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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.
2448:
22:
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3496:...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'...
2497:
5950:
2749:
2462:, 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.
3437:
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
2508:
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
1805:
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
337:
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
2307:
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
1869:
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
2677:
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
2312:
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
369:
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
1817:. 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
2882:
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
2219:
1513:
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
2521:, 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.
277:
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
3459:
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
3178:: "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."
357:
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
2555:
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
317:, 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
180:
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
1727:
2376:
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
1767:
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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
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2509:
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".
2337:
2116:
434:
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
2733:. 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 (∎)
2077:
2008:
1971:
1290:
228:, 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.
1483:
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.
177:(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.
3767:
2151:
2435:
2384:. Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of
238:, 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
2488:, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
551:
the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually
4975:
1839:
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a
547:. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case
1809:
A probabilistic proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work toward the
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3392:"these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E"
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2600:, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to
362:
studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certain
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3510:"Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century"
185:, which originated in practical problems of land measurement. The development of mathematical proof is primarily the product of
5560:
2558:
1345:. Since the expression on the left is an integer multiple of 2, the right expression is by definition divisible by 2. That is,
566:
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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3377:"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
1469:, 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
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provides two different expressions for the size of a single set, again showing that the two expressions are equal.
209:(384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known.
202:
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38:, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.
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The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with
556:
144:
140:
5345:
3960:
2645:
Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as
1806:
not specify which candidates have the property, but the probability could not be positive without at least one.
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65:; 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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391:
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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
374:, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are
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An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in
2459:
1860:
1794:
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of
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4440:
4407:
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4059:
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3119:
3108:
Matvievskaya, Galina (1987), "The Theory of
Quadratic Irrationals in Medieval Oriental Mathematics",
2829:
2812:
2620:
2485:
2401:
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "
2013:
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which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of
34:
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1307:
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The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".
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82:
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2285:. It is less commonly used to refer to a mathematical proof in the branch of mathematics known as
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2574:. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as
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Mathematicians have shown there are many statements that are neither provable nor disprovable in
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1234:
1202:
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3366:"in number theory and commutative algebra... in particular the statistical proof of the lemma."
3282:
Mathematik für das
Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis
415:, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.
89:
possible cases. A proposition that has not been proved but is believed to be true is known as a
61:
guarantee the conclusion. The argument may use other previously established statements, such as
3287:
Mathematics for the
Bachelor's degree I: Fundamentals and basics of linear algebra and analysis
3163:, Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, pp. 1–78,
2340:(ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see
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3689:
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3426:
3321:
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279:
255:
198:
186:
3628:"What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians"
3049:"The genesis of proof in ancient Greece The pedagogical implications of a Husserlian reading"
165:(to test). Related modern words are English "probe", "probation", and "probability", Spanish
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are both even, they have 2 as a common factor. This contradicts our previous statement that
560:
299:
217:
156:
101:
2854:
2722:. A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "
1884:
1127:
1100:
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5448:
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4568:
4536:
4337:
4139:
4082:
4032:
3997:
3955:
3408:
3340:
Davis, Philip J. (1972), "Fidelity in
Mathematical Discourse: Is One and One Really Two?"
3253:
2687:
2597:
2587:
2426:
1909:
371:
350:
229:
3592:
3048:
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shows that many axiom systems of mathematical interest will have undecidable statements.
3123:
1534:
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
2477:
2278:
1847:
is used to show that the expressions for their two sizes are equal. Alternatively, a
1199:
occurs, hence the statement must be false. A famous example involves the proof that
1196:
406:
383:
247:
139:, oral traditions in the mainstream mathematical community or in other cultures. The
93:, or a hypothesis if frequently used as an assumption for further mathematical work.
3548:
1195:(by reduction to the absurd), it is shown that if some statement is assumed true, a
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359:
346:
310:
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259:
124:
120:
112:
2214:{\displaystyle \left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{2}=2}
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5829:
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3837:
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3156:
3152:
3073:
2968:
2730:
2601:
2544:
1798:. Probabilistic proof, like proof by construction, is one of many ways to prove
54:
2405:". The left-hand picture below is an example of a historic visual proof of the
5777:
5745:
5710:
4357:
4212:
4183:
3989:
3530:
3454:
2744:
2578:, when the data or diagram is adequately convincing without further analysis.
1813:
shows how far plausibility is from genuine proof, as does the disproof of the
706:
For example, we can prove by induction that all positive integers of the form
274:
197:(c. 470–410 BCE) gave some of the first known proofs of theorems in geometry.
90:
513:
and, by definition, is even. Hence, the sum of any two even integers is even.
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5611:
5509:
5412:
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3682:
How to Read and Do Proofs: An
Introduction to Mathematical Thought Processes
2683:
2679:
2605:
2496:
1840:
947:
For example, contraposition can be used to establish that, given an integer
295:
206:
70:
3438:
the conventions of that day dictated that journals only published theorems.
3236:
Examples of simple proofs by mathematical induction for all natural numbers
517:
This proof uses the definition of even integers, the integer properties of
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3002:
21:
5760:
5494:
5292:
4740:
4445:
4039:
3539:
2624:
2410:
1466:
334:
333:
As practiced, a proof is expressed in natural language and is a rigorous
182:
2710:
is written to indicate the end of a proof. This abbreviation stands for
5824:
5755:
5090:
3882:
3081:
2674:
2535:"Statistical proof" from data refers to the application of statistics,
2329:, which is neither provable nor refutable from the remaining axioms of
1301:
510:
438:
62:
5662:
3780:
3627:
2727:
2701:
2654:
2369:
1937:
1499:
to disprove a proposition that all elements have a certain property.
213:
190:
105:
29:
3367:
2900:
The Nuts and Bolts of Proofs: An
Introduction to Mathematical Proofs
1583:
are each pairwise equivalent, proofs are given for the implications
1369:
is also an integer. Substitution into the original equation yields 2
1255:
were a rational number. Then it could be written in lowest terms as
3353:
Fallis, Don (1997), "The
Epistemic Status of Probabilistic Proof."
2441:
Animated visual proof for the
Pythagorean theorem by rearrangement.
2321:
A statement that is neither provable nor disprovable from a set of
5854:
5569:
4634:
3980:
3825:
3318:
Mathematics for Computer Scientists: Fundamentals and Applications
2715:
2495:
2322:
225:
97:
66:
58:
20:
2958:"proof" New Shorter Oxford English Dictionary, 1993, OUP, Oxford.
2290:
1772:
The pairwise equivalence of the statements then results from the
5814:
143:
is concerned with the role of language and logic in proofs, and
5542:
3784:
3078:
An Introduction to the History of Mathematics (Saunders Series)
3047:
Moutsios-Rentzos, Andreas; Spyrou, Panagiotis (February 2015).
2484:. For some time it was thought that certain theorems, like the
1973:
is irrational (this is true, but the proof is not elementary).
119:
without the involvement of natural language, are considered in
2661:, whereby standards of mathematical proof might be applied to
2325:
is called undecidable (from those axioms). One example is the
353:
instead of natural language. A formal proof is a sequence of
3256:, University of Warwick Glossary of Mathematical Terminology
2878:
The Concise Oxford Dictionary of Mathematics, Fourth edition
77:
which establish logical certainty, to be distinguished from
2669:
Influence of mathematical proof methods outside mathematics
757: − 1 = 2(1) − 1 = 1
3775:
1189:
In proof by contradiction, also known by the Latin phrase
205:(417–369 BCE) formulated theorems but did not prove them.
2682:
of propositions deduced in a mathematical proof, such as
588:
be a mathematical statement involving the natural number
169:(to smell or taste, or sometimes touch or test), Italian
5538:
3310:
Struckmann, Werner; Wätjen, Dietmar (October 20, 2016).
1353:
must also be even, as seen in the proposition above (in
539:
Despite its name, mathematical induction is a method of
5974:
3313:
Mathematik für Informatiker: Grundlagen und Anwendungen
3096:
No work, except The Bible, has been more widely used...
3762:
Proofs in Mathematics: Simple, Charming and Fallacious
3279:
Plaue, Matthias; Scherfner, Mike (February 11, 2019).
