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1395:
373:. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics.
523:", in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example is
504:
46:
1145:
567:. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.
358:), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed.
571:
2328:
2434:
1381:
1214:
Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express. What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be
594:
Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing
299:
In this context, the validity of a theorem depends only on the correctness of its proof. It is independent from the truth, or even the significance of the axioms. This does not mean that the significance of the axioms is uninteresting, but only that the validity of a theorem is independent from the
1060:
is regarded by some to be the longest proof of a theorem. It comprises tens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to give a complete proof, and several ongoing projects hope to shorten and simplify this proof. Another theorem of this
1180:
that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from the context. The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system.
1017:
describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs
546:
has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities.
487:
In order for a theorem to be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in a completely symbolic formâwith the presumption that a formal statement can be derived from the informal one.
664:). There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable.
361:
In addition to the better readability, informal arguments are typically easier to check than purely symbolic onesâindeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way
230:. All theorems were proved by using implicitly or explicitly these basic properties, and, because of the evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof. For example, the sum of the
542:. Both of these theorems are only known to be true by reducing them to a computational search that is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted. The mathematician
672:
A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time.
693:, which gives the meaning of a word or a phrase in terms of known concepts. Classical geometry discerns between axioms, which are general statements; and postulates, which are statements about geometrical objects. Historically, axioms were regarded as "
491:
It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses form the foundational basis of the theory and are called
595:
the proof. It is also possible to find a single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs.
1452:
In general, the distinction is weak, as the standard way to prove that a statement is provable consists of proving it. However, in mathematical logic, one considers often the set of all theorems of a theory, although one cannot prove them
1470:, whose existence requires the addition of a new axiom to set theory. This reliance on a new axiom of set theory has since been removed. Nevertheless, it is rather astonishing that the first proof of a statement expressed in elementary
1496:
Often, when the less general or "corollary"-like theorem is proven first, it is because the proof of the more general form requires the simpler, corollary-like form, for use as a what is functionally a lemma, or "helper"
1021:
Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem.
1089:
with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of
750:
is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in
343:
of the hypotheses. Namely, that the conclusion is true in case the hypotheses are trueâwithout any further assumptions. However, the conditional could also be interpreted differently in certain
805:
is a proposition that follows immediately from another theorem or axiom, with little or no required proof. A corollary may also be a restatement of a theorem in a simpler form, or for a
645:
of 1.59 Ă 10, which is approximately 10 to the power 4.3 Ă 10. Since the number of particles in the universe is generally considered less than 10 to the power 100 (a
1918:
1172:. A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.
112:. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as
2622:
311:. In particular, there are well-formed assertions than can be proved to not be a theorem of the ambient theory, although they can be proved in a wider theory. An example is
159:(sometimes included in the axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. This formalization led to
780:
is an "accessory proposition" - a proposition with little applicability outside its use in a particular proof. Over time a lemma may gain in importance and be considered a
1065:
whose computer generated proof is too long for a human to read. It is among the longest known proofs of a theorem whose statement can be easily understood by a layman.
3297:
1115:
1139:
292:
cannot be expressed with a well-formed formula. More precisely, if the set of all sets can be expressed with a well-formed formula, this implies that the theory is
171:
theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory (that is they cannot be proved inside the theory).
3380:
2521:
1463:
641:) is known: all numbers less than 10 have the Mertens property, and the smallest number that does not have this property is only known to be less than the
618:. Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved.
300:
significance of the axioms. This independence may be useful by allowing the use of results of some area of mathematics in apparently unrelated areas.
515:
states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
2387:
245:
that do not lead to any contradiction, although, in such geometries, the sum of the angles of a triangle is different from 180°. So, the property
3694:
1552:
1203:, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in
1057:
249:
is either true or false, depending whether Euclid's fifth postulate is assumed or denied. Similarly, the use of "evident" basic properties of
3852:
793:
100:
In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of
2640:
1246:
308:
164:
3707:
3030:
1545:"The Pythagorean proposition: its demonstrations analyzed and classified, and bibliography of sources for data of the four kinds of proofs"
210:, all mathematical theories were built from a few basic properties that were considered as self-evident; for example, the facts that every
511:
map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The
1418:
1335:. A validity is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are
1207:, which is concerned with the relationship between formal theories and structures that are able to provide a semantics for them through
1184:
In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be
710:
is an unproved statement that is believed to be true. Conjectures are usually made in public, and named after their maker (for example,
3292:
3712:
3702:
3439:
2645:
1256:
207:
3190:
2636:
1241:
656:
The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example,
538:
Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the
4287:
3848:
2292:
2262:
2227:
2196:
303:
An important consequence of this way of thinking about mathematics is that it allows defining mathematical theories and theorems as
578:: one way to illustrate its complexity is to extend the iteration from the natural numbers to the complex numbers. The result is a
909:
3945:
3689:
2514:
1921:
583:
3250:
2943:
2684:
1271:
726:), which should not be confused with "hypothesis" as the premise of a proof. Other terms are also used on occasion, for example
320:
101:
1557:
4206:
3908:
3671:
3666:
3491:
2912:
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2186:
218:
that passes through two given distinct points. These basic properties that were considered as absolutely evident were called
1251:
836:
of a theorem is a theorem with a similar statement but a broader scope, from which the original theorem can be deduced as a
1483:
A theory is often identified with the set of its theorems. This is avoided here for clarity, and also for not depending on
1315:. Different deductive systems can yield other interpretations, depending on the presumptions of the derivation rules (i.e.
625:
is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number
4201:
3984:
3901:
3614:
3545:
3422:
2664:
1606:
785:
687:
is a fundamental assumption regarding the object of study, that is accepted without proof. A related concept is that of a
3272:
4282:
4277:
4272:
4126:
3952:
3638:
2871:
1635:
McLarty, Colin (2010). "What does it take to prove Fermat's last theorem? Grothendieck and the logic of number theory".
1561:
606:
are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The
3277:
366:
it is obviously true. In some cases, one might even be able to substantiate a theorem by using a picture as its proof.
