1682:
conjecture (in the former case) or a constructive proof that
Goldbach's conjecture is false (in the latter case). Because no such proof is known, the quoted statement must also not have a known constructive proof. However, it is entirely possible that Goldbach's conjecture may have a constructive proof (as we do not know at present whether it does), in which case the quoted statement would have a constructive proof as well, albeit one that is unknown at present. The main practical use of weak counterexamples is to identify the "hardness" of a problem. For example, the counterexample just shown shows that the quoted statement is "at least as hard to prove" as Goldbach's conjecture. Weak counterexamples of this sort are often related to the
1645:
1476:
1434:
to show that the statement is non-constructive. This sort of counterexample shows that the statement implies some principle that is known to be non-constructive. If it can be proved constructively that the statement implies some principle that is not constructively provable, then the statement itself
1681:
Several facts about the real number α can be proved constructively. However, based on the different meaning of the words in constructive mathematics, if there is a constructive proof that "α = 0 or α ≠ 0" then this would mean that there is a constructive proof of
Goldbach's
1640:{\displaystyle a(n)={\begin{cases}(1/2)^{n}&{\mbox{if every even natural number in the interval }}{\mbox{ is the sum of two primes}},\\(1/2)^{k}&{\mbox{if }}k{\mbox{ is the least even natural number in the interval }}{\mbox{ which is not the sum of two primes}}\end{cases}}}
1457:
Brouwer also provided "weak" counterexamples. Such counterexamples do not disprove a statement, however; they only show that, at present, no constructive proof of the statement is known. One weak counterexample begins by taking some unsolved problem of mathematics, such as
1160:
1320:
563:
is constructive. But a common way of simplifying Euclid's proof postulates that, contrary to the assertion in the theorem, there are only a finite number of them, in which case there is a largest one, denoted
49:), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an
1055:
763:
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develops this idea further by classifying various principles in terms of "how nonconstructive" they are, by showing they are equivalent to various fragments of the law of the excluded middle.
353:
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1002:
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is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the
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1414:". However, the proof of the existence of this finite set is not constructive, and the forbidden minors are not actually specified. They are still unknown.
80:, and induces a different meaning for some terminology (for example, the term "or" has a stronger meaning in constructive mathematics than in classical).
1763:
Circles disturbed: the interplay of mathematics and narrative — Chapter 4. Hilbert on
Theology and Its Discontents The Origin Myth of Modern Mathematics
1810:
Hermann, Grete (1926). "Die Frage der endlich vielen
Schritte in der Theorie der Polynomideale: Unter Benutzung nachgelassener Sätze von K. Hentzelt".
107:
1438:
For example, a particular statement may be shown to imply the law of the excluded middle. An example of a
Brouwerian counterexample of this type is
1181:; it merely gives a number of possibilities (in this case, two mutually exclusive possibilities) and shows that one of them—but does not show
146:
Until the end of 19th century, all mathematical proofs were essentially constructive. The first non-constructive constructions appeared with
83:
Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues; consequently the proposition must be true (
1243:
1711:
1859:
1695:
1237:
proof of the theorem that a power of an irrational number to an irrational exponent may be rational gives an actual example, such as:
169:. From a philosophical point of view, the former is especially interesting, as implying the existence of a well specified object.
1155:{\displaystyle \left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{({\sqrt {2}}\cdot {\sqrt {2}})}={\sqrt {2}}^{2}=2.}
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The following 1953 proof by Dov Jarden has been widely used as an example of a non-constructive proof since at least 1970:
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converges to some real number α, according to the usual treatment of real numbers in constructive mathematics.
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Such a non-constructive existence theorem was such a surprise for mathematicians of that time that one of them,
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Dov Jarden, "A simple proof that a power of an irrational number to an irrational exponent may be rational",
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2014:
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1173:, which is not valid within a constructive proof. The non-constructive proof does not construct an example
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1462:, which asks whether every even natural number larger than 4 is the sum of two primes. Define a sequence
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which is not a constructive proof in the strong sense, as she used
Hilbert's result. She proved that, if
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is either rational or irrational. If it is rational, our statement is proved. If it is irrational,
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580:. Without establishing a specific prime number, this proves that one exists that is greater than
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661:." This theorem can be proven by using both a constructive proof, and a non-constructive proof.
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The first use of non-constructive proofs for solving previously considered problems seems to be
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1983:
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1170:
471:
88:
84:
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A Simple Proof That a Power of an
Irrational Number to an Irrational Exponent May Be Rational.
1855:, "The Root-2 Proof as an Example of Non-constructivity", unpublished paper, September 2014,
1450:, since the axiom of choice implies the law of excluded middle in such systems. The field of
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At its core, this proof is non-constructive because it relies on the statement "Either
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1765:. Doxiadēs, Apostolos K., 1953-, Mazur, Barry. Princeton: Princeton University Press.
