1693:
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
1656:
1487:
1445:
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
1692:
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
1651:{\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}}}
1468:
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
1171:
1331:
574:
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
60:), 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
1066:
774:
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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.
364:
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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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1425:". 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.
1722:
91:, and induces a different meaning for some terminology (for example, the term "or" has a stronger meaning in constructive mathematics than in classical).
1706:
1774:
Circles disturbed: the interplay of mathematics and narrative — Chapter 4. Hilbert on
Theology and Its Discontents The Origin Myth of Modern Mathematics
1821:
Hermann, Grete (1926). "Die Frage der endlich vielen
Schritte in der Theorie der Polynomideale: Unter Benutzung nachgelassener Sätze von K. Hentzelt".
1449:
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
1192:; it merely gives a number of possibilities (in this case, two mutually exclusive possibilities) and shows that one of them—but does not show
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2078:
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Until the end of 19th century, all mathematical proofs were essentially constructive. The first non-constructive constructions appeared with
94:
Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues; consequently the proposition must be true (
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1870:
1248:
proof of the theorem that a power of an irrational number to an irrational exponent may be rational gives an actual example, such as:
2061:
2039:
1789:
180:. From a philosophical point of view, the former is especially interesting, as implying the existence of a well specified object.
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1462:
1166:{\displaystyle \left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{({\sqrt {2}}\cdot {\sqrt {2}})}={\sqrt {2}}^{2}=2.}
126:
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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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1184:, which is not valid within a constructive proof. The non-constructive proof does not construct an example
1473:, 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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591:. Without establishing a specific prime number, this proves that one exists that is greater than
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2008:
1909:
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672:." 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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A Simple Proof That a Power of an
Irrational Number to an Irrational Exponent May Be Rational.
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130:
84:
55:
1866:, "The Root-2 Proof as an Example of Non-constructivity", unpublished paper, September 2014,
1461:, 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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1776:. Doxiadēs, Apostolos K., 1953-, Mazur, Barry. Princeton: Princeton University Press.
17:
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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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778: Dov Jarden Jerusalem
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106:) has been accepted in some varieties of constructive mathematics, including
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1402:, 9 would be equal to 2, but the former is odd, and the latter is even.
1979:
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1441:, as in classical mathematics. However, it is also possible to give a
520:, by considering as unknowns the finite number of coefficients of the
1755:(Summer 2018 ed.), Metaphysics Research Lab, Stanford University
1326:{\displaystyle a={\sqrt {2}}\,,\quad b=\log _{2}9\,,\quad a^{b}=3\,.}
567:
1963:
2073:(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
1414:
1910:"Nonconstructive tools for proving polynomial-time decidability"
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is a well defined sequence, constructively. Moreover, because
1714:- author of the book "Foundations of Constructive Analysis".
1615: is the least even natural number in the interval
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562:
First consider the theorem that there are an infinitude of
1180:
is rational or it is irrational"—an instance of the
129:
between proofs and programs, and such logical systems as
1908:
Fellows, Michael R.; Langston, Michael A. (1988-06-01).
1950:
769:{\displaystyle ({\sqrt {2}}^{\sqrt {2}})^{\sqrt {2}}=2}
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1718:Existence theorem § 'Pure' existence results
1673:) can be determined by exhaustive search, and so
479:exist, they can be found with degrees less than
183:The Nullstellensatz may be stated as follows: If
359:{\displaystyle f_{1}g_{1}+\ldots +f_{k}g_{k}=1.}
1998:, Lecture Notes in Mathematics 95, 1969, p. 102
962:is irrational, then the theorem is true, with
8:
875:is rational, then the theorem is true, with
852:. Either it is rational or it is irrational.
1723:Non-constructive algorithm existence proofs
1437:, a statement may be disproved by giving a
241:coefficients, which have no common complex
2056:(Fifth Edition). Oxford University Press.
1707:Constructivism (philosophy of mathematics)
1409:. A consequence of this theorem is that a
1196:one—must yield the desired example.
1747:Bridges, Douglas; Palmgren, Erik (2018),
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845:{\displaystyle q={\sqrt {2}}^{\sqrt {2}}}
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813:, and 2 is rational. Consider the number
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119:Brouwer–Heyting–Kolmogorov interpretation
2054:An Introduction to the Theory of Numbers
1637: which is not the sum of two primes
1373:is also irrational: if it were equal to
375:"this is not mathematics, it is theology
2011:", Stanford Encyclopedia of Mathematics
1753:The Stanford Encyclopedia of Philosophy
1739:
1225:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
1008:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
715:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
598:Now consider the theorem "there exist
595:, contrary to the original postulate.