1722:{\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}}
1389:. But then, by the same argument as before, 2 divides
390:, believed mathematical proofs are synthetic, whereas
127:
has led to much examination of current and historical
2227:
2154:
2124:
2085:
2052:
2016:
1983:
1946:
1918:
1887:
1735:
1689:
1669:
1629:
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1447:
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1310:
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1237:
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1103:
1064:
1058:
is odd. The product of two odd numbers is odd, hence
1044:
1024:
1000:
973:
953:
3673:
Proof and Other Dilemmas: Mathematics and Philosophy
2338:
Zermelo–Fraenkel set theory with the axiom of choice
1762:{\displaystyle \varphi _{n}\Rightarrow \varphi _{1}}
1656:{\displaystyle \varphi _{2}\Rightarrow \varphi _{3}}
1616:{\displaystyle \varphi _{1}\Rightarrow \varphi _{2}}
5903:
5875:
5868:
5723:
5688:
5640:
5594:
5436:
5331:
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5056:
4908:
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4524:
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4138:
4073:
3988:
3979:
3901:
3818:
2453:
A second animated proof of the Pythagorean theorem.
341:The concept of proof is formalized in the field of
3473:
3267:Four color theorem#Simplification and verification
2678:statements outside of mathematics, but having the
2354:Heuristic mathematics and experimental mathematics
2243:
2213:
2140:
2110:
2071:
2038:
2002:
1965:
1928:
1900:
1761:
1721:
1675:
1655:
1615:
1575:
1457:
1429:
1329:
1284:
1247:
1215:
1166:
1143:
1116:
1089:
1050:
1030:
1006:
986:
959:
1576:{\displaystyle \varphi _{1},\ldots ,\varphi _{n}}
3320:] (in German). Springer-Verlag. p. 28.
3289:] (in German). Springer-Verlag. p. 26.
2425:Visual proof for the (3,4,5) triangle as in the
1413:have no common factor, so we must conclude that
821:to an odd number results in an odd number. But
2855:"One of the Oldest Extant Diagrams from Euclid"
559:, which can be used, for example, to prove the
3155:(1998), "An introduction to proof theory", in
2608:. Inductive logic should not be confused with
2221:, which is thus a rational number of the form
1825:) are as good as genuine mathematical proofs.
453:. Since they are even, they can be written as
100:expressed in mathematical symbols, along with
5554:
3796:
3003:The History and Concept of Mathematical Proof
2947:Definition 3.1. Proof: An Informal Definition
8:
2582:Inductive logic proofs and Bayesian analysis
1936:is irrational (an easy proof is known since
1354:
2010:is a rational number and we are done (take
1865:A nonconstructive proof establishes that a
827: − 1) + 2 = 2
521:under addition and multiplication, and the
258:and a proof that there are infinitely many
28:, one of the oldest surviving fragments of
5872:
5637:
5561:
5547:
5539:
4622:
4217:
3985:
3803:
3789:
3781:
3197:Universität Zürich – Theologische Fakultät
3111:Annals of the New York Academy of Sciences
3606:
3538:
3418:Indra's Pearls: The Vision of Felix Klein
2458:Some illusory visual proofs, such as the
2232:
2226:
2199:
2192:
2180:
2168:
2161:
2153:
2131:
2123:
2111:{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}
2100:
2093:
2084:
2061:
2054:
2051:
2029:
2015:
1992:
1985:
1982:
1955:
1948:
1945:
1919:
1917:
1892:
1886:
1753:
1740:
1734:
1713:
1694:
1688:
1668:
1647:
1634:
1628:
1607:
1594:
1588:
1567:
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1238:
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1159:
1135:
1129:
1108:
1102:
1069:
1063:
1043:
1023:
999:
978:
972:
952:
577:} be the set of natural numbers, and let
212:Mathematical proof was revolutionized by
2998:
2996:
1487:, for instance, proved the existence of
1154:even, the supposition must be false, so
398:" that such a distinction is untenable.
189:, and one of its greatest achievements.
3005:, Steven G. Krantz. 1. February 5, 2007
2842:
2480:to refer to proofs that make no use of
2415:
2380:beyond the proof-theorem framework, in
2072:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
2003:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
1966:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
1285:{\displaystyle {\sqrt {2}}={a \over b}}
763:is odd, since it leaves a remainder of
561:irrationality of the square root of two
3705:How to Prove It: A Structured Approach
3564:"Introduction to the Two-Column Proof"
2513:Colloquial use of "mathematical proof"
2348:Gödel's (first) incompleteness theorem
2255:Statistical proofs in pure mathematics
1537:In order to prove that the statements
1495:. It can also be used to construct a
161:The word "proof" comes from the Latin
57:, showing that the stated assumptions
3650:Proof in Mathematics: An Introduction
2342:List of statements undecidable in ZFC
338:vague or incomplete may be rejected.
313:treats proofs as inductively defined
16:Reasoning for mathematical statements
7:
2543:to infer propositions regarding the
2500:A two-column proof published in 1913
2388:, which was ultimately so resolved.
265:Further advances also took place in
73:. Proofs are examples of exhaustive
5736:Analytic and synthetic propositions
5607:Formal semantics (natural language)
3598:Mathematics and Plausible Reasoning
3451:"A Note on the History of Fractals"
2720:"that which was to be demonstrated"
2673:Philosopher-mathematicians such as
2364:While early mathematicians such as
220:still in use today. It starts with
69:, along with the accepted rules of
3518:Educational Studies in Mathematics
3132:10.1111/j.1749-6632.1987.tb37206.x
2876:Clapham, C. & Nicholson, J.N.
2824:What the Tortoise Said to Achilles
1441:To paraphrase: if one could write
1381:. Dividing both sides by 2 yields
891:is odd, for all positive integers
469:, respectively, for some integers
14:
3188:Quine, Willard Van Orman (1961).
817:must also be odd, because adding
5948:
5522:
3749:
2857:. University of British Columbia
2761:
2747:
2446:
2434:
2418:
695:is true for all natural numbers
401:Proofs may be admired for their
366:not provable within the system.
2929:Discrete Mathematics with Proof
2559:mathematical methods of physics
2039:{\displaystyle a=b={\sqrt {2}}}
529:Proof by mathematical induction
133:quasi-empiricism in mathematics
3707:, Cambridge University Press,
3601:, Princeton University Press,
3472:Lesmoir-Gordon, Nigel (2000).
3018:; Kneale, Martha (May 1985) .
2079:is irrational so we can write
1746:
1706:
1640:
1600:
1337:. Squaring both sides yields 2
1330:{\displaystyle b{\sqrt {2}}=a}
1090:{\displaystyle x^{2}=x\cdot x}
388:analytic–synthetic distinction
269:. In the 10th century CE, the
216:(300 BCE), who introduced the
1:
5483:History of mathematical logic
3342:American Mathematical Monthly
2562:applied to analyze data in a
2141:{\displaystyle b={\sqrt {2}}}
831: + 1 = 2(
815: − 1) + 2
250:, including a proof that the
5408:Primitive recursive function
3671:; Simons, Rogers A. (2008).
3508:Herbst, Patricio G. (2002).
3476:Introducing Fractal Geometry
2706:Sometimes, the abbreviation
2627:or information is acquired.
2604:, and may be less than full
2525:Statistical proof using data
2291:Statistical proof using data
1349:is even, which implies that
267:medieval Islamic mathematics
81:arguments or non-exhaustive
2927:Gossett, Eric (July 2009).
2798:List of mathematical proofs
2409:in the case of the (3,4,5)
2279:probabilistic number theory
1929:{\displaystyle {\sqrt {2}}}
1458:{\displaystyle {\sqrt {2}}}
1430:{\displaystyle {\sqrt {2}}}
1300:are non-zero integers with
1248:{\displaystyle {\sqrt {2}}}
1216:{\displaystyle {\sqrt {2}}}
445:Consider two even integers
298:, who used it to prove the
6034:
4472:Schröder–Bernstein theorem
4199:Monadic predicate calculus
3858:Foundations of mathematics
3423:Cambridge University Press
3252:February 18, 2012, at the
3190:"Two Dogmas of Empiricism"
2977:Cambridge University Press
2699:
2634:
2585:
2528:
2469:
2357:
2300:
2258:
1858:
1832:
1787:
1506:
1476:
1397:must be even. However, if
1182:
908:
532:
427:
154:
125:formal and informal proofs
123:. The distinction between
5943:
5820:Necessity and sufficiency
5576:
5518:
5505:Philosophy of mathematics
5454:Automated theorem proving
4625:
4579:Von Neumann–Bernays–Gödel
4220:
3722:Hammack, Richard (2018),
3381:proof"" (Derogatory use.)
