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computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of a proof by humans either, especially if the proof contains natural language and requires deep mathematical insight to uncover the potential hidden assumptions and fallacies involved.
33:
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3505:...brought home again to Benoit that there was a 'mathematics of the eye', that visualization of a problem was as valid a method as any for finding a solution. Amazingly, he found himself alone with this conjecture. The teaching of mathematics in France was dominated by a handful of dogmatic mathematicians hiding behind the pseudonym 'Bourbaki'...
2471:, can be constructed in a way which appear to prove a supposed mathematical fact but only do so by neglecting tiny errors (for example, supposedly straight lines which actually bend slightly) which are unnoticeable until the entire picture is closely examined, with lengths and angles precisely measured or calculated.
3446:
What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were coins of the realm and
2517:
in elementary geometry classes in the United States. The proof is written as a series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an
1814:
In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates. One assigns a certain probability for each candidate to be chosen, and then proves that there is a non-zero probability that a chosen candidate will have the desired property. This does
348:
intended to convince the audience of the truth of a statement. The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. To gain acceptance, a proof has to meet communal standards of rigor; an argument considered
2316:
Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity. However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first
1878:
with a certain property exists—without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a
2686:
have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at
2321:
is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a
380:
The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated
1826:. While most mathematicians do not think that probabilistic evidence for the properties of a given object counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's
2891:
A statement whose truth is either to be taken as self-evident or to be assumed. Certain areas of mathematics involve choosing a set of axioms and discovering what results can be derived from them, providing proofs for the theorems that are
2228:
1524:
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the
2530:, such as numbers, to demonstrate something about everyday life, or when data used in an argument is numerical. It is sometimes also used to mean a "statistical proof" (below), especially when used to argue from data.
288:
worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of
3468:
Mandelbrot, working at the IBM Research
Laboratory, did some computer simulations for these sets on the reasonable assumption that, if you wanted to prove something, it might be helpful to know the answer ahead of
3187:: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the original formulation of first-order logic by Frege was the first successful step in this direction."
368:
in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones. This definition makes the concept of proof amenable to study. Indeed, the field of
2564:
from which probability statements are derived require empirical evidence from outside mathematics to verify. In physics, in addition to statistical methods, "statistical proof" can refer to the specialized
328:, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example
191:
Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with
1736:
2385:
developments of the late 19th and 20th centuries, proofs were an essential part of mathematics. With the increase in computing power in the 1960s, significant work began to be done investigating
1776:
1670:
1630:
420:
was known for describing proofs which he found to be particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book
1590:
2518:
axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typically headed "Statements" and the right-hand column is typically headed "Reasons".
2125:
445:
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to prove that the sum of two
2742:. Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" during an oral presentation. Unicode explicitly provides the "end of proof" character, U+220E (∎)
2086:
2017:
1980:
1301:
239:, propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek "axios", something worthy). From this basis, the method proves theorems using
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which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in
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was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand.
1494:
Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists.
3912:
188:(to try). The legal term "probity" means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status.
3776:
2160:
2393:. Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of
249:, was read by anyone who was considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the
2283:
2497:, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
562:
the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually
1848:
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a
558:. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case
4587:
1818:
A probabilistic proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work toward the
3780:
2444:
4670:
3811:
3401:"these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E"
2609:, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to
373:
studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certain
3255:
4984:
3519:"Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century"
196:, which originated in practical problems of land measurement. The development of mathematical proof is primarily the product of
1356:. Since the expression on the left is an integer multiple of 2, the right expression is by definition divisible by 2. That is,
577:
A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all
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3386:"Whether constant π (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some
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The
Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference
1480:, this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator.
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many) cases are provable. This avoids having to prove each case individually. A variant of mathematical induction is
374:
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1860:
provides two different expressions for the size of a single set, again showing that the two expressions are equal.
220:(384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known.
49:, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.
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4233:
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The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with
143:
127:
17:
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Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as
1815:
not specify which candidates have the property, but the probability could not be positive without at least one.
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76:; but every proof can, in principle, be constructed using only certain basic or original assumptions known as
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have variously criticized this view and attempted to develop a semantics for what they considered to be the
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mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the
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method of finding it. The following famous example of a nonconstructive proof shows that there exist two
385:, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are
3402:
2485:
An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in
1803:
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of
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3117:
Matvievskaya, Galina (1987), "The Theory of
Quadratic Irrationals in Medieval Oriental Mathematics",
3024:
2796:
2786:
2655:
2410:
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "
2022:
567:
155:
151:
135:
115:
which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of
3969:
1318:
1072:
912:
The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".
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5354:
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4823:
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4381:
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2294:. It is less commonly used to refer to a mathematical proof in the branch of mathematics known as
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4597:
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4361:
4305:
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2583:. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as
2345:
Mathematicians have shown there are many statements that are neither provable nor disprovable in
2311:
2295:
1924:
1827:
1455:
1427:
1245:
1213:
926:
402:
3375:"in number theory and commutative algebra... in particular the statistical proof of the lemma."
3291:
Mathematik für das
Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis
426:, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.
100:
possible cases. A proposition that has not been proved but is believed to be true is known as a
72:
guarantee the conclusion. The argument may use other previously established statements, such as
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Mathematics for the
Bachelor's degree I: Fundamentals and basics of linear algebra and analysis
3172:, Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, pp. 1–78,
2732:
2618:
2349:(ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see
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1195:
545:
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298:
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36:
6019:
5939:
5406:
5259:
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4769:
4505:
4411:
4270:
4255:
4136:
4111:
3637:"What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians"
3058:"The genesis of proof in ancient Greece The pedagogical implications of a Husserlian reading"
2816:
2811:
2514:
2468:
1869:
1499:
1489:
533:
529:
390:
176:(to test). Related modern words are English "probe", "probation", and "probability", Spanish
147:
139:
3184:
2233:
1675:
1416:
are both even, they have 2 as a common factor. This contradicts our previous statement that
5858:
5704:
5379:
5341:
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5022:
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4786:
4764:
4592:
4550:
4449:
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2731:. A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "
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1893:
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1138:
1111:
981:
941:
726:
559:
446:
422:
386:
329:
314:
244:
205:
45:
3349:
Davis, Philip J. (1972), "Fidelity in
Mathematical Discourse: Is One and One Really Two?"
8:
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2801:
2667:
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shows that many axiom systems of mathematical interest will have undecidable statements.
1875:
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1782:
1519:
555:
551:
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365:
250:
93:
85:
58:
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A closed chain inference shows that a collection of statements are pairwise equivalent.
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The expression "statistical proof" may be used technically or colloquially in areas of
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A particular way of organising a proof using two parallel columns is often used as a
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1880:
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is used to show that the expressions for their two sizes are equal. Alternatively, a
1853:
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1235:
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occurs, hence the statement must be false. A famous example involves the proof that
929:
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209:
197:
150:, oral traditions in the mainstream mathematical community or in other cultures. The
104:, or a hypothesis if frequently used as an assumption for further mathematical work.
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1206:(by reduction to the absurd), it is shown that if some statement is assumed true, a
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2223:{\displaystyle \left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{2}=2}
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1918:
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1807:. Probabilistic proof, like proof by construction, is one of many ways to prove
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2414:". The left-hand picture below is an example of a historic visual proof of the
1831:
1507:
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921:
578:
325:
232:
119:
3539:
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2587:, when the data or diagram is adequately convincing without further analysis.
1822:
shows how far plausibility is from genuine proof, as does the disproof of the
717:
For example, we can prove by induction that all positive integers of the form
208:(c. 470–410 BCE) gave some of the first known proofs of theorems in geometry.
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and, by definition, is even. Hence, the sum of any two even integers is even.
394:
258:
3691:
How to Read and Do Proofs: An
Introduction to Mathematical Thought Processes
958:
For example, contraposition can be used to establish that, given an integer
417:
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the conventions of that day dictated that journals only published theorems.
3245:
Examples of simple proofs by mathematical induction for all natural numbers
2645:
2584:
2279:
563:
528:
This proof uses the definition of even integers, the integer properties of
440:
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321:
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131:
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116:
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2977:
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2553:
344:
As practiced, a proof is expressed in natural language and is a rigorous
65:
2719:
is written to indicate the end of a proof. This abbreviation stands for
5786:
5754:
5719:
4366:
4221:
4192:
3998:
2544:"Statistical proof" from data refers to the application of statistics,
2338:, which is neither provable nor refutable from the remaining axioms of
285:
101:
5848:
5709:
5620:
5518:
5421:
4474:
4391:
4351:
4315:
4251:
4063:
4053:
4026:
3789:
3636:
2692:
2688:
2614:
2362:
1849:
1510:
to disprove a proposition that all elements have a certain property.