2031:Proof in Mathematics: An Introduction
42:that demonstrates the existence of a
7:
1446:cannot be constructively provable.
583:! + 1 (1 + the product of the first
426:{\displaystyle g_{1},\ldots ,g_{k},}
384:provided an algorithm for computing
1421:belong to a certain finite set of "
472:{\displaystyle g_{1},\ldots ,g_{k}}
284:{\displaystyle g_{1},\ldots ,g_{k}}
222:{\displaystyle f_{1},\ldots ,f_{k}}
1685:with a fixed rate of convergence,
1481:) of rational numbers as follows:
1457:is non-constructive in systems of
1405:A more substantial example is the
1340:is irrational, and 3 is rational.
25:
1772:McLarty, Colin (April 15, 2008).
1695:limited principle of omniscience
1463:constructive reverse mathematics
1302:
1275:
165:, and the formal definition of
117:: this idea is explored in the
1631:
1619:
1591:
1576:
1565: is the sum of two primes
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1529:
1514:
1500:
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1135:
1115:
749:
729:
27:Method of proof in mathematics
1:
1751:, in Zalta, Edward N. (ed.),
1417:if, and only if, none of its
1398:, then, by the properties of
1232:is irrational because of the
245:, then there are polynomials
2115:Constructivism (mathematics)
1453:, which shows that the full
1052:{\displaystyle {\sqrt {2}}}
932:{\displaystyle {\sqrt {2}}}
803:{\displaystyle {\sqrt {2}}}
579:. Then consider the number
127:Curry–Howard correspondence
2131:
1995:Principles of Intuitionism
1749:"Constructive Mathematics"
1429:Brouwerian counterexamples
1366:{\displaystyle \log _{2}9}
518:system of linear equations
135:intuitionistic type theory
81:law of the excluded middle
1782:10.1515/9781400842681.105
1443:Brouwerian counterexample
1391:{\displaystyle m \over n}
1234:Gelfond–Schneider theorem
506:{\displaystyle 2^{2^{n}}}
380:Twenty five years later,
174:Hilbert's Nullstellensatz
147:calculus of constructions
1435:constructive mathematics
73:constructive mathematics
2007:Mark van Atten, 2015, "
1459:constructive set theory
558:Non-constructive proofs
178:Hilbert's basis theorem
1652:
1392:
1367:
1327:
1226:
1182:law of excluded middle
1167:
1053:
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782:In a bit more detail:
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635:
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544:
543:{\displaystyle g_{i}.}
507:
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427:
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223:
100:principle of explosion
96:proof by contradiction
57:pure existence theorem
48:non-constructive proof
18:Non-constructive proof
1823:Mathematische Annalen
1653:
1471:Goldbach's conjecture
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1168:
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776:proves our statement.
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661:{\displaystyle a^{b}}
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286:
224:
2093:Weak counterexamples
2067:Anne Sjerp Troelstra
2028:and A. Daoud (2011)
2009:Weak Counterexamples
1952:Mathematics Magazine
1728:Probabilistic method
1488:
1451:Diaconescu's theorem
1413:can be drawn on the
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237:indeterminates with
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153:A historical example
2110:Mathematical proofs
1929:10.1145/44483.44491
1892:Scripta Mathematica
1407:graph minor theorem
1240:Constructive proofs
44:mathematical object
1917:Journal of the ACM
1873:2014-10-23 at the
1835:10.1007/BF01206635
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600:irrational numbers
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123:constructive logic
104:ex falso quodlibet
69:constructive proof
50:(also known as an
36:constructive proof
2079:978-0-444-70506-8
1992:A. S. Troelstra,
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1028:{\displaystyle b}
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888:{\displaystyle a}
868:{\displaystyle q}
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634:{\displaystyle b}
614:{\displaystyle a}
85:axiom of infinity
16:(Redirected from
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62:effective proof
52:existence proof
38:is a method of
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2071:Dirk van Dalen
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1074:
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1046:
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1001:
994:
971:
951:
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926:
904:
884:
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838:
831:
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822:
797:
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756:
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737:
731:
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655:
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559:
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131:Per Martin-Löf
77:Constructivism
26:
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2062:0-19-853171-0
2059:
2055:
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2050:Wright, E. M.