2788:List of incomplete proofs
2778:Automated theorem proving
2712:"quod erat demonstrandum"
2621:assessment of likelihoods
2368:did not use proofs, from
557:proof by infinite descent
187:ancient Greek mathematics
145:mathematics as a language
141:philosophy of mathematics
6001:Mathematical terminology
3161:Handbook of Proof Theory
3020:The development of logic
2726:" or "halmos" after its
2631:Proofs as mental objects
2382:experimental mathematics
2374:foundational mathematics
2360:Experimental mathematics
2297:Computer-assisted proofs
1849:double counting argument
1437:is an irrational number.
1355:#Proof by contraposition
934:contrapositive statement
396:Two Dogmas of Empiricism
5155:Self-verifying theories
4976:Tarski's axiomatization
3927:Tarski's undefinability
3922:incompleteness theorems
3608:2027/mdp.39015008206248
3531:10.1023/A:1020264906740
3024:Oxford University Press
2735:(220E(hex) = 8718(dec))
2623:of hypotheses when new
2615:Bayesian analysis uses
2303:Computer-assisted proof
2287:mathematical statistics
1912:. This proof uses that
1819:probabilistic algorithm
916:Proof by contraposition
905:Proof by contraposition
5529:Mathematics portal
5140:Proof of impossibility
4788:propositional variable
4098:Propositional calculus
3568:onemathematicalcat.org
3457:on February 15, 2009.
2610:mathematical induction
2501:
2317:Undecidable statements
2283:analytic number theory
2245:
2244:{\displaystyle a^{b}.}
2215:
2142:
2112:
2073:
2040:
2004:
1967:
1930:
1902:
1763:
1723:
1677:
1676:{\displaystyle \dots }
1657:
1617:
1577:
1529:Closed chain inference
1522:Closed chain inference
1489:transcendental numbers
1459:
1431:
1331:
1286:
1249:
1217:
1185:Proof by contradiction
1179:Proof by contradiction
1168:
1145:
1124:is not even. Thus, if
1118:
1091:
1052:
1032:
1008:
988:
961:
929:" by establishing the
535:Mathematical induction
364:undecidable statements
323:Non-Euclidean geometry
290:was introduced in the
55:mathematical statement
39:
5955:Philosophy portal
5398:Kolmogorov complexity
5351:Computably enumerable
5251:Model complete theory
5043:Principia Mathematica
4103:Propositional formula
3932:Banach–Tarski paradox
3703:Velleman, D. (2006),
3355:Journal of Philosophy
2933:John Wiley & Sons
2808:Proof by intimidation
2803:Nonconstructive proof
2619:to update a person's
2506:mathematical exercise
2499:
2460:missing square puzzle
2246:
2216:
2143:
2113:
2074:
2041:
2005:
1968:
1931:
1903:
1901:{\displaystyle a^{b}}
1861:Nonconstructive proof
1855:Nonconstructive proof
1764:
1724:
1678:
1658:
1618:
1578:
1479:Proof by construction
1473:Proof by construction
1460:
1432:
1332:
1287:
1250:
1218:
1197:logical contradiction
1169:
1146:
1144:{\displaystyle x^{2}}
1119:
1117:{\displaystyle x^{2}}
1092:
1053:
1033:
1009:
989:
987:{\displaystyle x^{2}}
962:
669:is true implies that
523:distributive property
386:, who introduced the
232:. Euclid's book, the
173:(to try), and German
151:History and etymology
129:mathematical practice
24:
6016:Sources of knowledge
5346:Church–Turing thesis
5333:Computability theory
4542:continuum hypothesis
4060:Square of opposition
3918:Gödel's completeness
3758:at Wikimedia Commons
3647:; Daoud, A. (2011),
3626:Fallis, Don (2002),
3174:. See in particular
2896:Cupillari, Antonella
2830:Zero-knowledge proof
2813:Termination analysis
2519:mathematical objects
2486:prime number theorem
2378:mathematical objects
2269:, such as involving
2225:
2152:
2122:
2083:
2050:
2014:
1981:
1944:
1916:
1885:
1790:Probabilistic method
1778:material conditional
1733:
1687:
1667:
1627:
1587:
1541:
1445:
1417:
1308:
1259:
1235:
1203:
1192:reductio ad absurdum
1158:
1128:
1101:
1062:
1042:
1022:
998:
971:
951:
931:logically equivalent
412:Proofs from THE BOOK
405:. The mathematician
394:argued in his 1951 "
319:Axiomatic set theory
288:arithmetic sequences
195:Hippocrates of Chios
6006:Mathematical proofs
5617:Philosophy of logic
5500:Mathematical object
5391:P versus NP problem
5356:Computable function
5150:Reverse mathematics
5076:Logical consequence
4953:primitive recursive
4948:elementary function
4721:Free/bound variable
4574:Tarski–Grothendieck
4093:Logical connectives
4023:Logical equivalence
3873:Logical consequence
3770:about proofs, in a
3728:, Richard Hammack,
3124:1987NYASA.500..253M
3053:Archive ouverte HAL
2793:List of long proofs
2659:language of thought
2641:Language of thought
2568:observational study
2407:Pythagorean theorem
2403:proof without words
1867:mathematical object
1835:Combinatorial proof
1829:Combinatorial proof
1784:Probabilistic proof
1509:Proof by exhaustion
1503:Proof by exhaustion
1491:by constructing an
1357:). So we can write
545:inductive reasoning
403:mathematical beauty
240:Pythagorean theorem
115:, written fully in
83:inductive reasoning
75:deductive reasoning
5996:Mathematical logic
5916:Rules of inference
5885:Mathematical logic
5627:Semantics of logic
5298:Transfer principle
5261:Semantics of logic
5246:Categorical theory
5222:Non-standard model
4736:Logical connective
3863:Information theory
3812:Mathematical logic
3756:Mathematical proof
3680:Solow, D. (2004),
3632:Logique et Analyse
3562:Dr. Fisher Burns.
3247:Proof by induction
2902:(Third ed.).
2818:Thought experiment
2769:Mathematics portal
2572:physical cosmology
2502:
2331:Euclidean geometry
2327:parallel postulate
2310:four color theorem
2241:
2211:
2148:. This then gives
2138:
2108:
2069:
2036:
2000:
1963:
1926:
1898:
1872:irrational numbers
1815:Mertens conjecture
1811:Collatz conjecture
1800:existence theorems
1796:probability theory
1759:
1719:
1673:
1653:
1613:
1573:
1516:four color theorem
1455:
1427:
1327:
1282:
1245:
1213:
1164:
1141:
1114:
1087:
1048:
1038:is not even. Then
1028:
1004:
984:
957:
921:the statement "if
575:= {1, 2, 3, 4, ...
477:. Then the sum is
343:mathematical logic
329:Nature and purpose
302:and properties of
280:irrational numbers
252:square root of two
201:(408–355 BCE) and
193:(624–546 BCE) and
44:mathematical proof
40:
5983:
5982:
5939:
5938:
5773:Deductive closure
5719:
5718:
5658:Critical thinking
5536:
5535:
5468:Abstract category
5271:Theories of truth
5081:Rule of inference
5071:Natural deduction
5052:
5051:
4597:
4596:
4302:Cartesian product
4207:
4206:
4113:Many-valued logic
4088:Boolean functions
3971:Russell's paradox
3946:diagonal argument
3843:First-order logic
3754:Media related to
3735:978-0-9894721-3-5
3714:978-0-521-67599-4
3695:978-0-471-68058-1
3660:978-0-646-54509-7
3491:978-1-84046-123-7
3432:978-0-521-35253-6
3405:Mumford, David B.