306:
217:
81:
3376:
2909:
The Nuts and Bolts of Proofs: An
Introduction to Mathematical Proofs
1592:
are each pairwise equivalent, proofs are given for the implications
1380:
is also an integer. Substitution into the original equation yields 2
1266:
were a rational number. Then it could be written in lowest terms as
5769:
5503:
5301:
4749:
4454:
4048:
3362:
Fallis, Don (1997), "The
Epistemic Status of Probabilistic Proof."
2633:
2521:
2450:
Animated visual proof for the
Pythagorean theorem by rearrangement.
2419:
2330:
A statement that is neither provable nor disprovable from a set of
1477:
345:
193:
3327:
Mathematics for Computer Scientists: Fundamentals and Applications
5833:
5764:
5099:
3891:
3090:
2967:"proof" New Shorter Oxford English Dictionary, 1993, OUP, Oxford.
2683:
2505:
2299:
1781:
The pairwise equivalence of the statements then results from the
1312:
521:
449:
73:
154:
is concerned with the role of language and logic in proofs, and
5671:
3759:
3087:
An Introduction to the History of Mathematics (Saunders Series)
3056:
Moutsios-Rentzos, Andreas; Spyrou, Panagiotis (February 2015).
2736:
2710:
2663:
2590:
2493:. For some time it was thought that certain theorems, like the
2378:
1982:
is irrational (this is true, but the proof is not elementary).
1946:
224:
201:
130:
without the involvement of natural language, are considered in
40:
32:
2670:, whereby standards of mathematical proof might be applied to
2334:
is called undecidable (from those axioms). One example is the
5863:
5578:
4643:
3989:
3834:
3676:
2724:
2331:
364:
instead of natural language. A formal proof is a sequence of
236:
108:
77:
69:
3265:, University of Warwick Glossary of Mathematical Terminology
2887:
The Concise Oxford Dictionary of Mathematics, Fourth edition
88:
which establish logical certainty, to be distinguished from
5823:
2678:
Influence of mathematical proof methods outside mathematics
768: − 1 = 2(1) − 1 = 1
3784:
1200:
In proof by contradiction, also known by the Latin phrase
216:(417–369 BCE) formulated theorems but did not prove them.
2691:
of propositions deduced in a mathematical proof, such as
2263:
599:
be a mathematical statement involving the natural number
180:(to smell or taste, or sometimes touch or test), Italian
5547:
3319:
Struckmann, Werner; Wätjen, Dietmar (October 20, 2016).
1364:
must also be even, as seen in the proposition above (in
550:
Despite its name, mathematical induction is a method of
3322:
Mathematik für Informatiker: Grundlagen und Anwendungen
3105:
No work, except The Bible, has been more widely used...
3771:
Proofs in Mathematics: Simple, Charming and Fallacious
3288:
Plaue, Matthias; Scherfner, Mike (February 11, 2019).
3055:
2884:
1731:{\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}}
1400:. But then, by the same argument as before, 2 divides
401:, believed mathematical proofs are synthetic, whereas
138:
has led to much examination of current and historical
2236:
2163:
2133:
2094:
2061:
2025:
1992:
1955:
1927:
1896:
1744:
1698:
1678:
1638:
1598:
1552:
1458:
1430:
1321:
1272:
1248:
1216:
1171:
1141:
1114:
1075:
1069:
is odd. The product of two odd numbers is odd, hence
1055:
1035:
1011:
984:
964:
3682:
Proof and Other Dilemmas: Mathematics and Philosophy
2753:
2347:
Zermelo–Fraenkel set theory with the axiom of choice
1771:{\displaystyle \varphi _{n}\Rightarrow \varphi _{1}}
1665:{\displaystyle \varphi _{2}\Rightarrow \varphi _{3}}
1625:{\displaystyle \varphi _{1}\Rightarrow \varphi _{2}}
3412:
2462:
A second animated proof of the Pythagorean theorem.
352:The concept of proof is formalized in the field of
3482:
3276:Four color theorem#Simplification and verification
2687:statements outside of mathematics, but having the
2363:Heuristic mathematics and experimental mathematics
2252:
2222:
2149:
2119:
2080:
2047:
2011:
1974:
1937:
1909:
1770:
1730:
1684:
1664:
1624:
1584:
1468:
1440:
1340:
1295:
1258:
1226:
1177:
1154:
1127:
1100:
1061:
1041:
1017:
997:
970:
539:
1585:{\displaystyle \varphi _{1},\ldots ,\varphi _{n}}
5996:
3329:] (in German). Springer-Verlag. p. 28.
3318:
3298:] (in German). Springer-Verlag. p. 26.
2434:Visual proof for the (3,4,5) triangle as in the
1424:have no common factor, so we must conclude that
832:to an odd number results in an odd number. But
2864:"One of the Oldest Extant Diagrams from Euclid"
2533:
570:, which can be used, for example, to prove the
3480:
3287:
3164:(1998), "An introduction to proof theory", in
2617:. Inductive logic should not be confused with
2230:, which is thus a rational number of the form
1834:) are as good as genuine mathematical proofs.
464:. Since they are even, they can be written as
111:expressed in mathematical symbols, along with
5563:
3805:
3652:
3012:The History and Concept of Mathematical Proof
2956:Definition 3.1. Proof: An Informal Definition
3570:
3116:
3023:
2591:Inductive logic proofs and Bayesian analysis
1945:is irrational (an easy proof is known since
1365:
2897:
2639:
2305:
2019:is a rational number and we are done (take
1874:A nonconstructive proof establishes that a
838: − 1) + 2 = 2
532:under addition and multiplication, and the
269:and a proof that there are infinitely many
39:, one of the oldest surviving fragments of
5570:
5556:
3997:
3812:
3798:
3206:Universität Zürich – Theologische Fakultät
3120:Annals of the New York Academy of Sciences
2858:
915:
3615:
3547:
3427:Indra's Pearls: The Vision of Felix Klein
2903:
2467:Some illusory visual proofs, such as the
2325:
2120:{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}
1532:
1189:
588:} be the set of natural numbers, and let
223:Mathematical proof was revolutionized by
3711:
3007:
3005:
2504:
1863:
1498:, for instance, proved the existence of
1483:
1165:even, the supposition must be false, so
409:" that such a distinction is untenable.
200:, and one of its greatest achievements.
161:
31:
3730:
3014:, Steven G. Krantz. 1. February 5, 2007
2976:
2935:
2489:to refer to proofs that make no use of
2389:beyond the proof-theorem framework, in
2081:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
2012:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
1975:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
1296:{\displaystyle {\sqrt {2}}={a \over b}}
774:is odd, since it leaves a remainder of
572:irrationality of the square root of two
14:
5997:
3819:
3714:How to Prove It: A Structured Approach
3634:
3573:"Introduction to the Two-Column Proof"
3516:
2522:Colloquial use of "mathematical proof"
2357:Gödel's (first) incompleteness theorem
2264:Statistical proofs in pure mathematics
1837:
1792:
1546:In order to prove that the statements
1513:
1506:. It can also be used to construct a
172:The word "proof" comes from the Latin
68:, showing that the stated assumptions
5551:
3793:
3688:
3659:Proof in Mathematics: An Introduction
3600:
3196:
3002:
2351:List of statements undecidable in ZFC
349:vague or incomplete may be rejected.
339:
324:treats proofs as inductively defined
27:Reasoning for mathematical statements
3160:
3081:
2552:to infer propositions regarding the
2509:A two-column proof published in 1913
2397:, which was ultimately so resolved.
276:Further advances also took place in
84:. Proofs are examples of exhaustive
5745:Analytic and synthetic propositions
5616:Formal semantics (natural language)
3607:Mathematics and Plausible Reasoning
3460:"A Note on the History of Fractals"
2729:"that which was to be demonstrated"
2682:Philosopher-mathematicians such as
2500:
2474:
2400:
2373:While early mathematicians such as
429:
231:still in use today. It starts with
80:, along with the accepted rules of
24:
3594:
3527:Educational Studies in Mathematics
3141:10.1111/j.1749-6632.1987.tb37206.x
2885:Clapham, C. & Nicholson, J.N.
2833:What the Tortoise Said to Achilles
1452:To paraphrase: if one could write
1392:. Dividing both sides by 2 yields
902:is odd, for all positive integers
480:, respectively, for some integers
25:
6036:
3752:
3197:Quine, Willard Van Orman (1961).
2704:
828:must also be odd, because adding
5957:
5531:
3758:
2866:. University of British Columbia
2770:
2756:
2455:
2443:
2427:
706:is true for all natural numbers
412:Proofs may be admired for their
377:not provable within the system.
3564:
3510:
3474:
3452:
3406:
3395:
3380:
3369:
3356:
3343:
3312:
3281:
3268:
3249:
3238:
3229:
3220:
3190:
3154:
3110:
3075:
2938:Discrete Mathematics with Proof
2568:mathematical methods of physics
2405:
2048:{\displaystyle a=b={\sqrt {2}}}
540:Proof by mathematical induction
434:
144:quasi-empiricism in mathematics
3716:, Cambridge University Press,
3610:, Princeton University Press,
3481:Lesmoir-Gordon, Nigel (2000).