2047:
2044:
2041:
2040:0-646-54509-4
2037:
2034:. Kew Books,
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1825:(in German).
1824:
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1791:9781400842681
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1712:Errett Bishop
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1140:
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1125:
1120:
1109:
1103:
1097:
1091:
1085:
1078:
1072:
1063:
1062:
1044:
1022:
999:
992:
969:
949:
941:
924:
902:
882:
862:
854:
836:
829:
823:
820:
812:
811:is irrational
795:
785:
784:
783:
779:
763:
760:
754:
742:
735:
706:
699:
688:
685:
681:
676:
673:
671:
653:
649:
628:
608:
601:
596:
594:
590:
586:
582:
578:
573:
569:
565:
564:prime numbers
557:
552:
550:
537:
532:
528:
519:
514:
496:
492:
487:
464:
460:
456:
453:
450:
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420:
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411:
407:
404:
401:
396:
392:
383:
382:Grete Hermann
378:
376:
372:
353:
350:
345:
341:
335:
331:
327:
324:
321:
316:
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306:
302:
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276:
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265:
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244:
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232:
214:
210:
206:
203:
200:
195:
191:
181:
179:
175:
170:
168:
164:
163:infinite sets
161:’s theory of
160:
152:
150:
148:
144:
140:
136:
132:
128:
124:
120:
116:
111:
109:
105:
101:
97:
92:
90:
86:
82:
78:
74:
70:
65:
63:
59:
58:
53:
49:
45:
41:
37:
33:
19:
2091:
2053:
2046:Hardy, G. H.
2029:
2003:
1993:
1988:
1955:
1951:
1945:
1920:
1916:
1903:
1895:
1890:
1886:
1881:
1859:
1826:
1822:
1816:
1773:
1767:
1757:, retrieved
1752:
1742:
1691:
1686:
1678:
1674:
1670:
1666:
1662:
1660:
1478:
1474:
1467:
1448:
1442:
1432:
1404:
1335:
1246:constructive
1245:
1243:
1198:
1193:
1189:
1185:
1177:
1175:
786:Recall that
781:
686:
683:
679:
678:
674:
597:
592:
588:
584:
580:
576:
561:
515:
379:
374:
368:
182:
171:
167:real numbers
159:Georg Cantor
156:
112:
108:intuitionism
103:
93:
68:
66:
61:
56:
51:
47:
35:
29:
2026:J. Franklin
1958:(1): 3–27.
1898::229 (1953)
1889:No. 339 in
915:both being
371:Paul Gordan
291:such that
231:polynomials
143:GĂ©rard Huet
32:mathematics
2104:Categories
1759:2019-10-25
1734:References
1400:logarithms
641:such that
115:algorithms
87:, and the
1972:0025-570X
1868:full text
1851:115897210
1843:0025-5831
1808:170826113
1800:775873004
1661:For each
1358:
1293:
1126:⋅
454:…
405:…
373:, wrote:
325:…
266:…
204:…
1937:16587284
1871:Archived
1701:See also
1605:if
670:rational
553:Examples
2052:(1979)
2042:, ch. 4
1980:2689939
1887:Curiosa
1059:, since
680:CURIOSA
239:complex
2077:
2060:
2048:&
2038:
1978:
1970:
1935:
1849:
1841:
1806:
1798:
1788:
1419:minors
1035:being
982:being
568:Euclid
137:, and
125:, the
83:, the
1976:JSTOR
1933:S2CID
1913:(PDF)
1847:S2CID
1804:S2CID
1681:is a
1415:torus
1411:graph
1194:which
572:proof
243:zeros
40:proof
2075:ISBN
2069:and
2058:ISBN
2036:ISBN
1968:ISSN
1839:ISSN
1796:OCLC
1786:ISBN
1336:The
1188:and
1015:and
895:and
684:339.
621:and
229:are
176:and
141:and
34:, a
1960:doi
1925:doi
1831:doi
1778:doi
1433:In
1349:log
1284:log
942:If
855:If
668:is
570:'s
377:".
233:in
145:'s
133:'s
121:of
54:or
30:In
2106::
1974:.
1966:.
1956:62
1954:.
1931:.
1921:35
1919:.
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1896:19
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1837:.
1827:95
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1784:.
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1244:A
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67:A
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1982:.
1962::
1939:.
1927::
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1780::
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