3327:978-3-662-49870-5
3296:978-3-662-58352-4
3226:Cupillari, p. 46.
3217:Cupillari, p. 20.
3170:978-0-08-053318-6
3076:(January 1990) .
3033:978-0-19-824773-9
2986:978-0-521-31803-7
2913:978-0-12-088509-1
2755:Philosophy portal
2736:
2663:empirical science
2592:Bayesian analysis
2541:Bayesian analysis
2531:Statistical proof
2429:500–200 BCE.
2366:Eudoxus of Cnidus
2293:" section below.
2289:. See also the "
2261:Statistical proof
2197:
2185:
2173:
2166:
2136:
2105:
2098:
2066:
2059:
2034:
1997:
1990:
1960:
1953:
1924:
1823:testing primality
1453:
1425:
1319:
1280:
1267:
1243:
1225:irrational number
1211:
1167:{\displaystyle x}
1051:{\displaystyle x}
1031:{\displaystyle x}
1007:{\displaystyle x}
960:{\displaystyle x}
843:+1) − 1
835:+1) − 1
647:is true whenever
304:Pascal's triangle
117:symbolic language
6023:
5953:
5952:
5951:
5873:
5638:
5602:Computer science
5563:
5556:
5549:
5540:
5527:
5526:
5478:History of logic
5473:Category of sets
5366:Decision problem
5145:Ordinal analysis
5086:Sequent calculus
4984:Boolean algebras
4924:
4923:
4898:
4869:logical/constant
4623:
4609:
4532:Zermelo–Fraenkel
4283:Set operations:
4218:
4155:
3986:
3966:Löwenheim–Skolem
3853:Formal semantics
3805:
3798:
3791:
3782:
3753:
3738:
3717:
3698:
3676:
3663:
3639:
3621:
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3579:
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3553:
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3514:
3505:
3499:
3498:
3479:
3469:
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3462:
3453:. Archived from
3447:
3441:
3440:
3409:Series, Caroline
3401:
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3390:
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3149:
3143:
3142:
3105:
3099:
3098:
3080:(6th ed.).
3070:
3064:
3063:
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3044:
3038:
3037:
3022:(New ed.).
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2771:
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2750:
2734:
2564:particle physics
2492:Two-column proof
2482:complex analysis
2472:Elementary proof
2466:Elementary proof
2450:
2438:
2422:
2392:Related concepts
2386:fractal geometry
2267:pure mathematics
2250:
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2218:
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1961:
1956:
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1940:), but not that
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1552:
1493:explicit example
1485:Joseph Liouville
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1315:
1302:no common factor
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1035:
1034:
1029:
1013:
1011:
1010:
1005:
993:
991:
990:
985:
983:
982:
966:
964:
963:
958:
896:
890:
877:
866:
855:
844:
836:
820:
816:
808:
797:
789:
777:
770:
767:when divided by
766:
762:
758:
750:
736:
728:
713:
700:
694:
679:
668:
657:
646:
630:
623:
612:
599:
593:
587:
576:
543:, not a form of
441:is always even:
419:Methods of proof
372:proof assistants
349:is written in a
300:binomial theorem
218:axiomatic method
157:History of logic
137:folk mathematics
135:, and so-called
102:natural language
6033:
6032:
6026:
6025:
6024:
6022:
6021:
6020:
5986:
5985:
5984:
5979:
5949:
5947:
5935:
5899:
5890:Boolean algebra
5864:
5715:
5706:Metamathematics
5684:
5636:
5590:
5572:
5567:
5537:
5532:
5521:
5514:
5459:Category theory
5449:Algebraic logic
5432:
5403:Lambda calculus
5341:Church encoding
5327:
5303:Truth predicate
5159:
5125:Complete theory
5048:
4917:
4913:
4909:
4904:
4896:
4616: and
4612:
4607:
4593:
4569:New Foundations
4537:axiom of choice
4520:
4482:Gödel numbering
4422: and
4414:
4318:
4203:
4153:
4134:
4083:Boolean algebra
4069:
4033:Equiconsistency
3998:Classical logic
3975:
3956:Halting problem
3944: and
3920: and
3908: and
3907:
3902:Theorems (
3897:
3814:
3809:
3746:
3736:
3721:
3715:
3702:
3696:
3679:
3667:
3661:
3643:
3625:
3619:
3591:
3588:
3586:Further reading
3583:
3582:
3572:
3570:
3561:
3560:
3556:
3512:
3507:
3506:
3502:
3492:
3471:
3470:
3466:
3449:
3448:
3444:
3433:
3403:
3402:
3398:
3391:
3387:
3376:
3372:
3365:
3361:
3352:
3348:
3339:
3335:
3328:
3309:
3308:
3304:
3297:
3278:
3277:
3273:
3264:
3260:
3254:Wayback Machine
3245:
3241:
3234:
3230:
3225:
3221:
3216:
3212:
3202:
3200:
3192:
3187:
3186:
3182:
3171:
3157:Buss, Samuel R.
3153:Buss, Samuel R.
3151:
3150:
3146:
3107:
3106:
3102:
3092:
3084:. p. 141.
3074:Eves, Howard W.
3072:
3071:
3067:
3057:
3055:
3046:
3045:
3041:
3034:
3016:Kneale, William
3014:
3013:
3009:
3001:
2994:
2987:
2967:
2966:
2962:
2957:
2953:
2943:
2926:
2925:
2921:
2914:
2894:
2893:
2889:
2875:
2874:
2870:
2860:
2858:
2849:
2848:
2844:
2839:
2834:
2767:
2760:
2753:
2748:
2746:
2743:
2704:
2698:
2671:
2643:
2635:Main articles:
2633:
2598:inductive logic
2594:
2588:Inductive logic
2586:Main articles:
2584:
2547:of data. While
2533:
2527:
2515:
2494:
2474:
2468:
2454:
2451:
2442:
2439:
2430:
2427:Zhoubi Suanjing
2423:
2399:
2394:
2362:
2356:
2319:
2305:
2299:
2263:
2257:
2228:
2223:
2222:
2191:
2160:
2156:
2155:
2150:
2149:
2120:
2119:
2092:
2081:
2080:
2053:
2048:
2047:
2012:
2011:
1984:
1979:
1978:
1947:
1942:
1941:
1914:
1913:
1910:rational number
1888:
1883:
1882:
1863:
1857:
1837:
1831:
1792:
1786:
1749:
1736:
1731:
1730:
1709:
1690:
1685:
1684:
1665:
1664:
1643:
1630:
1625:
1624:
1603:
1590:
1585:
1584:
1563:
1544:
1539:
1538:
1524:
1511:
1505:
1481:
1475:
1443:
1442:
1415:
1414:
1306:
1305:
1257:
1256:
1233:
1232:
1201:
1200:
1187:
1181:
1174:has to be even.