3049:
3027:; Kneale, Martha (May 1985) .
3017:
2970:
2961:
2929:
2878:
2852:
2088:is irrational so we can write
1755:
1715:
1649:
1609:
1348:. Squaring both sides yields 2
1341:{\displaystyle b{\sqrt {2}}=a}
1101:{\displaystyle x^{2}=x\cdot x}
399:analytic–synthetic distinction
280:. In the 10th century CE, the
227:(300 BCE), who introduced the
13:
1:
5492:History of mathematical logic
3351:American Mathematical Monthly
2845:
2571:applied to analyze data in a
2150:{\displaystyle b={\sqrt {2}}}
842: + 1 = 2(
826: − 1) + 2
261:, including a proof that the
5417:Primitive recursive function
3680:; Simons, Rogers A. (2008).
3517:Herbst, Patricio G. (2002).
3485:Introducing Fractal Geometry
2715:Sometimes, the abbreviation
2636:or information is acquired.
2613:, and may be less than full
2534:Statistical proof using data
2300:Statistical proof using data
1360:is even, which implies that
278:medieval Islamic mathematics
92:arguments or non-exhaustive
7:
2936:Gossett, Eric (July 2009).
2807:List of mathematical proofs
2749:
2418:in the case of the (3,4,5)
2288:probabilistic number theory
1938:{\displaystyle {\sqrt {2}}}
1469:{\displaystyle {\sqrt {2}}}
1441:{\displaystyle {\sqrt {2}}}
1311:are non-zero integers with
1259:{\displaystyle {\sqrt {2}}}
1227:{\displaystyle {\sqrt {2}}}
456:Consider two even integers
309:, who used it to prove the
10:
6041:
4481:Schröder–Bernstein theorem
4208:Monadic predicate calculus
3867:Foundations of mathematics
3432:Cambridge University Press
3261:February 18, 2012, at the
3199:"Two Dogmas of Empiricism"
2986:Cambridge University Press
2708:
2643:
2594:
2537:
2478:
2366:
2309:
2267:
1867:
1841:
1796:
1536:
1517:
1487:
1408:must be even. However, if
1193:
919:
543:
438:
165:
136:formal and informal proofs
134:. The distinction between
5952:
5912:
5884:
5877:
5829:Necessity and sufficiency
5732:
5697:
5649:
5603:
5585:
5577:
5527:
5514:Philosophy of mathematics
5463:Automated theorem proving
5445:
5340:
5172:
5065:
4917:
4634:
4610:
4588:Von Neumann–Bernays–Gödel
4533:
4427:
4331:
4229:
4220:
4147:
4082:
3988:
3910:
3827:
3731:Hammack, Richard (2018),
3390:proof"" (Derogatory use.)
2797:List of incomplete proofs
2787:Automated theorem proving
2721:"quod erat demonstrandum"
2630:assessment of likelihoods
2377:did not use proofs, from
568:proof by infinite descent
198:ancient Greek mathematics
156:mathematics as a language
152:philosophy of mathematics
6010:Mathematical terminology
3170:Handbook of Proof Theory
3029:The development of logic
2735:" or "halmos" after its
2640:Proofs as mental objects
2391:experimental mathematics
2383:foundational mathematics
2369:Experimental mathematics
2306:Computer-assisted proofs
1858:double counting argument
1448:is an irrational number.
1366:#Proof by contraposition
945:contrapositive statement
407:Two Dogmas of Empiricism
5164:Self-verifying theories
4985:Tarski's axiomatization
3936:Tarski's undefinability
3931:incompleteness theorems
3617:2027/mdp.39015008206248
3540:10.1023/A:1020264906740
3033:Oxford University Press
2744:(220E(hex) = 8718(dec))
2632:of hypotheses when new
2624:Bayesian analysis uses
2312:Computer-assisted proof
2296:mathematical statistics
1921:. This proof uses that
1828:probabilistic algorithm
927:Proof by contraposition
916:Proof by contraposition
5538:Mathematics portal
5149:Proof of impossibility
4797:propositional variable
4107:Propositional calculus
3577:onemathematicalcat.org
3466:on February 15, 2009.
2619:mathematical induction
2510:
2326:Undecidable statements
2292:analytic number theory
2254:
2253:{\displaystyle a^{b}.}
2224:
2151:
2121:
2082:
2049:
2013:
1976:
1939:
1911:
1772:
1732:
1686:
1685:{\displaystyle \dots }
1666:
1626:
1586:
1539:Closed chain inference
1533:Closed chain inference
1500:transcendental numbers
1470:
1442:
1342:
1297:
1260:
1228:
1196:Proof by contradiction
1190:Proof by contradiction
1179:
1156:
1135:is not even. Thus, if
1129:
1102:
1063:
1043:
1019:
999:
972:
940:" by establishing the
546:Mathematical induction
375:undecidable statements
334:Non-Euclidean geometry
301:was introduced in the
66:mathematical statement
50:
5964:Philosophy portal
5407:Kolmogorov complexity
5360:Computably enumerable
5260:Model complete theory
5052:Principia Mathematica
4112:Propositional formula
3941:Banach–Tarski paradox
3712:Velleman, D. (2006),
3364:Journal of Philosophy
2942:John Wiley & Sons
2817:Proof by intimidation
2812:Nonconstructive proof
2628:to update a person's
2515:mathematical exercise
2508:
2469:missing square puzzle
2255:
2225:
2152:
2122:
2083:
2050:
2014:
1977:
1940:
1912:
1910:{\displaystyle a^{b}}
1870:Nonconstructive proof
1864:Nonconstructive proof
1773:
1733:
1687:
1667:
1627:
1587:
1490:Proof by construction
1484:Proof by construction
1471:
1443:
1343:
1298:
1261:
1229:
1208:logical contradiction
1180:
1157:
1155:{\displaystyle x^{2}}
1130:
1128:{\displaystyle x^{2}}
1103:
1064:
1044:
1020:
1000:
998:{\displaystyle x^{2}}
973:
680:is true implies that
534:distributive property
397:, who introduced the
243:. Euclid's book, the
184:(to try), and German
162:History and etymology
140:mathematical practice
35:
6025:Sources of knowledge
5355:Church–Turing thesis
5342:Computability theory
4551:continuum hypothesis
4069:Square of opposition
3927:Gödel's completeness
3767:at Wikimedia Commons
3656:; Daoud, A. (2011),
3635:Fallis, Don (2002),
3183:. See in particular
2905:Cupillari, Antonella
2839:Zero-knowledge proof
2822:Termination analysis
2528:mathematical objects
2495:prime number theorem
2387:mathematical objects
2278:, such as involving
2234:
2161:
2131:
2092:
2059:
2023:
1990:
1953:
1925:
1894:
1799:Probabilistic method
1787:material conditional
1742:
1696:
1676:
1636:
1596:
1550:
1456:
1428:
1319:
1270:
1246:
1214:
1203:reductio ad absurdum
1169:
1139:
1112:
1073:
1053:
1033:
1009:
982:
962:
942:logically equivalent
423:Proofs from THE BOOK
416:. The mathematician
405:argued in his 1951 "
330:Axiomatic set theory
299:arithmetic sequences
206:Hippocrates of Chios
6015:Mathematical proofs
5626:Philosophy of logic
5509:Mathematical object
5400:P versus NP problem
5365:Computable function
5159:Reverse mathematics
5085:Logical consequence
4962:primitive recursive
4957:elementary function
4730:Free/bound variable
4583:Tarski–Grothendieck
4102:Logical connectives
4032:Logical equivalence
3882:Logical consequence
3779:about proofs, in a
3737:, Richard Hammack,
3133:1987NYASA.500..253M
3062:Archive ouverte HAL
2802:List of long proofs
2668:language of thought
2650:Language of thought
2577:observational study
2416:Pythagorean theorem
2412:proof without words
1876:mathematical object
1844:Combinatorial proof
1838:Combinatorial proof
1793:Probabilistic proof
1520:Proof by exhaustion
1514:Proof by exhaustion
1502:by constructing an
1368:). So we can write
556:inductive reasoning
414:mathematical beauty
251:Pythagorean theorem
126:, written fully in
94:inductive reasoning
86:deductive reasoning
6005:Mathematical logic
5925:Rules of inference
5894:Mathematical logic
5636:Semantics of logic
5307:Transfer principle
5270:Semantics of logic
5255:Categorical theory
5231:Non-standard model
4745:Logical connective
3872:Information theory
3821:Mathematical logic
3765:Mathematical proof
3689:Solow, D. (2004),
3641:Logique et Analyse
3571:Dr. Fisher Burns.
3256:Proof by induction
2911:(Third ed.).