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numbers). Either this number is prime, or all of its prime factors are greater than
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by creating or providing a method for creating the object. This is in contrast to a
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767: Dov Jarden Jerusalem
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95:) has been accepted in some varieties of constructive mathematics, including
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1430:, as in classical mathematics. However, it is also possible to give a
509:, by considering as unknowns the finite number of coefficients of the
1744:(Summer 2018 ed.), Metaphysics Research Lab, Stanford University
1315:{\displaystyle a={\sqrt {2}}\,,\quad b=\log _{2}9\,,\quad a^{b}=3\,.}
103:
1952:
2062:(1988) "Constructivism in Mathematics: Volume 1" Elsevier Science.
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This provides an algorithm, as the problem is reduced to solving a
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Constructive proofs can be seen as defining certified mathematical
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may also refer to the stronger concept of a proof that is valid in
1899:"Nonconstructive tools for proving polynomial-time decidability"
556:
1666:
is a well defined sequence, constructively. Moreover, because
1403:
1703:- author of the book "Foundations of Constructive Analysis".
1604: is the least even natural number in the interval
1633:
551:
First consider the theorem that there are an infinitude of
1169:
is rational or it is irrational"—an instance of the
118:
between proofs and programs, and such logical systems as
1897:
Fellows, Michael R.; Langston, Michael A. (1988-06-01).
1939:
758:{\displaystyle ({\sqrt {2}}^{\sqrt {2}})^{\sqrt {2}}=2}
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1707:Existence theorem § 'Pure' existence results
1662:) can be determined by exhaustive search, and so
468:exist, they can be found with degrees less than
172:The Nullstellensatz may be stated as follows: If
2090:
348:{\displaystyle f_{1}g_{1}+\ldots +f_{k}g_{k}=1.}
1987:, Lecture Notes in Mathematics 95, 1969, p. 102
1896:
1735:
951:is irrational, then the theorem is true, with
1417:
864:is rational, then the theorem is true, with
841:. Either it is rational or it is irrational.
1712:Non-constructive algorithm existence proofs
1426:, a statement may be disproved by giving a
230:coefficients, which have no common complex
2045:(Fifth Edition). Oxford University Press.
1938:
1696:Constructivism (philosophy of mathematics)
1398:. A consequence of this theorem is that a
1185:one—must yield the desired example.
546:
1736:Bridges, Douglas; Palmgren, Erik (2018),
1308:
1287:
1260:
834:{\displaystyle q={\sqrt {2}}^{\sqrt {2}}}
802:, and 2 is rational. Consider the number
108:Brouwer–Heyting–Kolmogorov interpretation
2043:An Introduction to the Theory of Numbers
1626: which is not the sum of two primes
1362:is also irrational: if it were equal to
364:"this is not mathematics, it is theology
2000:", Stanford Encyclopedia of Mathematics
1809:
1760:
1742:The Stanford Encyclopedia of Philosophy
1214:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
997:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
704:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
141:
2091:
1228:
587:Now consider the theorem "there exist
584:, contrary to the original postulate.
2020:Proof in Mathematics: An Introduction
31:that demonstrates the existence of a
1435:cannot be constructively provable.
572:! + 1 (1 + the product of the first
415:{\displaystyle g_{1},\ldots ,g_{k},}
373:provided an algorithm for computing
1410:belong to a certain finite set of "
461:{\displaystyle g_{1},\ldots ,g_{k}}
273:{\displaystyle g_{1},\ldots ,g_{k}}
211:{\displaystyle f_{1},\ldots ,f_{k}}
13:
2007:
1674:with a fixed rate of convergence,
1470:) of rational numbers as follows:
1446:is non-constructive in systems of
1394:A more substantial example is the
1329:is irrational, and 3 is rational.
14:
2115:
2074:
1761:McLarty, Colin (April 15, 2008).
1684:limited principle of omniscience
1452:constructive reverse mathematics
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154:, and the formal definition of
106:: this idea is explored in the
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16:Method of proof in mathematics
1:
1740:, in Zalta, Edward N. (ed.),
1722:
1406:if, and only if, none of its
1387:, then, by the properties of
1221:is irrational because of the
234:, then there are polynomials
2104:Constructivism (mathematics)
1442:, which shows that the full
7:
1689:
1041:{\displaystyle {\sqrt {2}}}
921:{\displaystyle {\sqrt {2}}}
792:{\displaystyle {\sqrt {2}}}
568:. Then consider the number
541:
116:Curry–Howard correspondence
10:
2120:
1984:Principles of Intuitionism
1738:"Constructive Mathematics"
1418:Brouwerian counterexamples
1355:{\displaystyle \log _{2}9}
507:system of linear equations
124:intuitionistic type theory
70:law of the excluded middle
1771:10.1515/9781400842681.105
1432:Brouwerian counterexample
1380:{\displaystyle m \over n}
1223:Gelfond–Schneider theorem
495:{\displaystyle 2^{2^{n}}}
369:Twenty five years later,
163:Hilbert's Nullstellensatz
136:calculus of constructions
1424:constructive mathematics
62:constructive mathematics
1996:Mark van Atten, 2015, "
1448:constructive set theory
547:Non-constructive proofs
167:Hilbert's basis theorem
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771:In a bit more detail:
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532:{\displaystyle g_{i}.}
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89:principle of explosion
85:proof by contradiction
46:pure existence theorem
37:non-constructive proof
1812:Mathematische Annalen
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1460:Goldbach's conjecture
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765:proves our statement.