1156:
1155:
1131:
1126:
1125:
1104:
1099:
1098:
1065:
1060:
1059:
1040:
1039:
1020:
1019:
996:
995:
974:
969:
968:
949:
948:
913:
907:
892:
884:
868:
857:
846:
838:
822:
818:
810:
799:
791:
785:
772:
768:
764:
760:
752:
745:
730:
719:
707:
696:
685:
670:
659:
658:is true, i.e.,
648:
637:
625:
614:
613:is true, i.e.,
607:
595:
589:
578:
571:
568:natural numbers
537:
531:
432:
426:
421:
351:formal language
331:
315:data structures
284:inductive proof
230:deductive logic
222:undefined terms
159:
153:
17:
12:
11:
5:
6031:
6030:
6027:
6019:
6018:
6013:
6008:
6003:
5998:
5988:
5987:
5981:
5980:
5978:
5977:
5972:
5962:
5957:
5944:
5941:
5940:
5937:
5936:
5934:
5933:
5928:
5923:
5918:
5913:
5907:
5905:
5901:
5900:
5898:
5897:
5892:
5887:
5881:
5879:
5870:
5866:
5865:
5863:
5862:
5857:
5852:
5847:
5842:
5837:
5832:
5827:
5822:
5817:
5812:
5807:
5802:
5797:
5796:
5795:
5785:
5780:
5775:
5770:
5765:
5764:
5763:
5758:
5748:
5743:
5738:
5733:
5727:
5725:
5721:
5720:
5717:
5716:
5714:
5713:
5708:
5703:
5698:
5692:
5690:
5686:
5685:
5683:
5682:
5677:
5672:
5667:
5666:
5665:
5660:
5650:
5644:
5642:
5635:
5634:
5629:
5624:
5619:
5614:
5609:
5604:
5598:
5596:
5592:
5591:
5589:
5588:
5583:
5577:
5574:
5573:
5568:
5566:
5565:
5558:
5551:
5543:
5534:
5533:
5519:
5516:
5515:
5513:
5512:
5507:
5502:
5497:
5492:
5491:
5490:
5480:
5475:
5470:
5461:
5456:
5451:
5446:
5444:Abstract logic
5440:
5438:
5434:
5433:
5431:
5430:
5425:
5423:Turing machine
5420:
5415:
5410:
5405:
5400:
5395:
5394:
5393:
5388:
5383:
5378:
5373:
5363:
5361:Computable set
5358:
5353:
5348:
5343:
5337:
5335:
5329:
5328:
5326:
5325:
5320:
5315:
5310:
5305:
5300:
5295:
5290:
5289:
5288:
5283:
5278:
5268:
5263:
5258:
5256:Satisfiability
5253:
5248:
5243:
5242:
5241:
5231:
5230:
5229:
5219:
5218:
5217:
5212:
5207:
5202:
5197:
5187:
5186:
5185:
5180:
5173:Interpretation
5169:
5167:
5161:
5160:
5158:
5157:
5152:
5147:
5142:
5137:
5127:
5122:
5121:
5120:
5119:
5118:
5108:
5103:
5093:
5088:
5083:
5078:
5073:
5068:
5062:
5060:
5054:
5053:
5050:
5049:
5047:
5046:
5038:
5037:
5036:
5035:
5030:
5029:
5028:
5023:
5018:
4998:
4997:
4996:
4994:minimal axioms
4991:
4980:
4979:
4978:
4967:
4966:
4965:
4960:
4955:
4950:
4945:
4940:
4927:
4925:
4906:
4905:
4903:
4902:
4901:
4900:
4888:
4883:
4882:
4881:
4876:
4871:
4866:
4856:
4851:
4846:
4841:
4840:
4839:
4834:
4824:
4823:
4822:
4817:
4812:
4807:
4797:
4792:
4791:
4790:
4785:
4780:
4770:
4769:
4768:
4763:
4758:
4753:
4748:
4743:
4733:
4728:
4723:
4718:
4717:
4716:
4711:
4706:
4701:
4691:
4686:
4684:Formation rule
4681:
4676:
4675:
4674:
4669:
4659:
4658:
4657:
4647:
4642:
4637:
4632:
4626:
4620:
4603:Formal systems
4599:
4598:
4595:
4594:
4592:
4591:
4586:
4581:
4576:
4571:
4566:
4561:
4556:
4551:
4546:
4545:
4544:
4539:
4528:
4526:
4522:
4521:
4519:
4518:
4517:
4516:
4506:
4501:
4500:
4499:
4492:Large cardinal
4489:
4484:
4479:
4474:
4469:
4455:
4454:
4453:
4448:
4443:
4428:
4426:
4416:
4415:
4413:
4412:
4411:
4410:
4405:
4400:
4390:
4385:
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4329:
4327:
4320:
4319:
4317:
4316:
4315:
4314:
4309:
4304:
4299:
4294:
4289:
4281:
4280:
4279:
4274:
4264:
4259:
4257:Extensionality
4254:
4252:Ordinal number
4249:
4239:
4234:
4233:
4232:
4221:
4215:
4209:
4208:
4205:
4204:
4202:
4201:
4196:
4191:
4186:
4181:
4176:
4171:
4170:
4169:
4159:
4158:
4157:
4144:
4142:
4136:
4135:
4133:
4132:
4131:
4130:
4125:
4120:
4110:
4105:
4100:
4095:
4090:
4085:
4079:
4077:
4071:
4070:
4068:
4067:
4062:
4057:
4052:
4047:
4042:
4037:
4036:
4035:
4025:
4020:
4015:
4010:
4005:
4000:
3994:
3992:
3983:
3977:
3976:
3974:
3973:
3968:
3963:
3958:
3953:
3948:
3936:Cantor's
3934:
3929:
3924:
3914:
3912:
3899:
3898:
3896:
3895:
3890:
3885:
3880:
3875:
3870:
3865:
3860:
3855:
3850:
3845:
3840:
3835:
3834:
3833:
3822:
3820:
3816:
3815:
3810:
3808:
3807:
3800:
3793:
3785:
3779:
3778:
3764:
3759:
3745:
3744:External links
3742:
3741:
3740:
3734:
3719:
3713:
3700:
3694:
3677:
3665:
3659:
3641:
3623:
3617:
3587:
3584:
3581:
3580:
3554:
3525:(3): 283–312.
3500:
3490:
3464:
3442:
3431:
3396:
3385:
3370:
3359:
3346:
3333:
3326:
3302:
3295:
3271:
3258:
3239:
3228:
3219:
3210:
3180:
3169:
3144:
3118:(1): 253–77 ,
3100:
3091:978-0030295584
3090:
3065:
3039:
3032:
3007:
2992:
2985:
2960:
2951:
2942:978-0470457931
2941:
2935:. p. 86.
2919:
2912:
2904:Academic Press
2887:
2868:
2851:Bill Casselman
2841:
2840:
2838:
2835:
2833:
2832:
2827:
2820:
2815:
2810:
2805:
2800:
2795:
2790:
2785:
2780:
2774:
2773:
2772:
2758:
2742:
2739:
2700:Main article:
2697:
2696:Ending a proof
2694:
2670:
2667:
2632:
2629:
2617:Bayes' theorem
2583:
2580:
2566:experiment or
2529:Main article:
2526:
2523:
2514:
2511:
2493:
2490:
2470:Main article:
2467:
2464:
2456:
2455:
2452:
2445:
2443:
2440:
2433:
2431:
2424:
2417:
2398:
2395:
2393:
2390:
2358:Main article:
2355:
2352:
2318:
2315:
2301:Main article:
2298:
2295:
2275:chaotic series
2259:Main article:
2256:
2253:
2252:
2251:
2240:
2235:
2231:
2210:
2207:
2202:
2196:
2190:
2184:
2178:
2172:
2165:
2159:
2135:
2130:
2127:
2104:
2097:
2091:
2088:
2065:
2058:
2033:
2028:
2025:
2022:
2019:
1996:
1989:
1959:
1952:
1923:
1895:
1891:
1859:Main article:
1856:
1853:
1833:Main article:
1830:
1827:
1788:Main article:
1785:
1782:
1756:
1752:
1748:
1743:
1739:
1716:
1712:
1708:
1703:
1700:
1697:
1693:
1672:
1650:
1646:
1642:
1637:
1633:
1610:
1606:
1602:
1597:
1593:
1570:
1566:
1562:
1559:
1556:
1551:
1547:
1527:Main article:
1523:
1520:
1507:Main article:
1504:
1501:
1497:counterexample
1477:Main article:
1474:
1471:
1452:
1439:
1438:
1424:
1326:
1323:
1318:
1313:
1279:
1276:
1271:
1266:
1242:
1210:
1183:Main article:
1180:
1177:
1176:
1175:
1163:
1138:
1134:
1111:
1107:
1086:
1083:
1080:
1077:
1072:
1068:
1047:
1027:
1003:
994:is even, then
981:
977:
956:
911:Contraposition
909:Main article:
906:
903:
899:
898:
889: − 1
879:
796: − 1
779:
735: − 1
712: − 1
704:
703:
681:
632:
533:Main article:
530:
527:
515:
514:
489: + 2
465: = 2
457: = 2
428:Main article:
425:
422:
420:
417:
330:
327:
273:mathematician
152:
149:
109:informal logic
96:Proofs employ
15:
13:
10:
9:
6:
4:
3:
2:
6029:
6028:
6017:
6014:
6012:
6009:
6007:
6004:
6002:
5999:
5997:
5994:
5993:
5991:
5976:
5973:
5970:
5966:
5963:
5961:
5958:
5956:
5946:
5945:
5942:
5932:
5931:Logic symbols
5929:
5927:
5924:
5922:
5919:
5917:
5914:
5912:
5909:
5908:
5906:
5902:
5896:
5893:
5891:
5888:
5886:
5883:
5882:
5880:
5878:
5874:
5871:
5867:
5861:
5858:
5856:
5853:
5851:
5848:
5846:
5843:
5841:
5838:
5836:
5833:
5831:
5828:
5826:
5823:
5821:
5818:
5816:
5813:
5811:
5810:Logical truth
5808:
5806:
5803:
5801:
5798:
5794:
5791:
5790:
5789:
5786:
5784:
5781:
5779:
5776:
5774:
5771:
5769:
5766:
5762:
5759:
5757:
5754:
5753:
5752:
5751:Contradiction
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5732:
5729:
5728:
5726:
5722:
5712:
5709:
5707:
5704:
5702:
5699:
5697:
5696:Argumentation
5694:
5693:
5691:
5687:
5681:
5680:Philosophical
5678:
5676:
5675:Non-classical
5673:
5671:
5668:
5664:
5661:
5659:
5656:
5655:
5654:
5651:
5649:
5646:
5645:
5643:
5639:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5605:
5603:
5600:
5599:
5597:
5593:
5587:
5584:
5582:
5579:
5578:
5575:
5571:
5564:
5559:
5557:
5552:
5550:
5545:
5544:
5541:
5531:
5530:
5525:
5517:
5511:
5508:
5506:
5503:
5501:
5498:
5496:
5493:
5489:
5486:
5485:
5484:
5481:
5479:
5476:
5474:
5471:
5469:
5465:
5462:
5460:
5457:
5455:
5452:
5450:
5447:
5445:
5442:
5441:
5439:
5435:
5429:
5426:
5424:
5421:
5419:
5418:Recursive set
5416:
5414:
5411:
5409:
5406:
5404:
5401:
5399:
5396:
5392:
5389:
5387:
5384:
5382:
5379:
5377:
5374:
5372:
5369:
5368:
5367:
5364:
5362:
5359:
5357:
5354:
5352:
5349:
5347:
5344:
5342:
5339:
5338:
5336:
5334:
5330:
5324:
5321:
5319:
5316:
5314:
5311:
5309:
5306:
5304:
5301:
5299:
5296:
5294:
5291:
5287:
5284:
5282:
5279:
5277:
5274:
5273:
5272:
5269:
5267:
5264:
5262:
5259:
5257:
5254:
5252:
5249:
5247:
5244:
5240:
5237:
5236:
5235:
5232:
5228:
5227:of arithmetic
5225:
5224:
5223:
5220:
5216:
5213:
5211:
5208:
5206:
5203:
5201:
5198:
5196:
5193:
5192:
5191:
5188:
5184:
5181:
5179:
5176:
5175:
5174:
5171:
5170:
5168:
5166:
5162:
5156:
5153:
5151:
5148:
5146:
5143:
5141:
5138:
5135:
5134:from ZFC
5131:
5128:
5126:
5123:
5117:
5114:
5113:
5112:
5109:
5107:
5104:
5102:
5099:
5098:
5097:
5094:
5092:
5089:
5087:
5084:
5082:
5079:
5077:
5074:
5072:
5069:
5067:
5064:
5063:
5061:
5059:
5055:
5045:
5044:
5040:
5039:
5034:
5033:non-Euclidean
5031:
5027:
5024:
5022:
5019:
5017:
5016:
5012:
5011:
5009:
5006:
5005:
5003:
4999:
4995:
4992:
4990:
4987:
4986:
4985:
4981:
4977:
4974:
4973:
4972:
4968:
4964:
4961:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4939:
4936:
4935:
4933:
4929:
4928:
4926:
4921:
4915:
4910:Example
4907:
4899:
4894:
4893:
4892:
4889:
4887:
4884:
4880:
4877:
4875:
4872:
4870:
4867:
4865:
4862:
4861:
4860:
4857:
4855:
4852:
4850:
4847:
4845:
4842:
4838:
4835:
4833:
4830:
4829:
4828:
4825:
4821:
4818:
4816:
4813:
4811:
4808:
4806:
4803:
4802:
4801:
4798:
4796:
4793:
4789:
4786:
4784:
4781:
4779:
4776:
4775:
4774:
4771:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4738:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4696:
4695:
4692:
4690:
4687:
4685:
4682:
4680:
4677:
4673:
4670:
4668:
4667:by definition
4665:
4664:
4663:
4660:
4656:
4653:
4652:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4624:
4621:
4619:
4615:
4610:
4604:
4600:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4555:
4554:Kripke–Platek
4552:
4550:
4547:
4543:
4540:
4538:
4535:
4534:
4533:
4530:
4529:
4527:
4523:
4515:
4512:
4511:
4510:
4507:
4505:
4502:
4498:
4495:
4494:
4493:
4490:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4467:
4463:
4459:
4456:
4452:
4449:
4447:
4444:
4442:
4439:
4438:
4437:
4433:
4430:
4429:
4427:
4425:
4421:
4417:
4409:
4406:
4404:
4401:
4399:
4398:constructible
4396:
4395:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4331:
4330:
4328:
4326:
4321:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4284:
4282:
4278:
4275:
4273:
4270:
4269:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4244:
4240:
4238:
4235:
4231:
4228:
4227:
4226:
4223:
4222:
4219:
4216:
4214:
4210:
4200:
4197:
4195:
4192:
4190:
4187:
4185:
4182:
4180:
4177:
4175:
4172:
4168:
4165:
4164:
4163:
4160:
4156:
4151:
4150:
4149:
4146:
4145:
4143:
4141:
4137:
4129:
4126:
4124:
4121:
4119:
4116:
4115:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4080:
4078:
4076:
4075:Propositional
4072:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4043:
4041:
4038:
4034:
4031:
4030:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4003:Logical truth
4001:
3999:
3996:
3995:
3993:
3991:
3987:
3984:
3982:
3978:
3972:
3969:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3947:
3943:
3939:
3935:
3933:
3930:
3928:
3925:
3923:
3919:
3916:
3915:
3913:
3911:
3905:
3900:
3894:
3891:
3889:
3886:
3884:
3881:
3879:
3876:
3874:
3871:
3869:
3866:
3864:
3861:
3859:
3856:
3854:
3851:
3849:
3846:
3844:
3841:
3839:
3836:
3832:
3829:
3828:
3827:
3824:
3823:
3821:
3817:
3813:
3806:
3801:
3799:
3794:
3792:
3787:
3786:
3783:
3777:
3773:
3769:
3765:
3763:
3760:
3757:
3752:
3748:
3747:
3743:
3737:
3731:
3727:
3726:
3725:Book of Proof
3720:
3716:
3710:
3706:
3701:
3697:
3691:
3687:
3683:
3678:
3674:
3670:
3666:
3662:
3656:
3653:, Kew Books,
3652:
3651:
3646:
3642:
3637:
3633:
3629:
3624:
3620:
3618:9780691080055
3614:
3609:
3604:
3600:
3599:
3594:
3590:
3589:
3585:
3569:
3565:
3558:
3555:
3550:
3546:
3541:
3540:2027.42/42653
3536:
3532:
3528:
3524:
3520:
3519:
3511:
3504:
3501:
3497:
3493:
3487:
3483:
3478:
3477:
3468:
3465:
3461:
3456:
3452:
3446:
3443:
3439:
3434:
3428:
3424:
3420:
3419:
3414:
3413:Wright, David
3410:
3406:
3400:
3397:
3394:
3389:
3386:
3383:
3380:
3374:
3371:
3368:
3363:
3360:
3356:
3350:
3347:
3343:
3337:
3334:
3329:
3323:
3319:
3315:
3314:
3306:
3303:
3298:
3292:
3288:
3284:
3283:
3275:
3272:
3268:
3262:
3259:
3255:
3251:
3248:
3243:
3240:
3237:
3232:
3229:
3223:
3220:
3214:
3211:
3198:
3191:
3184:
3181:
3177:
3172:
3166:
3162:
3158:
3154:
3148:
3145:
3141:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3112:
3104:
3101:
3097:
3093:
3087:
3083:
3079:
3075:
3069:
3066:
3054:
3050:
3043:
3040:
3035:
3029:
3026:. p. 3.