2827:Thought experiment
2778:Mathematics portal
2581:physical cosmology
2511:
2340:Euclidean geometry
2336:parallel postulate
2319:four color theorem
2250:
2220:
2157:. This then gives
2147:
2117:
2078:
2045:
2009:
1972:
1935:
1907:
1881:irrational numbers
1824:Mertens conjecture
1820:Collatz conjecture
1809:existence theorems
1805:probability theory
1768:
1728:
1682:
1662:
1622:
1582:
1527:four color theorem
1466:
1438:
1338:
1293:
1256:
1224:
1175:
1152:
1125:
1098:
1059:
1049:is not even. Then
1039:
1015:
995:
968:
932:the statement "if
586:= {1, 2, 3, 4, ...
488:. Then the sum is
354:mathematical logic
340:Nature and purpose
313:and properties of
291:irrational numbers
263:square root of two
212:(408–355 BCE) and
204:(624–546 BCE) and
55:mathematical proof
51:
5992:
5991:
5948:
5947:
5782:Deductive closure
5728:
5727:
5667:Critical thinking
5545:
5544:
5477:Abstract category
5280:Theories of truth
5090:Rule of inference
5080:Natural deduction
5061:
5060:
4606:
4605:
4311:Cartesian product
4216:
4215:
4122:Many-valued logic
4097:Boolean functions
3980:Russell's paradox
3955:diagonal argument
3852:First-order logic
3763:Media related to
3744:978-0-9894721-3-5
3723:978-0-521-67599-4
3704:978-0-471-68058-1
3669:978-0-646-54509-7
3500:978-1-84046-123-7
3441:978-0-521-35253-6
3414:Mumford, David B.
3336:978-3-662-49870-5
3305:978-3-662-58352-4
3235:Cupillari, p. 46.
3226:Cupillari, p. 20.
3179:978-0-08-053318-6
3085:(January 1990) .
3042:978-0-19-824773-9
2995:978-0-521-31803-7
2922:978-0-12-088509-1
2764:Philosophy portal
2745:
2672:empirical science
2601:Bayesian analysis
2550:Bayesian analysis
2540:Statistical proof
2438:500–200 BCE.
2375:Eudoxus of Cnidus
2302:" section below.
2298:. See also the "
2270:Statistical proof
2206:
2194:
2182:
2175:
2145:
2114:
2107:
2075:
2068:
2043:
2006:
1999:
1969:
1962:
1933:
1832:testing primality
1464:
1436:
1330:
1291:
1278:
1254:
1236:irrational number
1222:
1178:{\displaystyle x}
1062:{\displaystyle x}
1042:{\displaystyle x}
1018:{\displaystyle x}
971:{\displaystyle x}
854:+1) − 1
846:+1) − 1
658:is true whenever
315:Pascal's triangle
128:symbolic language
16:(Redirected from
6032:
5962:
5961:
5960:
5882:
5881:
5647:
5646:
5611:Computer science
5572:
5565:
5558:
5549:
5548:
5536:
5535:
5487:History of logic
5482:Category of sets
5375:Decision problem
5154:Ordinal analysis
5095:Sequent calculus
4993:Boolean algebras
4933:
4932:
4907:
4878:logical/constant
4632:
4631:
4618:
4541:Zermelo–Fraenkel
4292:Set operations:
4227:
4226:
4164:
3995:
3994:
3975:Löwenheim–Skolem
3862:Formal semantics
3814:
3807:
3800:
3791:
3790:
3762:
3747:
3726:
3707:
3685:
3672:
3648:
3630:
3619:
3588:
3587:
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3583:
3568:
3562:
3561:
3551:
3523:
3514:
3508:
3507:
3488:
3478:
3472:
3471:
3462:. Archived from
3456:
3450:
3449:
3418:Series, Caroline
3410:
3404:
3399:
3393:
3384:
3378:
3373:
3367:
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3213:
3203:
3194:
3188:
3182:
3158:
3152:
3151:
3114:
3108:
3107:
3089:(6th ed.).
3079:
3073:
3072:
3070:
3068:
3053:
3047:
3046:
3031:(New ed.).
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3015:
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2774:
2766:
2761:
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2759:
2743:
2573:particle physics
2501:Two-column proof
2491:complex analysis
2481:Elementary proof
2475:Elementary proof
2459:
2447:
2431:
2401:Related concepts
2395:fractal geometry
2276:pure mathematics
2259:
2257:
2256:
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2229:
2227:
2226:
2221:
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2002:
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1981:
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1973:
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1965:
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1949:), but not that
1944:
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1608:
1607:
1591:
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1588:
1583:
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1580:
1562:
1561:
1504:explicit example
1496:Joseph Liouville
1475:
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1472:
1467:
1465:
1460:
1447:
1445:
1444:
1439:
1437:
1432:
1347:
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1331:
1326:
1313:no common factor
1302:
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1150:
1134:
1132:
1131:
1126:
1124:
1123:
1107:
1105:
1104:
1099:
1085:
1084:
1068:
1066:
1065:
1060:
1048:
1046:
1045:
1040:
1024:
1022:
1021:
1016:
1004:
1002:
1001:
996:
994:
993:
977:
975:
974:
969:
907:
901:
888:
877:
866:
855:
847:
831:
827:
819:
808:
800:
788:
781:
778:when divided by
777:
773:
769:
761:
747:
739:
724:
711:
705:
690:
679:
668:
657:
641:
634:
623:
610:
604:
598:
587:
554:, not a form of
452:is always even:
430:Methods of proof
383:proof assistants
360:is written in a
311:binomial theorem
229:axiomatic method
168:History of logic
148:folk mathematics
146:, and so-called
113:natural language
21:
6040:
6039:
6035:
6034:
6033:
6031:
6030:
6029:
5995:
5994:
5993:
5988:
5958:
5956:
5944:
5908:
5899:Boolean algebra
5873:
5724:
5715:Metamathematics
5693:
5645:
5599:
5581:
5576:
5546:
5541:
5530:
5523:
5468:Category theory
5458:Algebraic logic
5441:
5412:Lambda calculus
5350:Church encoding
5336:
5312:Truth predicate
5168:
5134:Complete theory
5057:
4926:
4922:
4918:
4913:
4905:
4625: and
4621:
4616:
4602:
4578:New Foundations
4546:axiom of choice
4529:
4491:Gödel numbering
4431: and
4423:
4327:
4212:
4162:
4143:
4092:Boolean algebra
4078:
4042:Equiconsistency
4007:Classical logic
3984:
3965:Halting problem
3953: and
3929: and
3917: and
3916:
3911:Theorems (
3906:
3823:
3818:
3755:
3745:
3724:
3705:
3670:
3628:
3597:
3595:Further reading
3592:
3591:
3581:
3579:
3569:
3565:
3521:
3515:
3511:
3501:
3479:
3475:
3458:
3457:
3453:
3442:
3411:
3407:
3400:
3396:
3385:
3381:
3374:
3370:
3361:
3357:
3348:
3344:
3337:
3317:
3313:
3306:
3286:
3282:
3273:
3269:
3263:Wayback Machine
3254:
3250:
3243:
3239:
3234:
3230:
3225:
3221:
3211:
3209:
3201:
3195:
3191:
3180:
3166:Buss, Samuel R.
3162:Buss, Samuel R.
3159:
3155:
3115:
3111:
3101:
3093:. p. 141.
3083:Eves, Howard W.
3080:
3076:
3066:
3064:
3054:
3050:
3043:
3025:Kneale, William
3022:
3018:
3010:
3003:
2996:
2975:
2971:
2966:
2962:
2952:
2934:
2930:
2923:
2902:
2898:
2883:
2879:
2869:
2867:
2857:
2853:
2848:
2843:
2776:
2769:
2762:
2757:
2755:
2752:
2713:
2707:
2680:
2652:
2644:Main articles:
2642:
2607:inductive logic
2603:
2597:Inductive logic
2595:Main articles:
2593:
2556:of data. While
2542:
2536:
2524:
2503:
2483:
2477:
2463:
2460:
2451:
2448:
2439:
2436:Zhoubi Suanjing
2432:
2408:
2403:
2371:
2365:
2328:
2314:
2308:
2272:
2266:
2241:
2237:
2235:
2232:
2231:
2208:
2201:
2200:
2189:
2177:
2170:
2169:
2165:
2164:
2162:
2159:
2158:
2140:
2132:
2129:
2128:
2109:
2102:
2101:
2093:
2090:
2089:
2070:
2063:
2062:
2060:
2057:
2056:
2038:
2024:
2021:
2020:
2001:
1994:
1993:
1991:
1988:
1987:
1964:
1957:
1956:
1954:
1951:
1950:
1928:
1926:
1923:
1922:
1919:rational number
1901:
1897:
1895:
1892:
1891:
1872:
1866:
1846:
1840:
1801:
1795:
1762:
1758:
1749:
1745:
1743:
1740:
1739:
1722:
1718:
1703:
1699:
1697:
1694:
1693:
1677:
1674:
1673:
1656:
1652:
1643:
1639:
1637:
1634:
1633:
1616:
1612:
1603:
1599:
1597:
1594:
1593:
1576:
1572:
1557:
1553:
1551:
1548:
1547:
1541:
1535:
1522:
1516:
1492:
1486:
1459:
1457:
1454:
1453:
1431:
1429:
1426:
1425:
1325:
1320:
1317:
1316:
1283:
1273:
1271:
1268:
1267:
1249:
1247:
1244:
1243:
1217:
1215:
1212:
1211:
1198:
1192:
1185:has to be even.