319:, but is proved to be not provable in Peano arithmetic. However, it is provable in some more general theories, such as
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35:
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Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as
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339:. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a
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3479:
3215:
3121:
2980:
2965:
2846:
2821:
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615:
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771:), all theorems and geometric constructions were called "propositions" regardless of their importance.
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1049:(and RĂ©nyi may have been thinking of ErdĆs), who was famous for the many theorems he produced, the
776:
751:
661:
508:
348:
304:
269:
265:
140:
90:
50:
1670:
McLarty, Colin (2020). "The large structures of
Grothendieck founded on finite order arithmetic".
1572:
Originally published in 1940 and reprinted in 1968 by
National Council of Teachers of Mathematics.
1266:
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The exact style depends on the author or publication. Many publications provide instructions or
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2149:
347:, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g.,
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is a theorem stating an equality between two expressions, that holds for any value within its
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70:
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260:
This crisis has been resolved by revisiting the foundations of mathematics to make them more
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955:(name of the person who proved it, along with year of discovery or publication of the proof)
882:
543:
344:
316:
109:
86:
82:
1816:
1307:
The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a
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2011:
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interpreted as a proof of their truth. A theorem whose interpretation is a true statement
1157:
1153:
1082:
890:
503:
277:
257:. This has been resolved by elaborating the rules that are allowed for manipulating sets.
178:, theorems may be considered as expressing some truth, but in contrast to the notion of a
144:
105:
1050:
789:
354:
Although theorems can be written in a completely symbolic form (e.g., as propositions in
139:
in order to allow mathematical reasoning about them. In this context, statements become
4162:
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It has been estimated that over a quarter of a million theorems are proved every year.
905:
878:
832:
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469:
231:
211:
179:
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For a theory to be closed under a derivability relation, it must be associated with a
45:
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was historically called a theorem, although, for centuries, it was only a conjecture.
694:
555:
Theorems in mathematics and theories in science are fundamentally different in their
532:
528:
288:. Similarly, Russell's paradox disappears because, in an axiomatized set theory, the
280:
of the theory. So, the above theorem on the sum of the angles of a triangle becomes:
136:
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1656:
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3883:
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1200:
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has been verified to hold for the first 10 trillion non-trivial zeroes of the
556:
497:
293:
184:
160:
62:
1944:
741:
is a statement that has been proven to be true based on axioms and other theorems.
163:, which allows proving general theorems about theorems and proofs. In particular,
1844:
1038:
4146:
4026:
3205:
3195:
3142:
2826:
2746:
2731:
2611:
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2412:
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However, both theorems and scientific law are the result of investigations. See
1433:
1324:
1225:
1193:
1007:
996:
824:
814:
746:
560:
168:
58:
1614:
1144:
928:
A few well-known theorems have even more idiosyncratic names, for example, the
848:
Other terms may also be used for historical or customary reasons, for example:
198:
is a tentative proposition that may evolve to become a theorem if proven true.
3076:
2931:
2902:
2708:
2145:
1747:
1693:
1484:
1471:
1376:
1199:
The definition of theorems as sentences of a formal language is useful within
1014:
730:
when people are not sure whether the statement should be believed to be true.
706:
689:
564:
460:
417:
369:
Because theorems lie at the core of mathematics, they are also central to its
194:
2343:
1648:
563:, that is, it makes predictions about the natural world that are testable by
4228:
4131:
3184:
3101:
3061:
3025:
2961:
2773:
2763:
2736:
2348:
2282:
1849:
1761:
1428:
1328:
1312:
1185:
896:
810:
801:
570:
385:
219:
189:
1544:
2327:
2220:
Metalogic: An
Introduction to the Metatheory of Standard First Order Logic
1331:
of a formal system depends on whether or not all of its theorems are also
296:, and every well-formed assertion, as well as its negation, is a theorem.
241:
One aspect of the foundational crisis of mathematics was the discovery of
4213:
4011:
3459:
3164:
2758:
1165:
1034:
476:/2 is a natural number" is a typical example in which the hypothesis is "
235:
754:. In classical geometry the term "proposition" was used differently: in
610:
has been verified for start values up to about 2.88 Ă 10. The
559:. A scientific theory cannot be proved; its key attribute is that it is
416:
of the theorem ("hypothesis" here means something very different from a
17:
1423:
579:
336:
2499:
1969:
1363:
1316:
984:
819:
755:
646:
2433:
784:, though the term "lemma" is usually kept as part of its name (e.g.
428:
of the theorem. The two together (without the proof) are called the
2365:
1726:(Fall 2017 ed.), Metaphysics Research Lab, Stanford University
527:, and there are many other examples of simple yet deep theorems in
3353:
2699:
2544:
2357:
1684:
1143:
679:
569:
502:
493:
273:
223:
94:
44:
995:
marks, such as "âĄ" or "â", meaning "end of proof", introduced by
999:
following their use in magazines to mark the end of an article.
2503:
2369:
2189:: The Story of Paul ErdĆs and the Search for Mathematical Truth
2012:
An enormous theorem: the classification of finite simple groups
1039:"A mathematician is a device for turning coffee into theorems"
500:
studies formal languages, axioms and the structure of proofs.
238:
equals 180°, and this was considered as an undoubtable fact.
1797:
McGill
University â Department of Mathematics and Statistics
1788:
Darmon, Henri; Diamond, Fred; Taylor, Richard (2007-09-09).
1750:, p. clxxxii:"theorem (ΞΔᜌÏΜΌα) from ΞΔÏÏÎ”áŒłÎœ to investigate"
2281:
Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996).
1188:
relative to a given semantics, or relative to the standard
948:
A theorem and its proof are typically laid out as follows:
649:), there is no hope to find an explicit counterexample by
621:
Such evidence does not constitute proof. For example, the
286:, the sum of the interior angles of a triangle equals 180°
174:
As the axioms are often abstractions of properties of the
1247:
Gödel's incompleteness theorems of first-order arithmetic
2118:
Boolos, George; Burgess, John; Jeffrey, Richard (2007).
377:
is a particularly well-known example of such a theorem.
188:, the justification of the truth of a theorem is purely
2014:, Richard Elwes, Plus Magazine, Issue 41 December 2006.
