Supertask
5615:Recursion
5573:decidable
5407:saturated
5385:of models
5308:deductive
5303:axiomatic
5223:Hilbert's
5210:Euclidean
5191:canonical
5114:axiomatic
5046:Signature
4975:Predicate
4864:Extension
4786:Ackermann
4711:Operation
4590:Universal
4580:Recursive
4555:Singleton
4550:Inhabited
4535:Countable
4525:Types of
4509:power set
4479:partition
4396:Predicate
4342:Predicate
4257:Syllogism
4247:Soundness
4220:Inference
4210:Tautology
4112:paradoxes
3955:Textbooks
3900:Overspill
3632:EMS Press
3534:(1990) ,
3327:∀
3307:∃
3283:…
3272:∗
3264:→
3254:∗
3235:∀
3209:…
3201:→
3189:∀
3146:≤
3132:∈
3112:∗
3104:∈
3098:∀
3084:∗
3076:⊆
3065:∃
3051:∗
3043:∈
3037:∀
3002:≤
2988:∈
2971:∈
2965:∀
2954:⊆
2948:∃
2937:∈
2931:∀
2879:∖
2868:∗
2757:…
2746:∗
2738:⊆
2727:∃
2695:…
2684:∗
2676:⊆
2665:∀
2622:…
2614:⊆
2608:∃
2600:…
2592:⊆
2586:∀
2544:∖
2533:∗
2451:≤
2445:≤
2419:∈
2402:∈
2396:∀
2352:≤
2346:≤
2329:∗
2321:∈
2286:∗
2274:≤
2268:≤
2254:∈
2239:∗
2166:⊆
2136:∃
2102:∈
2096:∀
2055:∗
2047:∈
2041:∀
1978:∈
1972:∃
1961:∈
1955:∀
1915:∈
1909:∃
1896:∈
1890:∀
1781:∗
1773:→
1763:∗
1745:∗
1709:→
1654:⊆
1641:for some
1608:∗
1600:⊆
1505:⏟
1495:⋯
1417:⏟
1399:⋯
1327:Statement
1253:…
1247:ω
1216:ω
1191:ω
1172:ω
1092:⋆
1085:∈
1079:∀
1032:∈
1026:∀
980:⋆
973:∈
967:∃
956:⋆
949:∈
943:∀
899:∈
893:∃
884:∈
878:∀
701:⌋
695:⌊
689:∗
642:⌋
638:⋅
634:⌊
628:∗
601:⌋
595:⌊
589:∗
582:≥
535:⌋
531:⋅
527:⌊
504:⌋
498:⌊
495:≥
448:Keisler's
254:'s sense.
169:Jerzy ĆoĆ
167:In 1955,
6162:Category
6062:Validity
5963:Antinomy
5891:Theories
5855:Informal
5697:Logicism
5690:timeline
5666:Concrete
5525:Validity
5495:T-schema
5488:Kripke's
5483:Tarski's
5478:semantic
5468:Strength
5417:submodel
5412:spectrum
5380:function
5228:Tarski's
5217:Elements
5204:geometry
5160:Robinson
5081:variable
5066:function
5039:spectrum
5029:Sentence
4985:variable
4928:Language
4881:Relation
4842:Automata
4832:Alphabet
4816:language
4670:-jection
4648:codomain
4634:Function
4595:Universe
4565:Infinite
4469:Relation
4252:Validity
4242:Argument
4140:theorem,
3644:(1996),
3605:15104702
3567:(2004),
3540:Elsevier
3514:15104702
3411:See also
3392:internal
3380:*-finite
2839:internal
2820:internal
2790:internal
2482:The set
2304:must be
2082:replaces
1817:external
1687:internal
1675:standard
1370:, i.e.,
1297:topology
1283:form an
244:put it,
242:Robinson
140:". Here
6177:changes
6169: (
6027:Premise
5958:Paradox
5788:History
5783:Outline
5639:Related
5436:Diagram
5334: (
5313:Hilbert
5298:Systems
5293:Theorem
5171:of the
5116:systems
4896:Formula
4891:Grammar
4807: (
4751:General
4464:Forcing
4449:Element
4369:Monadic
4144:paradox
4085:Theorem
4021:General
3712:History
3668:0205854
3361:,
2208:The set
1840:,
1685:called
847:wrote:
845:Keisler
547:is the
460:Example
280:bounded
183:of the
134:Leibniz
128:History
58:improve
6079:topics
5865:Reason
5843:Logics
5834:Syntax
5402:finite
5165:Skolem
5118:
5093:Theory
5061:Symbol
5051:String
5034:atomic
4911:ground
4906:closed
4901:atomic
4857:ground
4820:syntax
4716:binary
4643:domain
4560:Finite
4325:finite
4183:Logics
4142:
4090:Theory
3666:
3656:
3603:
3546:
3512:
3351:ƒ
3123:
3095:
3062:
2979:
2962:
2945:
2763:
2625:
2410:
1986:
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