3025:
3021:
3017:
3011:
3008:
3004:
2999:
2997:
2993:
2988:
2982:
2978:
2974:
2970:
2964:
2961:
2955:
2952:
2948:
2944:
2938:
2934:
2930:
2923:
2920:
2915:
2909:
2906:. p. 3.
2905:
2901:
2897:
2891:
2888:
2884:
2879:
2872:
2869:
2861:September 26,
2856:
2852:
2846:
2843:
2836:
2831:
2828:
2826:
2825:
2821:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2783:Invalid proof
2781:
2779:
2776:
2775:
2770:
2764:
2759:
2756:
2745:
2740:
2738:
2732:
2729:
2725:
2721:
2717:
2713:
2709:
2703:
2695:
2693:
2691:
2690:
2685:
2681:
2676:
2668:
2666:
2664:
2660:
2656:
2652:
2648:
2642:
2638:
2630:
2628:
2626:
2622:
2618:
2613:
2611:
2607:
2603:
2599:
2596:Proofs using
2593:
2589:
2581:
2579:
2577:
2576:scatter plots
2573:
2569:
2565:
2561:
2560:
2554:
2550:
2546:
2542:
2538:
2537:data analysis
2532:
2524:
2522:
2520:
2512:
2510:
2507:
2498:
2491:
2489:
2487:
2483:
2479:
2478:number theory
2473:
2465:
2463:
2461:
2449:
2444:
2437:
2432:
2428:
2421:
2416:
2414:
2412:
2408:
2404:
2396:
2391:
2389:
2387:
2383:
2379:
2375:
2371:
2367:
2361:
2353:
2351:
2349:
2345:
2343:
2339:
2334:
2332:
2328:
2324:
2316:
2314:
2311:
2308:proof of the
2304:
2296:
2294:
2292:
2288:
2284:
2280:
2276:
2272:
2268:
2262:
2254:
2238:
2233:
2229:
2208:
2205:
2200:
2194:
2188:
2182:
2176:
2170:
2163:
2157:
2133:
2128:
2125:
2102:
2095:
2089:
2086:
2063:
2056:
2031:
2026:
2023:
2020:
2017:
1994:
1987:
1976:
1975:
1974:
1957:
1950:
1939:
1921:
1911:
1893:
1889:
1880:
1876:
1873:
1868:
1862:
1854:
1852:
1850:
1846:
1842:
1836:
1828:
1826:
1824:
1820:
1816:
1812:
1807:
1803:
1801:
1797:
1791:
1783:
1781:
1779:
1775:
1770:
1754:
1750:
1741:
1737:
1714:
1710:
1701:
1698:
1695:
1691:
1670:
1648:
1644:
1635:
1631:
1608:
1604:
1595:
1591:
1568:
1564:
1560:
1557:
1554:
1549:
1545:
1535:
1532:
1531:
1530:
1521:
1519:
1517:
1510:
1502:
1500:
1498:
1494:
1490:
1486:
1480:
1472:
1470:
1468:
1450:
1422:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1324:
1321:
1316:
1311:
1303:
1299:
1295:
1277:
1274:
1269:
1264:
1240:
1231:Suppose that
1230:
1229:
1228:
1226:
1208:
1198:
1194:
1193:
1186:
1178:
1161:
1153:
1136:
1132:
1109:
1105:
1097:is odd. Thus
1084:
1081:
1078:
1075:
1070:
1066:
1045:
1025:
1017:
1016:
1015:
1001:
979:
975:
954:
945:
943:
939:
935:
932:
928:
924:
920:
917:
912:
904:
902:
895:
888:
883:
880:
875:
871:
864:
860:
853:
849:
842:
834:
830:
826:
814:
806:
802:
795:
788:
783:
780:
775:
756:
748:
743:
740:
739:
738:
734:
726:
722:
717:
711:
702:
699:
692:
688:
682:
677:
673:
666:
662:
655:
651:
644:
640:
636:
633:
628:
621:
617:
610:
606:
603:
602:
601:
598:
594:belonging to
592:
585:
581:
574:
569:
564:
562:
558:
554:
550:
546:
542:
536:
528:
526:
524:
520:
512:
508:
504:
501:). Therefore
500:
496:
492:
488:
484:
481: +
480:
476:
472:
468:
464:
460:
456:
452:
448:
444:
443:
442:
440:
437:
431:
423:
418:
416:
414:
413:
408:
404:
399:
397:
393:
389:
385:
381:
377:
373:
367:
365:
361:
356:
352:
348:
344:
339:
336:
328:
326:
324:
320:
316:
312:
307:
305:
301:
297:
293:
289:
285:
281:
276:
272:
268:
263:
261:
260:prime numbers
257:
253:
249:
248:number theory
245:
241:
237:
236:
231:
227:
223:
219:
215:
210:
208:
204:
200:
196:
192:
188:
184:
178:
176:
172:
168:
164:
158:
150:
148:
146:
142:
138:
134:
130:
126:
122:
118:
114:
113:formal proofs
110:
107:
103:
99:
94:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
49:
45:
37:
36:
31:
27:
23:
19:
6011:Proof theory
5850:Substitution
5670:Mathematical
5595:Major fields
5520:
5318:Ultraproduct
5165:Model theory
5130:Independence
5066:Formal proof
5058:Proof theory
5041:
5014:
4971:real numbers
4943:second-order
4854:Substitution
4731:Metalanguage
4672:conservative
4645:Axiom schema
4589:Constructive
4559:Morse–Kelley
4525:Set theories
4504:Aleph number
4497:inaccessible
4403:Grothendieck
4287:intersection
4174:Higher-order
4162:Second-order
4108:Truth tables
4065:Venn diagram
3848:Formal proof
3724:
3704:
3681:
3672:
3669:Gold, Bonnie
3649:
3645:Franklin, J.
3635:
3631:
3597:
3571:. Retrieved
3567:
3557:
3522:
3516:
3503:
3495:
3475:
3467:
3458:
3455:the original
3445:
3436:
3416:
3399:
3388:
3378:
3373:
3362:
3354:
3349:
3341:
3336:
3317:
3312:
3305:
3286:
3281:
3274:
3261:
3242:
3231:
3222:
3213:
3201:. Retrieved
3199:. p. 12
3196:
3183:
3160:
3147:
3115:
3109:
3103:
3095:
3077:
3068:
3056:. Retrieved
3052:
3042:
3019:
3010:
2972:
2969:Hacking, Ian
2963:
2954:
2946:
2928:
2922:
2899:
2890:
2881:
2877:
2871:
2859:. Retrieved
2845:
2822:
2719:
2711:
2707:
2705:
2688:
2672:
2644:
2637:Psychologism
2614:
2595:
2557:
2552:
2548:
2534:
2516:
2503:
2475:
2457:
2400:
2397:Visual proof
2363:
2346:
2335:
2320:
2306:
2271:cryptography
2264:
1878:
1874:
1864:
1843:between two
1838:
1808:
1804:
1793:
1774:transitivity
1771:
1536:
1533:
1526:
1525:
1512:
1482:
1440:
1410:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1370:
1366:
1362:
1358:
1350:
1346:
1342:
1338:
1297:
1293:
1190:
1188:
1151:
946:
941:
937:
926:
922:
914:
900:
893:
886:
881:
873:
869:
862:
858:
851:
847:
840:
832:
828:
824:
812:
804:
800:
793:
786:
781:
773:
754:
746:
741:
732:
724:
720:
709:
705:
697:
690:
686:
683:
675:
671:
664:
660:
653:
649:
642:
638:
634:
626:
624:is true for
619:
615:
608:
604:
596:
590:
583:
579:
572:
565:
538:
516:
506:
502:
498:
494:
490:
486:
482:
478:
474:
470:
466:
462:
458:
454:
450:
446:
433:
430:Direct proof
424:Direct proof
410:
400:
368:
360:proof theory
347:formal proof
340:
332:
311:proof theory
308:
291:
264:
246:also covers
243:
234:
211:
179:
174:
170:
166:
162:
160:
121:proof theory
95:
86:
43:
41:
33:
18:
5965:WikiProject
5835:Proposition
5830:Probability
5783:Description
5724:Foundations
5428:Type theory
5376:undecidable
5308:Truth value
5195:equivalence
4874:non-logical
4487:Enumeration
4477:Isomorphism
4424:cardinality
4408:Von Neumann
4373:Ultrafilter
4338:Uncountable
4272:equivalence
4189:Quantifiers
4179:Fixed-point
4148:First-order
4028:Consistency
4013:Proposition
3990:Traditional
3961:Lindström's
3951:Compactness
3893:Type theory
3838:Cardinality
3776:Wikiversity
3573:October 15,
3379:statistical
3203:October 20,
3082:Brooks/Cole
3058:October 20,
2731:Paul Halmos
2714:, which is
2602:probability
2553:assumptions
2545:probability
729:represent "
509:has 2 as a
5990:Categories
5895:Set theory
5793:Linguistic
5788:Entailment
5778:Definition
5746:Consequent
5741:Antecedent
5239:elementary
4932:arithmetic
4800:Quantifier
4778:functional
4650:Expression
4368:Transitive
4312:identities
4297:complement
4230:hereditary
4213:Set theory
3482:Icon Books
3357:94:165–86.