1093:. Some accounts define a theory to be closed under the
1045:, although it is often attributed to RĂ©nyi's colleague
1888:
1127:
1103:
877:
is a theorem that establishes a useful formula (e.g.
809:: for example, the theorem "all internal angles in a
983:
The end of the proof may be signaled by the letters
4155:
4050:
3882:
3775:
3627:
3320:
3243:
3137:
3041:
2930:
2857:
2792:
2707:
2698:
2620:
2537:
2467:
2441:
2405:
1474:
involves the existence of very large infinite sets.
1232:Some important theorems in mathematical logic are:
480:
is an even natural number", and the conclusion is "
116:only the most important results, and use the terms
2250:
1970:"Earliest Uses of Symbols of Set Theory and Logic"
1912:
1133:
1109:
1117:), while others define it to be closed under the
827:" - a square being a special case of a rectangle.
817:" has a corollary that "all internal angles in a
496:or postulates. The field of mathematics known as
307:, and to prove theorems about them. Examples are
247:"the sum of the angles of a triangle equals 180°"
1053:of his collaborations, and his coffee drinking.
214:has a successor, and that there is exactly one
135:, the concepts of theorems and proofs have been
27:In mathematics, a statement that has been proven
1343:when all of its theorems are also tautologies.
900:is a theorem with wide applicability (e.g. the
2515:
2381:
1192:of the underlying language. A theory that is
1013:It is common for a theorem to be preceded by
108:(ZFC), or of a less powerful theory, such as
8:
2304:A Concise Introduction to Mathematical Logic
2122:(5th ed.). Cambridge University Press.
1607:"Theorem | Definition of Theorem by Lexico"
959:Statement of theorem (sometimes called the
264:. In these new foundations, a theorem is a
3341:
2936:
2704:
2522:
2508:
2500:
2388:
2374:
2366:
1882:Such as the derivation of the formula for
1678:(2). Cambridge University Press: 296â325.
1643:(3). Cambridge University Press: 359â377.
1018:presented after the proof of the theorem.
722:is also used in this sense (for example,
206:Until the end of the 19th century and the
2287:. A.K. Peters, Wellesley, Massachusetts.
1887:
1683:
1126:
1102:
151:consists of some basis statements called
2140:(2nd ed.). Harcourt Academic Press.
282:Under the axioms and inference rules of
2127:Chiswell, Ian; Hodges, Wilfred (2007).
1724:The Stanford Encyclopedia of Philosophy
1535:
1445:
1262:Church-Turing theorem of undecidability
637:) equals or exceeds the square root of
1949:. Ginn & Co. Articles 46, 47.
1553:Education Resources Information Center
1464:Wiles's proof of Fermat's Last Theorem
1058:classification of finite simple groups
388:, many theorems are of the form of an
1783:
1781:
1743:
1252:Consistency of first-order arithmetic
7:
2138:A Mathematical Introduction to Logic
1922:addition formulas of sine and cosine
1913:{\displaystyle \tan(\alpha +\beta )}
1713:
1711:
1943:Wentworth, G.; Smith, D.E. (1913).
1419:List of theorems called fundamental
89:to establish that the theorem is a
85:that uses the inference rules of a
2222:. University of California Press.
2177:Fundamentals of Mathematical Logic
1347:Interpretation of a formal theorem
208:foundational crisis of mathematics
25:
1959:Wentworth & Smith, article 51
1746:Introduction, The terminology of
1339:). A formal system is considered
1242:Completeness of first-order logic
551:Relation with scientific theories
396:. Such a theorem does not assert
4241:
2432:
2326:
2315:(3rd ed.). Springer-Verlag.
1393:
1379:
1237:Compactness of first-order logic
464:, respectively. The theorem "If
97:and previously proved theorems.
1558:Institute of Education Sciences
1257:Tarski's undefinability theorem
1219:a formal system (as opposed to
1196:has all sentences as theorems.
1081:is a set of sentences within a
629:for which the Mertens function
309:Gödel's incompleteness theorems
165:Gödel's incompleteness theorems
2187:The Man Who Loved Only Numbers
1907:
1895:
1510:can also refer to an axiom, a
484:/2 is also a natural number".
404:is a necessary consequence of
327:Epistemological considerations
253:leads to the contradiction of
53:has at least 370 known proofs.
1:
4202:History of mathematical logic
2302:Rautenberg, Wolfgang (2010).
2241:. Cambridge University Press.
2239:Notes on Logic and Set Theory
2210:. Cambridge University Press.
1722:, in Zalta, Edward N. (ed.),
1611:Lexico Dictionaries | English
1466:, which relies implicitly on
1462:An exception is the original
1223:a formal system) is called a
1152:that can be constructed from
765:
128:for less important theorems.
4127:Primitive recursive function
2179:. Wellesley, MA: A K Peters.
2032:Chiswell and Hodges, p. 172.
1720:"Rationalism vs. Empiricism"
1637:The Review of Symbolic Logic
1562:U.S. Department of Education
1164:may be broadly divided into
1121:, or derivability relation (
582:, which (in accordance with
381:Informal account of theorems
272:that can be proved from the
2257:. Oxford University Press.
2023:Boolos, et al 2007, p. 191.
1368:Theory (mathematical logic)
918:least-upper-bound principle
321:ZermeloâFraenkel set theory
102:ZermeloâFraenkel set theory
4314:
3191:SchröderâBernstein theorem
2918:Monadic predicate calculus
2577:Foundations of mathematics
2170:. Oxford University Press.
2136:Enderton, Herbert (2001).
2131:. Oxford University Press.
1672:Bulletin of Symbolic Logic
1361:
1350:
1296:
29:
4237:
4224:Philosophy of mathematics
4173:Automated theorem proving
3344:
3298:Von NeumannâBernaysâGödel
2939:
2430:
2306:(3rd ed.). Springer.
2237:Johnstone, P. T. (1987).
1694:10.1017/S1755020319000340
910:Kolmogorov's zeroâone law
697:"; today they are merely
315:, which can be stated in
4288:Mathematical terminology
2311:van Dalen, Dirk (1994).
2272:Monk, J. Donald (1976).
2206:Hodges, Wilfrid (1993).