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650:{\displaystyle a^{b}}
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2082:Weak counterexamples
2056:Anne Sjerp Troelstra
2017:and A. Daoud (2011)
1998:Weak Counterexamples
1941:Mathematics Magazine
1717:Probabilistic method
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1440:Diaconescu's theorem
1402:can be drawn on the
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142:A historical example
2099:Mathematical proofs
1918:10.1145/44483.44491
1881:Scripta Mathematica
1396:graph minor theorem
1229:Constructive proofs
33:mathematical object
1906:Journal of the ACM
1862:2014-10-23 at the
1824:10.1007/BF01206635
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112:constructive logic
93:ex falso quodlibet
58:constructive proof
39:(also known as an
25:constructive proof
2068:978-0-444-70506-8
1981:A. S. Troelstra,
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1017:{\displaystyle b}
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964:{\displaystyle a}
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877:{\displaystyle a}
857:{\displaystyle q}
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603:{\displaystyle a}
74:axiom of infinity
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128:Thierry Coquand
78:axiom of choice
51:effective proof
41:existence proof
27:is a method of
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2075:External links
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2060:Dirk van Dalen
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1428:counterexample
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1351:
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1323:
1322:
1311:
1307:
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1299:
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1267:
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1252:
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1230:
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1207:
1200:
1163:
1162:
1151:
1148:
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1137:
1131:
1126:
1121:
1116:
1111:
1106:
1100:
1094:
1088:
1082:
1076:
1069:
1063:
1050:
1049:
1035:
1013:
990:
983:
960:
940:
929:
915:
893:
873:
853:
842:
827:
820:
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786:
754:
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745:
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733:
726:
720:
697:
690:
644:
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619:
599:
548:
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528:
523:
519:
487:
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478:
455:
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143:
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120:Per Martin-Löf
66:Constructivism
15:
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2026:
2023:. Kew Books,
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752:
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731:
724:
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688:
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642:
638:
617:
597:
590:
585:
583:
579:
575:
571:
567:
562:
558:
554:
553:prime numbers
539:
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521:
517:
508:
503:
485:
481:
476:
453:
449:
445:
442:
439:
434:
430:
409:
404:
400:
396:
393:
390:
385:
381:
372:
371:Grete Hermann
367:
365:
361:
342:
339:
334:
330:
324:
320:
316:
313:
310:
305:
301:
295:
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265:
261:
257:
254:
251:
246:
242:
233:
229:
221:
203:
199:
195:
192:
189:
184:
180:
170:
168:
164:
159:
157:
153:
152:infinite sets
150:’s theory of
149:
139:
137:
133:
129:
125:
121:
117:
113:
109:
105:
100:
98:
94:
90:
86:
81:
79:
75:
71:
67:
63:
59:
54:
52:
48:
47:
42:
38:
34:
30:
26:
22:
2080:
2042:
2035:Hardy, G. H.
2018:
1992:
1982:
1977:
1944:
1940:
1934:
1909:
1905:
1892:
1884:
1879:
1875:
1870:
1848:
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1762:
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1467:
1463:
1456:
1437:
1431:
1421:
1393:
1324:
1235:constructive
1234:
1232:
1187:
1182:
1178:
1174:
1166:
1164:
775:Recall that
770:
675:
672:
668:
667:
663:
586:
581:
577:
573:
569:
565:
550:
504:
368:
363:
357:
171:
160:
156:real numbers
148:Georg Cantor
145:
101:
97:intuitionism
92:
82:
57:
55:
50:
45:
40:
36:
24:
18:
2015:J. Franklin
1947:(1): 3–27.
1887::229 (1953)
1878:No. 339 in
904:both being
360:Paul Gordan
280:such that
220:polynomials
132:GĂ©rard Huet
21:mathematics
2093:Categories
1748:2019-10-25
1723:References
1389:logarithms
630:such that
104:algorithms
76:, and the
1961:0025-570X
1857:full text
1840:115897210
1832:0025-5831
1797:170826113
1789:775873004
1650:For each
1347:
1282:
1115:⋅
443:…
394:…
362:, wrote:
314:…
255:…
193:…
1926:16587284
1860:Archived
1690:See also
1594:if
659:rational
542:Examples
2041:(1979)
2031:, ch. 4
1969:2689939
1876:Curiosa
1048:, since
669:CURIOSA
228:complex
2066:
2049:
2037:&
2027:
1967:
1959:
1924:
1838:
1830:
1795:
1787:
1777:
1408:minors
1024:being
971:being
557:Euclid
126:, and
114:, the
72:, the
1965:JSTOR
1922:S2CID
1902:(PDF)
1836:S2CID
1793:S2CID
1670:is a
1404:torus
1400:graph
1183:which
561:proof
232:zeros
29:proof
2064:ISBN
2058:and
2047:ISBN
2025:ISBN
1957:ISSN
1828:ISSN
1785:OCLC
1775:ISBN
1325:The
1177:and
1004:and
884:and
673:339.
610:and
218:are
165:and
130:and
23:, a
1949:doi
1914:doi
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