1170:
1167:
1166:
1146:
1142:
1140:
1137:
1136:
1119:
1115:
1113:
1110:
1109:
1080:
1076:
1074:
1071:
1070:
1054:
1051:
1050:
1034:
1031:
1030:
1010:
1007:
1006:
989:
985:
983:
980:
979:
963:
960:
959:
924:
918:
903:
895:
879:
868:
857:
849:
833:
829:
821:
810:
802:
796:
783:
779:
775:
771:
763:
756:
741:
730:
718:
707:
696:
681:
670:
669:is true, i.e.,
659:
648:
636:
625:
624:is true, i.e.,
618:
606:
600:
589:
582:
579:natural numbers
548:
542:
443:
437:
432:
362:formal language
342:
326:data structures
295:inductive proof
241:deductive logic
233:undefined terms
170:
164:
28:
23:
22:
18:Theorem-proving
15:
12:
11:
5:
6038:
6028:
6027:
6022:
6017:
6012:
6007:
5990:
5989:
5987:
5986:
5981:
5971:
5966:
5953:
5950:
5949:
5946:
5945:
5943:
5942:
5937:
5932:
5927:
5922:
5916:
5914:
5910:
5909:
5907:
5906:
5901:
5896:
5890:
5888:
5879:
5875:
5874:
5872:
5871:
5866:
5861:
5856:
5851:
5846:
5841:
5836:
5831:
5826:
5821:
5816:
5811:
5806:
5805:
5804:
5794:
5789:
5784:
5779:
5774:
5773:
5772:
5767:
5757:
5752:
5747:
5742:
5736:
5734:
5730:
5729:
5726:
5725:
5723:
5722:
5717:
5712:
5707:
5701:
5699:
5695:
5694:
5692:
5691:
5686:
5681:
5676:
5675:
5674:
5669:
5659:
5653:
5651:
5644:
5643:
5638:
5633:
5628:
5623:
5618:
5613:
5607:
5605:
5601:
5600:
5598:
5597:
5592:
5586:
5583:
5582:
5575:
5574:
5567:
5560:
5552:
5543:
5542:
5528:
5525:
5524:
5522:
5521:
5516:
5511:
5506:
5501:
5500:
5499:
5489:
5484:
5479:
5470:
5465:
5460:
5455:
5453:Abstract logic
5449:
5447:
5443:
5442:
5440:
5439:
5434:
5432:Turing machine
5429:
5424:
5419:
5414:
5409:
5404:
5403:
5402:
5397:
5392:
5387:
5382:
5372:
5370:Computable set
5367:
5362:
5357:
5352:
5346:
5344:
5338:
5337:
5335:
5334:
5329:
5324:
5319:
5314:
5309:
5304:
5299:
5298:
5297:
5292:
5287:
5277:
5272:
5267:
5265:Satisfiability
5262:
5257:
5252:
5251:
5250:
5240:
5239:
5238:
5228:
5227:
5226:
5221:
5216:
5211:
5206:
5196:
5195:
5194:
5189:
5182:Interpretation
5178:
5176:
5170:
5169:
5167:
5166:
5161:
5156:
5151:
5146:
5136:
5131:
5130:
5129:
5128:
5127:
5117:
5112:
5102:
5097:
5092:
5087:
5082:
5077:
5071:
5069:
5063:
5062:
5059:
5058:
5056:
5055:
5047:
5046:
5045:
5044:
5039:
5038:
5037:
5032:
5027:
5007:
5006:
5005:
5003:minimal axioms
5000:
4989:
4988:
4987:
4976:
4975:
4974:
4969:
4964:
4959:
4954:
4949:
4936:
4934:
4915:
4914:
4912:
4911:
4910:
4909:
4897:
4892:
4891:
4890:
4885:
4880:
4875:
4865:
4860:
4855:
4850:
4849:
4848:
4843:
4833:
4832:
4831:
4826:
4821:
4816:
4806:
4801:
4800:
4799:
4794:
4789:
4779:
4778:
4777:
4772:
4767:
4762:
4757:
4752:
4742:
4737:
4732:
4727:
4726:
4725:
4720:
4715:
4710:
4700:
4695:
4693:Formation rule
4690:
4685:
4684:
4683:
4678:
4668:
4667:
4666:
4656:
4651:
4646:
4641:
4635:
4629:
4612:Formal systems
4608:
4607:
4604:
4603:
4601:
4600:
4595:
4590:
4585:
4580:
4575:
4570:
4565:
4560:
4555:
4554:
4553:
4548:
4537:
4535:
4531:
4530:
4528:
4527:
4526:
4525:
4515:
4510:
4509:
4508:
4501:Large cardinal
4498:
4493:
4488:
4483:
4478:
4464:
4463:
4462:
4457:
4452:
4437:
4435:
4425:
4424:
4422:
4421:
4420:
4419:
4414:
4409:
4399:
4394:
4389:
4384:
4379:
4374:
4369:
4364:
4359:
4354:
4349:
4344:
4338:
4336:
4329:
4328:
4326:
4325:
4324:
4323:
4318:
4313:
4308:
4303:
4298:
4290:
4289:
4288:
4283:
4273:
4268:
4266:Extensionality
4263:
4261:Ordinal number
4258:
4248:
4243:
4242:
4241:
4230:
4224:
4218:
4217:
4214:
4213:
4211:
4210:
4205:
4200:
4195:
4190:
4185:
4180:
4179:
4178:
4168:
4167:
4166:
4153:
4151:
4145:
4144:
4142:
4141:
4140:
4139:
4134:
4129:
4119:
4114:
4109:
4104:
4099:
4094:
4088:
4086:
4080:
4079:
4077:
4076:
4071:
4066:
4061:
4056:
4051:
4046:
4045:
4044:
4034:
4029:
4024:
4019:
4014:
4009:
4003:
4001:
3992:
3986:
3985:
3983:
3982:
3977:
3972:
3967:
3962:
3957:
3945:Cantor's
3943:
3938:
3933:
3923:
3921:
3908:
3907:
3905:
3904:
3899:
3894:
3889:
3884:
3879:
3874:
3869:
3864:
3859:
3854:
3849:
3844:
3843:
3842:
3831:
3829:
3825:
3824:
3817:
3816:
3809:
3802:
3794:
3788:
3787:
3773:
3768:
3754:
3753:External links
3751:
3750:
3749:
3743:
3728:
3722:
3709:
3703:
3686:
3674:
3668:
3650:
3632:
3626:
3596:
3593:
3590:
3589:
3563:
3534:(3): 283–312.
3509:
3499:
3473:
3451:
3440:
3405:
3394:
3379:
3368:
3355:
3342:
3335:
3311:
3304:
3280:
3267:
3248:
3237:
3228:
3219:
3189:
3178:
3153:
3127:(1): 253–77 ,
3109:
3100:978-0030295584
3099:
3074:
3048:
3041:
3016:
3001:
2994:
2969:
2960:
2951:978-0470457931
2950:
2944:. p. 86.