2146:Heath, Sir Thomas Little
1520:probability distribution
1303:Formal semantics (logic)
1272:LöwenheimâSkolem theorem
1110:{\displaystyle \models }
243:non-Euclidean geometries
77:, or can be proven. The
30:Not to be confused with
3874:Self-verifying theories
3695:Tarski's axiomatization
2646:Tarski's undefinability
2641:incompleteness theorems
2168:A First Course in Logic
2151:The works of Archimedes
2120:Computability and Logic
1821:intrologic.stanford.edu
1790:"Fermat's Last Theorem"
1583:"Definition of THEOREM"
1287:Cut-elimination theorem
1148:This diagram shows the
1134:{\displaystyle \vdash }
1006:for typesetting in the
989:quod erat demonstrandum
452:can be also termed the
4248:Mathematics portal
3859:Proof of impossibility
3507:propositional variable
2817:Propositional calculus
2191:. Hyperion, New York.
2175:Hinman, Peter (2005).
2166:Hedman, Shawn (2004).
1993:Hoffman 1998, p. 204.
1933:Petkovsek et al. 1996.
1914:
1718:Markie, Peter (2017),
1649:10.2178/bsl/1286284558
1468:Grothendieck universes
1353:Interpretation (logic)
1173:
1135:
1111:
868:Vandermonde's identity
598:For example, both the
591:
516:
436:of the theorem (e.g. "
390:indicative conditional
356:propositional calculus
54:
4117:Kolmogorov complexity
4070:Computably enumerable
3970:Model complete theory
3762:Principia Mathematica
2822:Propositional formula
2651:BanachâTarski paradox
1915:
1766:mathworld.wolfram.com
1543:Elisha Scott Loomis.
1358:Theorems and theories
1341:semantically complete
1147:
1136:
1119:syntactic consequence
1112:
1041:, is probably due to
938:BanachâTarski paradox
794:the fundamental lemma
732:Fermat's Last Theorem
712:Goldbach's conjecture
573:
535:, among other areas.
525:Fermat's Last Theorem
506:
375:Fermat's Last Theorem
341:necessary consequence
202:Theoremhood and truth
48:
4065:ChurchâTuring thesis
4052:Computability theory
3261:continuum hypothesis
2779:Square of opposition
2637:Gödel's completeness
2335:at Wikimedia Commons
2184:Hoffman, P. (1998).
1886:
1293:Syntax and semantics
1170:well-formed formulas
1125:
1101:
1095:semantic consequence
973:Description of proof
922:pigeonhole principle
902:law of large numbers
305:mathematical objects
141:well-formed formulas
4283:Mathematical proofs
4278:Logical expressions
4273:Logical consequence
4219:Mathematical object
4110:P versus NP problem
4075:Computable function
3869:Reverse mathematics
3795:Logical consequence
3672:primitive recursive
3667:elementary function
3440:Free/bound variable
3293:TarskiâGrothendieck
2812:Logical connectives
2742:Logical equivalence
2592:Logical consequence
2313:Logic and Structure
2002:Hoffman 1998, p. 7.
1760:Weisstein, Eric W.
1617:on November 2, 2019
1277:Lindström's theorem
1091:logical consequence
1087:well-formed formula
991:) or by one of the
914:Harnack's principle
769: 300 BCE
752:propositional logic
662:mathematical theory
519:Some theorems are "
349:non-classical logic
313:Goodstein's theorem
270:mathematical theory
266:well-formed formula
228:Euclid's postulates
91:logical consequence
51:Pythagorean theorem
4017:Transfer principle
3980:Semantics of logic
3965:Categorical theory
3941:Non-standard model
3455:Logical connective
2582:Information theory
2531:Mathematical logic
2358:Theorem of the Day
2341:Weisstein, Eric W.
2276:. Springer-Verlag.
2274:Mathematical Logic
2129:Mathematical Logic
2104:van Dalen, p. 104.
2095:Rautenberg, p. 81.
1974:jeff560.tripod.com
1910:
1842:Weisstein, Eric W.
1516:probability theory
1401:Mathematics portal
1174:
1162:strings of symbols
1150:syntactic entities
1131:
1107:
1085:. A sentence is a
1075:mathematical logic
1063:four color theorem
930:division algorithm
724:Riemann hypothesis
716:Collatz conjecture
623:Mertens conjecture
612:Riemann hypothesis
608:Collatz conjecture
604:Riemann hypothesis
600:Collatz conjecture
592:
576:Collatz conjecture
517:
513:four color theorem
444:). Alternatively,
284:Euclidean geometry
133:mathematical logic
81:of a theorem is a
55:
4298:Concepts in logic
4255:
4254:
4187:Abstract category
3990:Theories of truth
3800:Rule of inference
3790:Natural deduction
3771:
3770:
3316:
3315:
3021:Cartesian product
2926:
2925:
2832:Many-valued logic
2807:Boolean functions
2690:Russell's paradox
2665:diagonal argument
2562:First-order logic
2497:
2496:
2331:Media related to
2077:Johnstone, p. 21.