3344:79:252–63.
2837:References
2692:argument.
1881:such that
600:such that
553:infinitely
493: = 2(
407:Paul Erdős
294:(1000) by
275:Al-Hashimi
256:irrational
203:Theaetetus
155:See also:
91:conjecture
26:P. Oxy. 29
5926:Fallacies
5921:Paradoxes
5911:Logicians
5845:Statement
5840:Reference
5805:Induction
5768:Deduction
5731:Abduction
5701:Metalogic
5648:Classical
5612:Inference
5510:Supertask
5413:Recursion
5371:decidable
5205:saturated
5183:of models
5106:deductive
5101:axiomatic
5021:Hilbert's
5008:Euclidean
4989:canonical
4912:axiomatic
4844:Signature
4773:Predicate
4662:Extension
4584:Ackermann
4509:Operation
4388:Universal
4378:Recursive
4353:Singleton
4348:Inhabited
4333:Countable
4323:Types of
4307:power set
4277:partition
4194:Predicate
4140:Predicate
4055:Syllogism
4045:Soundness
4018:Inference
4008:Tautology
3910:paradoxes
3593:Pólya, G.
3176:p. 3
3140:121416910
2971:(1984) .
2898:(2005) .
2883:obtained.
2724:tombstone
2684:Descartes
2680:certainty
2606:certainty
1841:bijection
1751:φ
1747:⇒
1738:φ
1711:φ
1707:⇒
1699:−
1692:φ
1671:…
1645:φ
1641:⇒
1632:φ
1605:φ
1601:⇒
1592:φ
1565:φ
1558:…
1546:φ
1082:⋅
1014:is even:
737:is odd":
541:deduction
485: = 2
380:synthetic
296:Al-Karaji
292:Al-Fakhri
207:Aristotle
175:probieren
111:. Purely
79:empirical
71:inference
59:logically
48:deductive
5960:Category
5860:Validity
5761:Antinomy
5689:Theories
5653:Informal
5495:Logicism
5488:timeline
5464:Concrete
5323:Validity
5293:T-schema
5286:Kripke's
5281:Tarski's
5276:semantic
5266:Strength
5215:submodel
5210:spectrum
5178:function
5026:Tarski's
5015:Elements
5002:geometry
4958:Robinson
4879:variable
4864:function
4837:spectrum
4827:Sentence
4783:variable
4726:Language
4679:Relation
4640:Automata
4630:Alphabet
4614:language
4468:-jection
4446:codomain
4432:Function
4393:Universe
4363:Infinite
4267:Relation
4050:Validity
4040:Argument
3938:theorem,
3638:: 373–88
3595:(1954),
3549:23084607
3415:(2002).
3250:Archived
2741:See also
2708:"Q.E.D."
2625:evidence
2411:triangle
1467:fraction
1365:, where
1304:. Thus,
1018:Suppose
867:implies
845:is odd (
809:), then
798:is odd (
784:For any
778:is true.
680:is true.
439:integers
376:analytic
355:formulas
335:argument
244:Elements
235:Elements
183:geometry
106:rigorous
63:theorems
51:argument
35:Elements
5975:changes
5967: (
5825:Premise
5756:Paradox
5586:History
5581:Outline
5437:Related
5234:Diagram
5132: (
5111:Hilbert
5096:Systems
5091:Theorem
4969:of the
4914:systems
4694:Formula
4689:Grammar
4605: (
4549:General
4262:Forcing
4247:Element
4167:Monadic
3942:paradox
3883:Theorem
3819:General
3159:(ed.),
3120:Bibcode
2675:Spinoza
2647:Leibniz
2372:to the
1977:Either
1776:of the
856:). So
771:. Thus
718:. Let
549:implies
519:closure
309:Modern
199:Eudoxus
171:provare
163:probare
5877:topics
5663:Reason
5641:Logics
5632:Syntax
5200:finite
4963:Skolem
4916:
4891:Theory
4859:Symbol
4849:String
4832:atomic
4709:ground
4704:closed
4699:atomic
4655:ground
4618:syntax
4514:binary
4441:domain
4358:Finite
4123:finite
3981:Logics
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3888:Theory
3772:course
3768:lesson
3732:
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2910:
2728:eponym
2702:Q.E.D.
2689:cogito
2655:Carnap
2653:, and
2370:Euclid
2323:axioms
2277:, and
2046:), or
1938:Euclid
1292:where
1223:is an
936:: "if
919:infers
759:, and
570:: Let
511:factor
242:, the
226:axioms
214:Euclid
191:Thales
167:probar
67:axioms
53:for a
30:Euclid
5904:other
5869:Lists
5855:Truth
5622:Proof
5570:Logic
5190:Model
4938:Peano
4795:Proof
4635:Arity
4564:Naive
4451:image
4383:Fuzzy
4343:Empty
4292:union
4237:Class
3878:Model
3868:Lemma
3826:Axiom
3774:from
3686:Wiley
3545:S2CID
3513:(PDF)
3460:time.
3316:[
3285:[
3193:(PDF)
3136:S2CID
2716:Latin
2651:Frege
2549:using
2539:, or
1908:is a
1465:as a
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967:, if
942:not p
940:then
938:not q
925:then
837:, so
790:, if
684:Then
392:Quine
282:. An
271:Iraqi
98:logic
46:is a
5969:talk
5815:Name
5800:Form
5313:Type
5116:list
4920:list
4897:list
4886:Term
4820:rank
4714:open
4608:list
4420:Maps
4325:sets
4184:Free
4154:list
3904:list
3831:list
3730:ISBN
3709:ISBN
3690:ISBN
3655:ISBN
3613:ISBN
3575:2009
3486:ISBN
3427:ISBN
3322:ISBN
3291:ISBN
3265:See
3205:2019
3165:ISBN
3086:ISBN
3060:2019
3028:ISBN
2981:ISBN
2937:ISBN
2908:ISBN
2863:2008
2718:for
2639:and
2590:and
2118:and
1877:and
1845:sets
1821:for
1729:and
1409:and
1401:and
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1296:and
882:Thus
782:(ii)
744:For
714:are
635:(ii)
473:and
461:and
449:and
436:even
384:Kant
345:. A
321:and
286:for
224:and
5711:Set
5000:of
4982:of
4930:of
4462:Sur
4436:Map
4243:Ur-
4225:Set
3603:hdl
3535:hdl
3527:doi
3128:doi
3116:500
2570:in
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716:odd
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611:(1)
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378:or
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874:n
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870:P
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863:n
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852:n
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841:n
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805:n
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801:P
794:n
792:2
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774:P
769:2
765:1
761:1
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725:n
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710:n
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505:+
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495:a
491:b
487:a
483:y
479:x
475:b
471:a
467:b
463:y
459:a
455:x
451:y
447:x
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