2928:
2921:
2913:Academic Press
2896:
2877:
2860:Bill Casselman
2850:
2849:
2847:
2844:
2842:
2841:
2836:
2829:
2824:
2819:
2814:
2809:
2804:
2799:
2794:
2789:
2783:
2782:
2781:
2767:
2751:
2748:
2709:Main article:
2706:
2705:Ending a proof
2703:
2679:
2676:
2641:
2638:
2626:Bayes' theorem
2592:
2589:
2575:experiment or
2538:Main article:
2535:
2532:
2523:
2520:
2502:
2499:
2479:Main article:
2476:
2473:
2465:
2464:
2461:
2454:
2452:
2449:
2442:
2440:
2433:
2426:
2407:
2404:
2402:
2399:
2367:Main article:
2364:
2361:
2327:
2324:
2310:Main article:
2307:
2304:
2284:chaotic series
2268:Main article:
2265:
2262:
2261:
2260:
2249:
2244:
2240:
2219:
2216:
2211:
2205:
2199:
2193:
2187:
2181:
2174:
2168:
2144:
2139:
2136:
2113:
2106:
2100:
2097:
2074:
2067:
2042:
2037:
2034:
2031:
2028:
2005:
1998:
1968:
1961:
1932:
1904:
1900:
1868:Main article:
1865:
1862:
1842:Main article:
1839:
1836:
1797:Main article:
1794:
1791:
1765:
1761:
1757:
1752:
1748:
1725:
1721:
1717:
1712:
1709:
1706:
1702:
1681:
1659:
1655:
1651:
1646:
1642:
1619:
1615:
1611:
1606:
1602:
1579:
1575:
1571:
1568:
1565:
1560:
1556:
1537:Main article:
1534:
1531:
1518:Main article:
1515:
1512:
1508:counterexample
1488:Main article:
1485:
1482:
1463:
1450:
1449:
1435:
1337:
1334:
1329:
1324:
1290:
1287:
1282:
1277:
1253:
1221:
1194:Main article:
1191:
1188:
1187:
1186:
1174:
1149:
1145:
1122:
1118:
1097:
1094:
1091:
1088:
1083:
1079:
1058:
1038:
1014:
1005:is even, then
992:
988:
967:
922:Contraposition
920:Main article:
917:
914:
910:
909:
900: − 1
890:
807: − 1
790:
746: − 1
723: − 1
715:
714:
692:
643:
544:Main article:
541:
538:
526:
525:
500: + 2
476: = 2
468: = 2
439:Main article:
436:
433:
431:
428:
341:
338:
284:mathematician
163:
160:
120:informal logic
107:Proofs employ
26:
9:
6:
4:
3:
2:
6037:
6026:
6023:
6021:
6018:
6016:
6013:
6011:
6008:
6006:
6003:
6002:
6000:
5985:
5982:
5979:
5975:
5972:
5970:
5967:
5965:
5955:
5954:
5951:
5941:
5940:Logic symbols
5938:
5936:
5933:
5931:
5928:
5926:
5923:
5921:
5918:
5917:
5915:
5911:
5905:
5902:
5900:
5897:
5895:
5892:
5891:
5889:
5887:
5883:
5880:
5876:
5870:
5867:
5865:
5862:
5860:
5857:
5855:
5852:
5850:
5847:
5845:
5842:
5840:
5837:
5835:
5832:
5830:
5827:
5825:
5822:
5820:
5819:Logical truth
5817:
5815:
5812:
5810:
5807:
5803:
5800:
5799:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5771:
5768:
5766:
5763:
5762:
5761:
5760:Contradiction
5758:
5756:
5753:
5751:
5748:
5746:
5743:
5741:
5738:
5737:
5735:
5731:
5721:
5718:
5716:
5713:
5711:
5708:
5706:
5705:Argumentation
5703:
5702:
5700:
5696:
5690:
5689:Philosophical
5687:
5685:
5684:Non-classical
5682:
5680:
5677:
5673:
5670:
5668:
5665:
5664:
5663:
5660:
5658:
5655:
5654:
5652:
5648:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5622:
5619:
5617:
5614:
5612:
5609:
5608:
5606:
5602:
5596:
5593:
5591:
5588:
5587:
5584:
5580:
5573:
5568:
5566:
5561:
5559:
5554:
5553:
5550:
5540:
5539:
5534:
5526:
5520:
5517:
5515:
5512:
5510:
5507:
5505:
5502:
5498:
5495:
5494:
5493:
5490:
5488:
5485:
5483:
5480:
5478:
5474:
5471:
5469:
5466:
5464:
5461:
5459:
5456:
5454:
5451:
5450:
5448:
5444:
5438:
5435:
5433:
5430:
5428:
5427:Recursive set
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5401:
5398:
5396:
5393:
5391:
5388:
5386:
5383:
5381:
5378:
5377:
5376:
5373:
5371:
5368:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5347:
5345:
5343:
5339:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5308:
5305:
5303:
5300:
5296:
5293:
5291:
5288:
5286:
5283:
5282:
5281:
5278:
5276:
5273:
5271:
5268:
5266:
5263:
5261:
5258:
5256:
5253:
5249:
5246:
5245:
5244:
5241:
5237:
5236:of arithmetic
5234:
5233:
5232:
5229:
5225:
5222:
5220:
5217:
5215:
5212:
5210:
5207:
5205:
5202:
5201:
5200:
5197:
5193:
5190:
5188:
5185:
5184:
5183:
5180:
5179:
5177:
5175:
5171:
5165:
5162:
5160:
5157:
5155:
5152:
5150:
5147:
5144:
5143:from ZFC
5140:
5137:
5135:
5132:
5126:
5123:
5122:
5121:
5118:
5116:
5113:
5111:
5108:
5107:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5072:
5070:
5068:
5064:
5054:
5053:
5049:
5048:
5043:
5042:non-Euclidean
5040:
5036:
5033:
5031:
5028:
5026:
5025:
5021:
5020:
5018:
5015:
5014:
5012:
5008:
5004:
5001:
4999:
4996:
4995:
4994:
4990:
4986:
4983:
4982:
4981:
4977:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4945:
4944:
4942:
4938:
4937:
4935:
4930:
4924:
4919:Example
4916:
4908:
4903:
4902:
4901:
4898:
4896:
4893:
4889:
4886:
4884:
4881:
4879:
4876:
4874:
4871:
4870:
4869:
4866:
4864:
4861:
4859:
4856:
4854:
4851:
4847:
4844:
4842:
4839:
4838:
4837:
4834:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4811:
4810:
4807:
4805:
4802:
4798:
4795:
4793:
4790:
4788:
4785:
4784:
4783:
4780:
4776:
4773:
4771:
4768:
4766:
4763:
4761:
4758:
4756:
4753:
4751:
4748:
4747:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4724:
4721:
4719:
4716:
4714:
4711:
4709:
4706:
4705:
4704:
4701:
4699:
4696:
4694:
4691:
4689:
4686:
4682:
4679:
4677:
4676:by definition
4674:
4673:
4672:
4669:
4665:
4662:
4661:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4636:
4633:
4630:
4628:
4624:
4619:
4613:
4609:
4599:
4596:
4594:
4591:
4589:
4586:
4584:
4581:
4579:
4576:
4574:
4571:
4569:
4566:
4564:
4563:Kripke–Platek
4561:
4559:
4556:
4552:
4549:
4547:
4544:
4543:
4542:
4539:
4538:
4536:
4532:
4524:
4521:
4520:
4519:
4516:
4514:
4511:
4507:
4504:
4503:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4476:
4472:
4468:
4465:
4461:
4458:
4456:
4453:
4451:
4448:
4447:
4446:
4442:
4439:
4438:
4436:
4434:
4430:
4426:
4418:
4415:
4413:
4410:
4408:
4407:constructible
4405:
4404:
4403:
4400:
4398:
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4339:
4337:
4335:
4330:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4293:
4291:
4287:
4284:
4282:
4279:
4278:
4277:
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4257:
4253:
4249:
4247:
4244:
4240:
4237:
4236:
4235:
4232:
4231:
4228:
4225:
4223:
4219:
4209:
4206:
4204:
4201:
4199:
4196:
4194:
4191:
4189:
4186:
4184:
4181:
4177:
4174:
4173:
4172:
4169:
4165:
4160:
4159:
4158:
4155:
4154:
4152:
4150:
4146:
4138:
4135:
4133:
4130:
4128:
4125:
4124:
4123:
4120:
4118:
4115:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4089:
4087:
4085:
4084:Propositional
4081:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4043:
4040:
4039:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4012:Logical truth
4010:
4008:
4005:
4004:
4002:
4000:
3996:
3993:
3991:
3987:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3952:
3948:
3944:
3942:
3939:
3937:
3934:
3932:
3928:
3925:
3924:
3922:
3920:
3914:
3909:
3903:
3900:
3898:
3895:
3893:
3890:
3888:
3885:
3883:
3880:
3878:
3875:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3841:
3838:
3837:
3836:
3833:
3832:
3830:
3826:
3822:
3815:
3810:
3808:
3803:
3801:
3796:
3795:
3792:
3786:
3782:
3778:
3774:
3772:
3769:
3766:
3761:
3757:
3756:
3746:
3740:
3736:
3735:
3734:Book of Proof
3729:
3725:
3719:
3715:
3710:
3706:
3700:
3696:
3692:
3687:
3683:
3679:
3675:
3671:
3665:
3662:, Kew Books,
3661:
3660:
3655:
3651:
3646:
3642:
3638:
3633:
3629:
3627:9780691080055
3623:
3618:
3613:
3609:
3608:
3603:
3599:
3598:
3578:
3574:
3567:
3559:
3555:
3550:
3549:2027.42/42653
3545:
3541:
3537:
3533:
3529:
3528:
3520:
3513:
3506:
3502:
3496:
3492:
3487:
3486:
3477:
3470:
3465:
3461:
3455:
3448:
3443:
3437:
3433:
3429:
3428:
3423:
3422:Wright, David
3419:
3415:
3409:
3403:
3398:
3392:
3389:
3383:
3377:
3372:
3365:
3359:
3352:
3346:
3338:
3332:
3328:
3324:
3323:
3315:
3307:
3301:
3297:
3293:
3292:
3284:
3277:
3271:
3264:
3260:
3257:
3252:
3246:
3241:
3232:
3223:
3207:
3200:
3193:
3186:
3181:
3175:
3171:
3167:
3163:
3157:
3150:
3146:
3142:
3138:
3134:
3130:
3126:
3122:
3121:
3113:
3106:
3102:
3096:
3092:
3088:
3084:
3078:
3063:
3059:
3052:
3044:
3038:
3035:. p. 3.