1512:rule of inference
1409:Law (mathematics)
1387:Philosophy portal
1311:which introduces
1309:true proposition,
1069:Theorems in logic
864:BĂ©zout's identity
651:exhaustive search
540:Kepler conjecture
345:deductive systems
255:Russell's paradox
16:(Redirected from
4305:
4246:
4245:
4197:History of logic
4192:Category of sets
4085:Decision problem
3864:Ordinal analysis
3805:Sequent calculus
3703:Boolean algebras
3643:
3642:
3617:
3588:logical/constant
3342:
3328:
3251:ZermeloâFraenkel
3002:Set operations:
2937:
2874:
2705:
2685:LöwenheimâSkolem
2572:Formal semantics
2524:
2517:
2510:
2501:
2436:
2390:
2383:
2376:
2367:
2354:
2353:
2330:
2316:
2307:
2298:
2277:
2268:
2256:
2253:Elementary Logic
2242:
2233:
2216:Hunter, Geoffrey
2211:
2202:
2180:
2171:
2162:
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2159:
2141:
2132:
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2102:
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2087:
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2060:
2057:
2051:
2048:
2042:
2041:Enderton, p. 148
2039:
2033:
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2015:
2009:
2003:
2000:
1994:
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1866:Doron Zeilberger
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1837:
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1623:
1622:
1613:. Archived from
1603:
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1540:
1523:
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1454:
1450:
1414:List of theorems
1403:
1398:
1397:
1389:
1384:
1383:
1382:
1178:deductive system
1154:formal languages
1140:
1138:
1137:
1132:
1116:
1114:
1113:
1108:
770:
767:
586:) resembles the
544:Doron Zeilberger
408:. In this case,
317:Peano arithmetic
167:show that every
110:Peano arithmetic
87:deductive system
83:logical argument
21:
4313:
4312:
4308:
4307:
4306:
4304:
4303:
4302:
4258:
4257:
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4251:
4240:
4233:
4178:Category theory
4168:Algebraic logic
4151:
4122:Lambda calculus
4060:Church encoding
4046:
4022:Truth predicate
3878:
3844:Complete theory
3767:
3636:
3632:
3628:
3623:
3615:
3335: and
3331:
3326:
3312:
3288:New Foundations
3256:axiom of choice
3239:
3201:Gödel numbering
3141: and
3133:
3037:
2922:
2872:
2853:
2802:Boolean algebra
2788:
2752:Equiconsistency
2717:Classical logic
2694:
2675:Halting problem
2663: and
2639: and
2627: and
2626:
2621:Theorems (
2616:
2533:
2528:
2498:
2493:
2463:
2437:
2428:
2401:
2394:
2363:
2339:
2338:
2323:
2310:
2301:
2295:
2280:
2271:
2265:
2245:
2236:
2230:
2214:
2205:
2199:
2183:
2174:
2165:
2157:
2155:
2144:
2135:
2126:
2117:
2114:
2109:
2108:
2103:
2099:
2094:
2090:
2085:
2081:
2076:
2072:
2067:
2063:
2059:Hinman, p. 139.
2058:
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2045:
2040:
2036:
2031:
2027:
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2018:
2010:
2006:
2001:
1997:
1992:
1988:
1978:
1976:
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1963:
1958:
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1932:
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1664:
1634:
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1618:
1605:
1604:
1600:
1591:
1589:
1587:Merriam-Webster
1581:
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1576:
1566:
1564:
1547:
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1537:
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1527:
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1399:
1392:
1385:
1380:
1378:
1375:
1370:
1362:Main articles:
1360:
1355:
1349:
1305:
1297:Main articles:
1295:
1282:Craig's theorem
1123:
1122:
1099:
1098:
1083:formal language
1071:
1033:The well-known
1028:
946:
934:Euler's formula
768:
670:
553:
383:
329:
290:set of all sets
278:inference rules
232:interior angles
204:
145:formal language
106:axiom of choice
43:
28:
23:
22:
15:
12:
11:
5:
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4170:
4165:
4163:Abstract logic
4159:
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4149:
4144:
4142:Turing machine
4139:
4134:
4129:
4124:
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4114:
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4107:
4102:
4097:
4092:
4082:
4080:Computable set
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4054:
4048:
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4045:
4044:
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4034:
4029:
4024:
4019:
4014:
4009:
4008:
4007:
4002:
3997:
3987:
3982:
3977:
3975:Satisfiability
3972:
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3962:
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3960:
3950:
3949:
3948:
3938:
3937:
3936:
3931:
3926:
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3904:
3899:
3892:Interpretation
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3717:
3716:
3715:
3713:minimal axioms
3710:
3699:
3698:
3697:
3686:
3685:
3684:
3679:
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3669:
3664:
3659:
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3504:
3499:
3489:
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3452:
3447:
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3405:
3403:Formation rule
3400:
3395:
3394:
3393:
3388:
3378:
3377:
3376:
3366:
3361:
3356:
3351:
3345:
3339:
3322:Formal systems
3318:
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3310:
3305:
3300:
3295:
3290:
3285:
3280:
3275:
3270:
3265:
3264:
3263:
3258:
3247:
3245:
3241:
3240:
3238:
3237:
3236:
3235:
3225:
3220:
3219:
3218:
3211:Large cardinal
3208:
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3198:
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3188:
3174:
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3162:
3147:
3145:
3135:
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3131:
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3084:
3079:
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3036:
3035:
3034:
3033:
3028:
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3018:
3013:
3008:
3000:
2999:
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2993:
2983:
2978:
2976:Extensionality
2973:
2971:Ordinal number
2968:
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2953:
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2951:
2940:
2934:
2928:
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2655:Cantor's
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2633:
2631:
2618:
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2468:Negation
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2427:
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2418:truth function
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2395:
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2321:External links
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2110:
2107:
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2088:
2079:
2070:
2068:Hodges, p. 33.
2061:
2052:
2050:Hedman, p. 89.
2043:
2034:
2025:
2016:
2004:
1995:
1986:
1961:
1952:
1946:Plane Geometry
1935:
1926:
1909:
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1845:"Deep Theorem"
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1374:
1371:
1359:
1356:
1351:Main article:
1348:
1345:
1299:Syntax (logic)
1294:
1291:
1290:
1289:
1284:
1279:
1274:
1269:
1264:
1259:
1254:
1249:
1244:
1239:
1209:interpretation
1190:interpretation
1130:
1106:
1070:
1067:
1027:
1024:
981:
980:
975:
970:
965:
956:
945:
942:
926:
925:
906:law of cosines
886:
871:
846:
845:
833:generalization
828:
797:
772:
742:
735:
702:
669:
666:
588:Mandelbrot set
552:
549:
470:natural number
412:is called the
382:
379:
328:
325:
226:; for example
212:natural number
203:
200:
180:scientific law
176:physical world
157:deducing rules
73:that has been
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4310:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4265:
4263:
4250:
4249:
4244:
4236:
4230:
4227:
4225:
4222:
4220:
4217:
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4212:
4208:
4205:
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4200:
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4184:
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4179:
4176:
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4171:
4169:
4166:
4164:
4161:
4160:
4158:
4154:
4148:
4145:
4143:
4140:
4138:
4137:Recursive set
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4091:
4088:
4087:
4086:
4083:
4081:
4078:
4076:
4073:
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4068:
4066:
4063:
4061:
4058:
4057:
4055:
4053:
4049:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4006:
4003:
4001:
3998:
3996:
3993:
3992:
3991:
3988:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3959:
3956:
3955:
3954:
3951:
3947:
3946:of arithmetic
3944:
3943:
3942:
3939:
3935:
3932:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
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3910:
3907:
3903:
3900:
3898:
3895:
3894:
3893:
3890:
3889:
3887:
3885:
3881:
3875:
3872:
3870:
3867:
3865:
3862:
3860:
3857:
3854:
3853:from ZFC
3850:
3847:
3845:
3842:
3836:
3833:
3832:
3831:
3828:
3826:
3823:
3821:
3818:
3817:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3786:
3783:
3782:
3780:
3778:
3774:
3764:
3763:
3759:
3758:
3753:
3752:non-Euclidean
3750:
3746:
3743:
3741:
3738:
3736:
3735:
3731:
3730:
3728:
3725:
3724:
3722:
3718:
3714:
3711:
3709:
3706:
3705:
3704:
3700:
3696:
3693:
3692:
3691:
3687:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3654:
3652:
3648:
3647:
3645:
3640:
3634:
3629:Example
3626:
3618:
3613:
3612:
3611:
3608:
3606:
3603:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
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3574:
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3557:
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3552:
3549:
3548:
3547:
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3540:
3537:
3535:
3532:
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3527:
3525:
3522:
3521:
3520:
3517:
3515:
3512:
3508:
3505:
3503:
3500:
3498:
3495:
3494:
3493:
3490:
3486:
3483:
3481:
3478:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3457:
3456:
3453:
3451:
3448:
3446:
3443:
3441:
3438:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3415:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3392:
3389:
3387:
3386:by definition
3384:
3383:
3382:
3379:
3375:
3372:
3371:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3346:
3343:
3340:
3338:
3334:
3329:
3323:
3319:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3274:
3273:KripkeâPlatek
3271:
3269:
3266:
3262:
3259:
3257:
3254:
3253:
3252:
3249:
3248:
3246:
3242:
3234:
3231:
3230:
3229:
3226:
3224:
3221:
3217:
3214:
3213:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3186:
3182:
3178:
3175:
3171:
3168:
3166:
3163:
3161:
3158:
3157:
3156:
3152:
3149:
3148:
3146:
3144:
3140:
3136:
3128:
3125:
3123:
3120:
3118:
3117:constructible
3115:
3114:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3049:
3047:
3045:
3040:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3003:
3001:
2997:
2994:
2992:
2989:
2988:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2963:
2959:
2957:
2954:
2950:
2947:
2946:
2945:
2942:
2941:
2938:
2935:
2933:
2929:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2887:
2884:
2883:
2882:
2879:
2875:
2870:
2869:
2868:
2865:
2864:
2862:
2860:
2856:
2848:
2845:
2843:
2840:
2838:
2835:
2834:
2833:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2813:
2810:
2808:
2805:
2803:
2800:
2799:
2797:
2795:
2794:Propositional
2791:
2785:
2782:
2780:
2777:
2775:
2772:
2770:
2767:
2765:
2762:
2760:
2757:
2753:
2750:
2749:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2722:Logical truth
2720:
2718:
2715:
2714:
2712:
2710:
2706:
2703:
2701:
2697:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2662:
2658:
2654:
2652:
2649:
2647:
2644:
2642:
2638:
2635:
2634:
2632:
2630:
2624:
2619:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2551:
2548:
2547:
2546:
2543:
2542:
2540:
2536:
2532:
2525:
2520:
2518:
2513:
2511:
2506:
2505:
2502:
2490:
2489:inconsistency
2487:
2485:
2484:contradiction
2482:
2480:
2476:
2473:
2472:
2470:
2466:
2460:
2457:
2455:
2452:
2450:
2447:
2446:
2444:
2440:
2435:
2425:
2422:⊨
2421:
2419:
2416:
2414:
2411:
2410:
2408:
2404:
2399:
2398:Logical truth
2391:
2386:
2384:
2379:
2377:
2372:
2371:
2368:
2364:
2359:
2356:
2351:
2350:
2345:
2342:
2337:
2334:
2329:
2325:
2324:
2320:
2314:
2309:
2305:
2300:
2296:
2294:1-56881-063-6
2290:
2286:
2285:
2279:
2275:
2270:
2266:
2264:0-19-501491-X
2260:
2255:
2254:
2248:
2247:Mates, Benson
2244:
2240:
2235:
2231:
2229:0-520-02356-0
2225:
2221:
2217:
2213:
2209:
2204:
2200:
2198:1-85702-829-5
2194:
2190:
2188:
2182:
2178:
2173:
2169:
2164:
2153:
2152:
2147:
2143:
2139:
2134:
2130:
2125:
2121:
2116:
2115:
2111:
2101:
2098:
2092:
2089:
2086:Monk, p. 208.
2083:
2080:
2074:
2071:
2065:
2062:
2056:
2053:
2047:
2044:
2038:
2035:
2029:
2026:
2020:
2017:
2013:
2008:
2005:
1999:
1996:
1990:
1987:
1975:
1971:
1965:
1962:
1956:
1953:
1948:
1947:
1939:
1936:
1930:
1927:
1923:
1904:
1901:
1898:
1892:
1889:
1879:
1876:
1871:
1867:
1861:
1858:
1852:
1851:
1846:
1843:
1836:
1833:
1822:
1818:
1817:"Implication"
1812:
1809:
1798:
1791:
1784:
1782:
1778:
1767:
1763:
1756:
1753:
1749:
1745:
1739:
1736:
1725:
1721:
1714:
1712:
1708:
1703:
1699:
1695:
1691:
1686:
1681:
1677:
1673:
1666:
1663:
1658:
1654:
1650:
1646:
1642:
1638:
1631:
1628:
1616:
1612:
1608:
1602:
1599:
1588:
1584:
1578:
1575:
1563:
1560:(IES) of the
1559:
1555:
1554:
1546:
1539:
1536:
1529:
1521:
1517:
1513:
1509:
1503:
1500:
1493:
1490:
1486:
1480:
1477:
1473:
1469:
1465:
1459:
1456:
1453:individually.