3034:
3030:
3026:
3020:
3013:
3008:
3006:
2997:
2991:
2987:
2983:
2979:
2973:
2964:
2957:
2953:
2947:
2943:
2939:
2932:
2924:
2918:
2915:. p. 3.
2914:
2910:
2906:
2900:
2893:
2888:
2881:
2870:September 26,
2865:
2861:
2855:
2851:
2840:
2837:
2835:
2834:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2813:
2810:
2808:
2805:
2803:
2800:
2798:
2795:
2793:
2792:Invalid proof
2790:
2788:
2785:
2784:
2779:
2773:
2768:
2765:
2754:
2747:
2741:
2738:
2734:
2730:
2726:
2722:
2718:
2712:
2702:
2700:
2699:
2694:
2690:
2685:
2675:
2673:
2669:
2665:
2661:
2657:
2651:
2647:
2637:
2635:
2631:
2627:
2622:
2620:
2616:
2612:
2608:
2605:Proofs using
2602:
2598:
2588:
2586:
2585:scatter plots
2582:
2578:
2574:
2570:
2569:
2563:
2559:
2555:
2551:
2547:
2546:data analysis
2541:
2531:
2529:
2519:
2516:
2507:
2498:
2496:
2492:
2488:
2487:number theory
2482:
2472:
2470:
2458:
2453:
2446:
2441:
2437:
2430:
2425:
2424:
2423:
2421:
2417:
2413:
2398:
2396:
2392:
2388:
2384:
2380:
2376:
2370:
2360:
2358:
2354:
2352:
2348:
2343:
2341:
2337:
2333:
2323:
2320:
2317:proof of the
2313:
2303:
2301:
2297:
2293:
2289:
2285:
2281:
2277:
2271:
2247:
2242:
2238:
2217:
2214:
2209:
2203:
2197:
2191:
2185:
2179:
2172:
2166:
2142:
2137:
2134:
2111:
2104:
2098:
2095:
2072:
2065:
2040:
2035:
2032:
2029:
2026:
2003:
1996:
1985:
1984:
1983:
1966:
1959:
1948:
1930:
1920:
1902:
1898:
1889:
1885:
1882:
1877:
1871:
1861:
1859:
1855:
1851:
1845:
1835:
1833:
1829:
1825:
1821:
1816:
1812:
1810:
1806:
1800:
1790:
1788:
1784:
1779:
1763:
1759:
1750:
1746:
1723:
1719:
1710:
1707:
1704:
1700:
1679:
1657:
1653:
1644:
1640:
1617:
1613:
1604:
1600:
1577:
1573:
1569:
1566:
1563:
1558:
1554:
1544:
1540:
1530:
1528:
1521:
1511:
1509:
1505:
1501:
1497:
1491:
1481:
1479:
1461:
1433:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1351:
1335:
1332:
1327:
1322:
1314:
1310:
1306:
1288:
1285:
1280:
1275:
1251:
1242:Suppose that
1241:
1240:
1239:
1237:
1219:
1209:
1205:
1204:
1197:
1172:
1164:
1147:
1143:
1120:
1116:
1108:is odd. Thus
1095:
1092:
1089:
1086:
1081:
1077:
1056:
1036:
1028:
1027:
1026:
1012:
990:
986:
965:
956:
954:
950:
946:
943:
939:
935:
931:
928:
923:
913:
906:
899:
894:
891:
886:
882:
875:
871:
864:
860:
853:
845:
841:
837:
825:
817:
813:
806:
799:
794:
791:
786:
767:
759:
754:
751:
750:
749:
745:
737:
733:
728:
722:
713:
710:
703:
699:
693:
688:
684:
677:
673:
666:
662:
655:
651:
647:
644:
639:
632:
628:
621:
617:
614:
613:
612:
609:
605:belonging to
603:
596:
592:
585:
580:
575:
573:
569:
565:
561:
557:
553:
547:
537:
535:
531:
523:
519:
515:
512:). Therefore
511:
507:
503:
499:
495:
492: +
491:
487:
483:
479:
475:
471:
467:
463:
459:
455:
454:
453:
451:
448:
442:
427:
425:
424:
419:
415:
410:
408:
404:
400:
396:
392:
388:
384:
378:
376:
372:
367:
363:
359:
355:
350:
347:
337:
335:
331:
327:
323:
318:
316:
312:
308:
304:
300:
296:
292:
287:
283:
279:
274:
272:
271:prime numbers
268:
264:
260:
259:number theory
256:
252:
248:
247:
242:
238:
234:
230:
226:
221:
219:
215:
211:
207:
203:
199:
195:
189:
187:
183:
179:
175:
169:
159:
157:
153:
149:
145:
141:
137:
133:
129:
125:
124:formal proofs
121:
118:
114:
110:
105:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
60:
56:
48:
47:
42:
38:
34:
30:
19:
6020:Proof theory
5859:Substitution
5679:Mathematical
5604:Major fields
5529:
5327:Ultraproduct
5174:Model theory
5139:Independence
5075:Formal proof
5067:Proof theory
5050:
5023:
4980:real numbers
4952:second-order
4863:Substitution
4740:Metalanguage
4681:conservative
4654:Axiom schema
4598:Constructive
4568:Morse–Kelley
4534:Set theories
4513:Aleph number
4506:inaccessible
4412:Grothendieck
4296:intersection
4183:Higher-order
4171:Second-order
4117:Truth tables
4074:Venn diagram
3857:Formal proof
3733:
3713:
3690:
3681:
3678:Gold, Bonnie
3658:
3654:Franklin, J.
3644:
3640:
3606:
3580:. Retrieved
3576:
3566:
3531:
3525:
3512:
3504:
3484:
3476:
3467:
3464:the original
3454:
3445:
3425:
3408:
3397:
3387:
3382:
3371:
3363:
3358:
3350:
3345:
3326:
3321:
3314:
3295:
3290:
3283:
3270:
3251:
3240:
3231:
3222:
3210:. Retrieved
3208:. p. 12
3205:
3192:
3169:
3156:
3124:
3118:
3112:
3104:
3086:
3077:
3065:. Retrieved
3061:
3051:
3028:
3019:
2981:
2978:Hacking, Ian
2972:
2963:
2955:
2937:
2931:
2908:
2899:
2890:
2886:
2880:
2868:. Retrieved
2854:
2831:
2728:
2720:
2716:
2714:
2697:
2681:
2653:
2646:Psychologism
2623:
2604:
2566:
2561:
2557:
2543:
2525:
2512:
2484:
2466:
2409:
2406:Visual proof
2372:
2355:
2344:
2329:
2315:
2280:cryptography
2273:
1887:
1883:
1873:
1852:between two
1847:
1817:
1813:
1802:
1783:transitivity
1780:
1545:
1542:
1523:
1493:
1451:
1421:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1361:
1357:
1353:
1349:
1308:
1304:
1201:
1199:
1162:
957:
952:
948:
937:
933:
925:
911:
904:
897:
892:
884:
880:
873:
869:
862:
858:
851:
843:
839:
835:
823:
815:
811:
804:
797:
792:
784:
765:
757:
752:
743:
735:
731:
720:
716:
708:
701:
697:
694:
686:
682:
675:
671:
664:
660:
653:
649:
645:
637:
635:is true for
630:
626:
619:
615:
607:
601:
594:
590:
583:
576:
549:
527:
517:
513:
509:
505:
501:
497:
493:
489:
485:
481:
477:
473:
469:
465:
461:
457:
444:
441:Direct proof
435:Direct proof
421:
411:
379:
371:proof theory
358:formal proof
351:
343:
322:proof theory
319:
302:
275:
257:also covers
254:
245:
222:
190:
185:
181:
177:
173:
171:
132:proof theory
106:
97:
54:
52:
44:
29:
5974:WikiProject
5844:Proposition
5839:Probability
5792:Description
5733:Foundations
5437:Type theory
5385:undecidable
5317:Truth value
5204:equivalence
4883:non-logical
4496:Enumeration
4486:Isomorphism
4433:cardinality
4417:Von Neumann
4382:Ultrafilter
4347:Uncountable
4281:equivalence
4198:Quantifiers
4188:Fixed-point
4157:First-order
4037:Consistency
4022:Proposition
3999:Traditional
3970:Lindström's
3960:Compactness
3902:Type theory
3847:Cardinality
3785:Wikiversity
3582:October 15,
3388:statistical
3212:October 20,
3091:Brooks/Cole
3067:October 20,
2740:Paul Halmos
2723:, which is
2611:probability
2562:assumptions
2554:probability
740:represent "
520:has 2 as a
5999:Categories
5904:Set theory
5802:Linguistic
5797:Entailment
5787:Definition
5755:Consequent
5750:Antecedent
5248:elementary
4941:arithmetic
4809:Quantifier
4787:functional
4659:Expression
4377:Transitive
4321:identities
4306:complement
4239:hereditary
4222:Set theory
3491:Icon Books
3366:94:165–86.