1449:
1446:
1439:
1435:
1432:
1430:
1427:
1425:
1422:
1420:
1417:
1415:
1412:
1410:
1407:
1406:
1402:
1396:
1391:
1388:
1377:
1372:
1369:
1365:
1357:
1354:
1346:
1344:
1342:
1338:
1334:
1330:
1326:
1322:
1321:justification
1318:
1314:
1310:
1304:
1300:
1292:
1288:
1285:
1283:
1280:
1278:
1275:
1273:
1270:
1268:
1267:Löb's theorem
1265:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1238:
1235:
1234:
1233:
1230:
1228:
1227:
1222:
1218:
1212:
1210:
1206:
1202:
1197:
1195:
1191:
1187:
1182:
1179:
1171:
1167:
1163:
1159:
1155:
1151:
1146:
1142:
1128:
1120:
1104:
1096:
1092:
1088:
1084:
1080:
1079:formal theory
1076:
1068:
1066:
1064:
1059:
1054:
1052:
1048:
1044:
1040:
1036:
1031:
1025:
1023:
1019:
1016:
1011:
1009:
1005:
1000:
998:
994:
990:
986:
979:
976:
974:
971:
969:
966:
964:
960:
957:
954:
951:
950:
949:
943:
941:
939:
935:
931:
923:
919:
915:
911:
907:
903:
899:
898:
893:
892:
887:
884:
883:Cramer's rule
880:
876:
872:
869:
865:
861:
857:
856:
851:
850:
849:
843:
839:
835:
834:
829:
826:
822:
821:
816:
812:
808:
804:
803:
798:
795:
791:
787:
786:Gauss's lemma
783:
779:
778:
773:
763:
762:
757:
753:
749:
748:
743:
740:
736:
733:
729:
725:
721:
717:
713:
709:
708:
703:
700:
696:
692:
691:
686:
682:
681:
676:
675:
674:
667:
665:
663:
659:
654:
652:
648:
644:
640:
636:
632:
628:
624:
619:
617:
616:zeta function
613:
609:
605:
601:
596:
589:
585:
581:
577:
572:
568:
566:
562:
558:
550:
548:
545:
541:
536:
534:
533:combinatorics
530:
529:number theory
526:
522:
514:
510:
505:
501:
499:
495:
489:
485:
483:
479:
475:
471:
467:
463:
462:
457:
456:
451:
447:
443:
439:
435:
431:
427:
423:
419:
415:
411:
407:
403:
399:
395:
391:
387:
380:
378:
376:
372:
367:
365:
359:
357:
352:
350:
346:
342:
338:
334:
326:
324:
322:
318:
314:
310:
306:
301:
297:
295:
291:
287:
285:
279:
275:
271:
267:
263:
258:
256:
252:
248:
244:
239:
237:
233:
229:
225:
221:
217:
213:
209:
201:
199:
197:
196:
191:
187:
186:
181:
177:
172:
170:
166:
162:
158:
154:
150:
146:
142:
138:
134:
129:
127:
123:
119:
115:
111:
107:
103:
98:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
52:
47:
41:
37:
33:
19:
4239:
4037:Ultraproduct
3884:Model theory
3849:Independence
3809:
3785:Formal proof
3777:Proof theory
3760:
3733:
3690:real numbers
3662:second-order
3573:Substitution
3450:Metalanguage
3391:conservative
3364:Axiom schema
3308:Constructive
3278:MorseâKelley
3244:Set theories
3223:Aleph number
3216:inaccessible
3122:Grothendieck
3006:intersection
2893:Higher-order
2881:Second-order
2827:Truth tables
2784:Venn diagram
2601:
2567:Formal proof
2474:
2458:
2454:formal proof
2362:
2347:
2312:
2303:
2283:
2273:
2252:
2238:
2219:
2208:Model Theory
2207:
2185:
2176:
2167:
2156:. Retrieved
2150:
2137:
2128:
2119:
2100:
2091:
2082:
2073:
2064:
2055:
2046:
2037:
2028:
2019:
2007:
1998:
1989:
1977:. Retrieved
1973:
1964:
1955:
1945:
1938:
1929:
1878:
1870:"Opinion 51"
1860:
1848:
1835:
1824:. Retrieved
1820:
1811:
1800:. Retrieved
1796:
1769:. Retrieved
1765:
1755:
1738:
1728:, retrieved
1723:
1675:
1671:
1665:
1640:
1636:
1630:
1619:. Retrieved
1615:the original
1610:
1601:
1590:. Retrieved
1586:
1577:
1565:. Retrieved
1551:
1538:
1507:
1502:
1492:
1479:
1458:
1448:
1308:
1306:
1231:
1224:
1220:
1216:
1213:
1205:model theory
1201:proof theory
1198:
1194:inconsistent
1183:
1175:
1072:
1061:type is the
1055:
1043:Alfréd Rényi
1032:
1029:
1020:
1012:
1001:
988:
982:
977:
972:
967:
962:
958:
952:
947:
927:
895:
889:
874:
853:
847:
841:
838:special case
831:
825:right angles
818:
815:right angles
807:special case
800:
790:Zorn's lemma
781:
775:
760:
745:
738:
727:
719:
718:). The term
705:
698:
695:self-evident
688:
684:
678:
671:
658:group theory
655:
638:
634:
630:
626:
620:
597:
593:
584:universality
557:epistemology
554:
537:
518:
498:proof theory
490:
486:
481:
477:
473:
465:
459:
453:
449:
445:
441:
438:If A, then B
437:
433:
429:
425:
421:
413:
409:
405:
401:
400:â only that
397:
394:If A, then B
393:
384:
368:
363:
360:
353:
332:
330:
302:
298:
294:inconsistent
289:
281:
259:
246:
240:
205:
193:
185:experimental
183:
173:
161:proof theory
156:
152:
130:
125:
121:
117:
113:
99:
78:
66:
63:formal logic
56:
4147:Type theory
4095:undecidable
4027:Truth value
3914:equivalence
3593:non-logical
3206:Enumeration
3196:Isomorphism
3143:cardinality
3127:Von Neumann
3092:Ultrafilter
3057:Uncountable
2991:equivalence
2908:Quantifiers
2898:Fixed-point
2867:First-order
2747:Consistency
2732:Proposition
2709:Traditional
2680:Lindström's
2670:Compactness
2612:Type theory
2557:Cardinality
2413:truth value
2406:Functional:
1434:Toy theorem
1337:tautologies
1226:metatheorem
1015:definitions
1008:house style
997:Paul Halmos
961:proposition
879:Bayes' rule
747:proposition
701:to be true.
668:Terminology
643:exponential
565:experiments
561:falsifiable
468:is an even
442:proposition
430:proposition
182:, which is
155:, and some
122:proposition
59:mathematics
4293:Statements
4262:Categories
3958:elementary
3651:arithmetic
3519:Quantifier
3497:functional
3369:Expression
3087:Transitive
3031:identities
3016:complement
2949:hereditary
2932:Set theory
2158:2009-11-15
2112:References
1979:2 November
1826:2019-11-02
1802:2019-11-01
1771:2019-11-02
1748:Archimedes
1744:Heath 1897
1730:2019-11-02
1621:2019-11-02
1592:2019-11-02
1567:2010-09-26
1530:References
1485:set theory
1472:arithmetic
1333:validities
1325:modalities
1097:relation (
1047:Paul ErdĆs
936:, and the
920:, and the
720:hypothesis
707:conjecture
690:definition
461:consequent
455:antecedent
426:conclusion
418:conjecture
414:hypothesis
371:aesthetics
333:hypotheses
220:postulates
195:conjecture
169:consistent
137:formalized
4229:Supertask
4132:Recursion
4090:decidable
3924:saturated
3902:of models
3825:deductive
3820:axiomatic
3740:Hilbert's
3727:Euclidean
3708:canonical
3631:axiomatic
3563:Signature
3492:Predicate
3381:Extension
3303:Ackermann
3228:Operation
3107:Universal
3097:Recursive
3072:Singleton
3067:Inhabited
3052:Countable
3042:Types of
3026:power set
2996:partition
2913:Predicate
2859:Predicate
2774:Syllogism
2764:Soundness
2737:Inference
2727:Tautology
2629:paradoxes
2424:tautology
2349:MathWorld
2344:"Theorem"
2218:(1996) .
1920:from the
1905:β
1899:α
1893:
1850:MathWorld
1762:"Theorem"
1702:118395028
1685:1102.1773
1514:, or, in
1506:The word
1429:Inference
1329:soundness
1323:or other
1313:semantics
1129:⊢
1105:⊨
993:tombstone
897:principle
842:corollary
811:rectangle
802:corollary
685:postulate
440:" is the
434:statement
386:Logically
190:deductive
126:corollary
104:with the
71:statement
4268:Theorems
4214:Logicism
4207:timeline
4183:Concrete
4042:Validity
4012:T-schema
4005:Kripke's
4000:Tarski's
3995:semantic
3985:Strength
3934:submodel
3929:spectrum
3897:function
3745:Tarski's
3734:Elements
3721:geometry
3677:Robinson
3598:variable
3583:function
3556:spectrum
3546:Sentence
3502:variable
3445:Language
3398:Relation
3359:Automata
3349:Alphabet
3333:language
3187:-jection
3165:codomain
3151:Function
3112:Universe
3082:Infinite
2986:Relation
2769:Validity
2759:Argument
2657:theorem,
2333:Theorems
2249:(1972).
2148:(1897).
1657:13475845
1497:theorem.
1373:See also
1166:nonsense
1035:aphorism
855:identity
761:Elements
602:and the
458:and the
337:premises
262:rigorous
236:triangle
143:of some
114:theorems
36:Theorema
18:Theorems
4156:Related
3953:Diagram
3851: (
3830:Hilbert
3815:Systems
3810:Theorem
3688:of the
3633:systems
3413:Formula
3408:Grammar
3324: (
3268:General
2981:Forcing
2966:Element
2886:Monadic
2661:paradox
2602:Theorem
2538:General
2459:theorem
2442:Formal:
2400: â€
2154:. Dover
1424:Formula
1327:). The
1186:unsound
1158:symbols
953:Theorem
782:theorem
739:theorem
728:problem
699:assumed
580:fractal
521:trivial
472:, then
420:), and
93:of the
67:theorem
32:Teorema
3919:finite
3682:Skolem
3635:
3610:Theory
3578:Symbol
3568:String
3551:atomic
3428:ground
3423:closed
3418:atomic
3374:ground
3337:syntax
3233:binary
3160:domain
3077:Finite
2842:finite
2700:Logics
2659:
2607:Theory
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2449:theory
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1317:belief
1221:within
1156:. The
1051:number
1004:macros
985:Q.E.D.
944:Layout
916:, the
862:(e.g.
860:domain
820:square
792:, and
756:Euclid
647:googol
509:planar
494:axioms
274:axioms
224:axioms
153:axioms
149:theory
95:axioms
75:proven
40:Theory
3909:Model
3657:Peano
3514:Proof
3354:Arity
3283:Naive
3170:image
3102:Fuzzy
3062:Empty
3011:union
2956:Class
2597:Model
2587:Lemma
2545:Axiom
2479:false
2284:A = B
1793:(PDF)
1698:S2CID
1680:arXiv
1653:S2CID
1548:(PDF)
1440:Notes
1217:about
968:Proof
777:lemma
680:axiom
660:(see
268:of a
234:of a
118:lemma
79:proof
69:is a
38:, or
4032:Type
3835:list
3639:list
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3605:Term
3539:rank
3433:open
3327:list
3139:Maps
3044:sets
2903:Free
2873:list
2623:list
2550:list
2289:ISBN
2259:ISBN
2224:ISBN
2193:ISBN
1981:2019
1518:, a
1366:and
1301:and
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1160:and
1077:, a
1056:The
1026:Lore
881:and
875:rule
866:and
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714:and
574:The
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894:or
891:law
852:An
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840:(a
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