3353:79:252–63.
2846:References
2701:argument.
1890:such that
611:such that
564:infinitely
504: = 2(
418:Paul Erdős
305:(1000) by
286:Al-Hashimi
267:irrational
214:Theaetetus
166:See also:
102:conjecture
37:P. Oxy. 29
5935:Fallacies
5930:Paradoxes
5920:Logicians
5854:Statement
5849:Reference
5814:Induction
5777:Deduction
5740:Abduction
5710:Metalogic
5657:Classical
5621:Inference
5519:Supertask
5422:Recursion
5380:decidable
5214:saturated
5192:of models
5115:deductive
5110:axiomatic
5030:Hilbert's
5017:Euclidean
4998:canonical
4921:axiomatic
4853:Signature
4782:Predicate
4671:Extension
4593:Ackermann
4518:Operation
4397:Universal
4387:Recursive
4362:Singleton
4357:Inhabited
4342:Countable
4332:Types of
4316:power set
4286:partition
4203:Predicate
4149:Predicate
4064:Syllogism
4054:Soundness
4027:Inference
4017:Tautology
3919:paradoxes
3602:Pólya, G.
3185:p. 3
3149:121416910
2980:(1984) .
2907:(2005) .
2892:obtained.
2733:tombstone
2693:Descartes
2689:certainty
2615:certainty
1850:bijection
1760:φ
1756:⇒
1747:φ
1720:φ
1716:⇒
1708:−
1701:φ
1680:…
1654:φ
1650:⇒
1641:φ
1614:φ
1610:⇒
1601:φ
1574:φ
1567:…
1555:φ
1093:⋅
1025:is even:
748:is odd":
552:deduction
496: = 2
391:synthetic
307:Al-Karaji
303:Al-Fakhri
218:Aristotle
186:probieren
122:. Purely
90:empirical
82:inference
70:logically
59:deductive
5969:Category
5869:Validity
5770:Antinomy
5698:Theories
5662:Informal
5504:Logicism
5497:timeline
5473:Concrete
5332:Validity
5302:T-schema
5295:Kripke's
5290:Tarski's
5285:semantic
5275:Strength
5224:submodel
5219:spectrum
5187:function
5035:Tarski's
5024:Elements
5011:geometry
4967:Robinson
4888:variable
4873:function
4846:spectrum
4836:Sentence
4792:variable
4735:Language
4688:Relation
4649:Automata
4639:Alphabet
4623:language
4477:-jection
4455:codomain
4441:Function
4402:Universe
4372:Infinite
4276:Relation
4059:Validity
4049:Argument
3947:theorem,
3647:: 373–88
3604:(1954),
3558:23084607
3424:(2002).
3259:Archived
2750:See also
2717:"Q.E.D."
2634:evidence
2420:triangle
1478:fraction
1376:, where
1315:. Thus,
1029:Suppose
878:implies
856:is odd (
820:), then
809:is odd (
795:For any
789:is true.
691:is true.
450:integers
387:analytic
366:formulas
346:argument
255:Elements
246:Elements
194:geometry
117:rigorous
74:theorems
62:argument
46:Elements
5984:changes
5976: (
5834:Premise
5765:Paradox
5595:History
5590:Outline
5446:Related
5243:Diagram
5141: (
5120:Hilbert
5105:Systems
5100:Theorem
4978:of the
4923:systems
4703:Formula
4698:Grammar
4614: (
4558:General
4271:Forcing
4256:Element
4176:Monadic
3951:paradox
3892:Theorem
3828:General
3168:(ed.),
3129:Bibcode
2684:Spinoza
2656:Leibniz
2381:to the
1986:Either
1785:of the
867:). So
782:. Thus
729:. Let
560:implies
530:closure
320:Modern
210:Eudoxus
182:provare
174:probare
5886:topics
5672:Reason
5650:Logics
5641:Syntax
5209:finite
4972:Skolem
4925:
4900:Theory
4868:Symbol
4858:String
4841:atomic
4718:ground
4713:closed
4708:atomic
4664:ground
4627:syntax
4523:binary
4450:domain
4367:Finite
4132:finite
3990:Logics
3949:
3897:Theory
3781:course
3777:lesson
3741:
3720:
3701:
3684:. MAA.
3666:
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3497:
3438:
3333:
3302:
3176:
3147:
3097:
3039:
2992:
2948:
2919:
2737:eponym
2711:Q.E.D.
2698:cogito
2664:Carnap
2662:, and
2379:Euclid
2332:axioms
2286:, and
2055:), or
1947:Euclid
1303:where
1234:is an
947:: "if
930:infers
770:, and
581:: Let
522:factor
253:, the
237:axioms
225:Euclid
202:Thales
178:probar
78:axioms
64:for a
41:Euclid
5913:other
5878:Lists
5864:Truth
5631:Proof
5579:Logic
5199:Model
4947:Peano
4804:Proof
4644:Arity
4573:Naive
4460:image
4392:Fuzzy
4352:Empty
4301:union
4246:Class
3887:Model
3877:Lemma
3835:Axiom
3783:from
3695:Wiley
3554:S2CID
3522:(PDF)
3469:time.
3325:[
3294:[
3202:(PDF)
3145:S2CID
2725:Latin
2660:Frege
2558:using
2548:, or
1917:is a
1476:as a
1404:, so
1388:) = 4
978:, if
953:not p
951:then
949:not q
936:then
848:, so
801:, if
695:Then
403:Quine
293:. An
282:Iraqi
109:logic
57:is a
5978:talk
5824:Name
5809:Form
5322:Type
5125:list
4929:list
4906:list
4895:Term
4829:rank
4723:open
4617:list
4429:Maps
4334:sets
4193:Free
4163:list
3913:list
3840:list
3739:ISBN
3718:ISBN
3699:ISBN
3664:ISBN
3622:ISBN
3584:2009
3495:ISBN
3436:ISBN
3331:ISBN
3300:ISBN
3274:See
3214:2019
3174:ISBN
3095:ISBN
3069:2019
3037:ISBN
2990:ISBN
2946:ISBN
2917:ISBN
2872:2008
2727:for
2648:and
2599:and
2127:and
1886:and
1854:sets
1830:for
1738:and
1420:and
1412:and
1384:= (2
1307:and
893:Thus
793:(ii)
755:For
725:are
646:(ii)
484:and
472:and
460:and
447:even
395:Kant
356:. A
332:and
297:for
235:and
5720:Set
5009:of
4991:of
4939:of
4471:Sur
4445:Map
4252:Ur-
4234:Set
3612:hdl
3544:hdl
3536:doi
3137:doi
3125:500
2579:in
2290:or
1396:= 2
1372:= 2
955:".
887:+1)
865:+1)
787:(1)
760:= 1
753:(i)
727:odd
689:+1)
656:+1)
640:= 1
622:(1)
616:(i)
389:or
265:is
98:all
43:'s
6001::
5395:NP
5019::
5013::
4943::
4620:),
4475:Bi
4467:In
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3643:,
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2180:2
2173:2
2167:(
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2138:=
2135:b
2112:2
2105:2
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2096:a
2073:2
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2036:=
2033:b
2030:=
2027:a
2004:2
1997:2
1967:2
1960:2
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1570:,
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1418:a
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1333:=
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1323:b
1309:b
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1281:=
1276:2
1252:2
1220:2
1173:x
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1090:x
1087:=
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1078:x
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1037:x
1013:x
991:2
987:x
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908:.
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898:n
896:2
889:.
885:n
883:(
881:P
876:)
874:n
872:(
870:P
863:n
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859:P
852:n
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840:n
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830:2
824:n
818:)
816:n
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812:P
805:n
803:2
798:n
785:P
780:2
776:1
772:1
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744:n
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738:)
736:n
734:(
732:P
721:n
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712:.
709:n
704:)
702:n
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698:P
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683:P
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676:n
674:(
672:P
667:)
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650:P
642:.
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627:P
620:P
608:N
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593:(
591:P
584:N
518:y
516:+
514:x
510:b
508:+
506:a
502:b
498:a
494:y
490:x
486:b
482:a
478:b
474:y
470:a
466:x
462:y
458:x
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