4564:
1282:
983:
58:, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and therefore by
1277:{\displaystyle {\begin{aligned}{\sqrt {k}}&={\frac {m}{n}}\\&={\frac {m\left({\sqrt {k}}-q\right)}{n\left({\sqrt {k}}-q\right)}}\\&={\frac {m{\sqrt {k}}-mq}{n{\sqrt {k}}-nq}}\\&={\frac {\left(n{\sqrt {k}}\right){\sqrt {k}}-mq}{n\left({\frac {m}{n}}\right)-nq}}\\&={\frac {nk-mq}{m-nq}}\end{aligned}}}
392:
of
Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called "Pythagoras' number" or "Pythagoras'
2235:
In any of these cases, one
Pythagorean triangle with two sides each of which is a square or twice a square has led to a smaller one, which in turn would lead to a smaller one, etc.; since such a sequence cannot go on infinitely, the original premise that such a triangle exists must be wrong.
42:
used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the
73:âcan then be inferred. Once there, one would try to prove that if a smallest solution exists, then it must imply the existence of a smaller solution (in some sense), which again proves that the existence of any solution would lead to a contradiction.
1501:
Suppose there exists such a
Pythagorean triangle. Then it can be scaled down to give a primitive (i.e., with no common factors other than 1) Pythagorean triangle with the same property. Primitive Pythagorean triangles' sides can be written as
1494:, and the historical proofs of the latter proceeded by more broadly proving the former using infinite descent. The following more recent proof demonstrates both of these impossibilities by proving still more broadly that a
988:
243:
roughly doubles the length of the numbers required to write it (as number of digits), so that "halving" a point gives a rational with smaller terms. Since the terms are positive, they cannot decrease forever.
830:
472:
540:
213:). In this way Fermat was able to show the non-existence of solutions in many cases of Diophantine equations of classical interest (for example, the problem of four perfect squares in
695:
203:
2353:
2296:
2408:
1488:
1435:
210:
149:
2943:
1583:
2229:
1908:
1628:
742:
587:
1727:
2135:
2052:
1988:
3618:
843:
could be written as a rational number, then it could always be written as a rational number with smaller parts, which itself could be written with yet-smaller parts,
1535:
1862:
1835:
1808:
1781:
1754:
2183:
2155:
2111:
2073:
2028:
2008:
1964:
1944:
3701:
2842:
152:
4015:
4173:
2460:
2961:
69:
An alternative way to express this is to assume one or more solutions or examples exists, from which a smallest solution or example—a
4028:
3351:
256:
of the twentieth century, the infinite descent method was taken up again, and pushed to a point where it connected with the main thrust of
1681:
is a square or twice a square. There are three cases, depending on which two sides are postulated to each be a square or twice a square:
3613:
4033:
4023:
3760:
2966:
2595:
1379:
3511:
2957:
1349:
that have the same ratio. But infinite descent on the natural numbers is impossible, so this disproves the original assumption that
4598:
4169:
2580:
772:
4266:
4010:
2835:
384:
discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is
273:
3571:
3264:
3005:
2413:
cannot have non-trivial solutions, since non-trivial solutions would give
Pythagorean triangles with two sides being squares.
4588:
4527:
4229:
3992:
3987:
3812:
3233:
2917:
4522:
4305:
4222:
3935:
3866:
3743:
2985:
2714:
881:
were rational, no "smallest" representation as a fraction could exist, as any attempt to find a "smallest" representation
3593:
4593:
4447:
4273:
3959:
3192:
3598:
3930:
3669:
2927:
2828:
965:
4325:
4320:
4254:
3844:
3238:
3206:
2897:
1498:
cannot have any two of its sides each either a square or twice a square, since there is no smallest such triangle:
2971:
4544:
4493:
4390:
3888:
3849:
3326:
4385:
3000:
440:
339:
4315:
3854:
3706:
3689:
3412:
2892:
495:
1491:
629:
54:
Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more
4217:
4194:
4155:
4041:
3982:
3628:
3548:
3392:
3336:
2949:
850:
257:
44:
87:
proves that every composite integer is divided (in Euclid's terminology "measured") by some prime number.
4507:
4234:
4212:
4179:
4072:
3918:
3903:
3876:
3827:
3711:
3646:
3471:
3437:
3432:
3306:
3137:
3114:
214:
166:
70:
59:
39:
2509:
4437:
4290:
4082:
3800:
3536:
3442:
3301:
3286:
3167:
3142:
2599:
2302:
2245:
416:
presented another geometric proof by infinite descent that may be more accessible). The following is an
4563:
2360:
1440:
1387:
1300:)âwhich is positive but less than 1âand then simplified independently. So, the resulting products, say
101:
4410:
4372:
4249:
4053:
3893:
3817:
3795:
3623:
3581:
3480:
3447:
3311:
3099:
3010:
2753:
2572:
2165:) each of which is a square or twice a square, with a smaller hypotenuse than the original triangle (
1495:
95:
77:
62:, the original premise—that any solution exists—is incorrect: its correctness produces a
48:
1539:
388:. Little is known with certainty about the time or circumstances of this discovery, but the name of
4539:
4430:
4415:
4395:
4352:
4239:
4189:
4115:
4060:
3997:
3790:
3785:
3733:
3501:
3490:
3162:
3062:
2990:
2981:
2977:
2912:
2907:
2188:
1867:
1587:
701:
546:
2482:
1705:
4568:
4337:
4300:
4285:
4278:
4261:
4047:
3913:
3839:
3822:
3775:
3588:
3497:
3331:
3316:
3276:
3228:
3213:
3201:
3157:
3132:
2902:
2851:
2731:
2116:
2033:
1969:
413:
221:
206:
4065:
3521:
4503:
4310:
4120:
4110:
4002:
3883:
3718:
3694:
3475:
3459:
3364:
3341:
3218:
3187:
3152:
3047:
2882:
2576:
2456:
605:
385:
373:
331:
318:) is finite, which is certainly a necessary condition for the finite generation of the group
220:
In some cases, to the modern eye, his "method of infinite descent" is an exploitation of the
4517:
4512:
4405:
4362:
4184:
4145:
4140:
4125:
3951:
3908:
3805:
3603:
3553:
3127:
3089:
2723:
1505:
409:
362:
2537:
1840:
1813:
1786:
1759:
1732:
4498:
4488:
4442:
4425:
4380:
4342:
4244:
4164:
3971:
3898:
3871:
3859:
3765:
3679:
3653:
3608:
3576:
3377:
3179:
3122:
3072:
3037:
2995:
431:
299:
288:
334:. In this way, abstractly-defined cohomology groups in the theory become identified with
4483:
4462:
4420:
4400:
4295:
4150:
3748:
3738:
3728:
3723:
3657:
3531:
3407:
3296:
3291:
3269:
2870:
2685:
2642:
2638:
2168:
2140:
2096:
2058:
2013:
1993:
1949:
1929:
381:
298:
had to make more explicit the way of quantifying the size of a solution, by means of a
229:
225:
55:
4582:
4457:
4135:
3642:
3427:
3417:
3387:
3372:
3042:
2810:
2564:
2429:
401:
377:
295:
253:
160:
63:
376:(i.e. cannot be expressed as a fraction of two whole numbers) was discovered by the
4357:
4204:
4105:
4097:
3977:
3925:
3834:
3770:
3753:
3684:
3543:
3402:
3104:
2887:
2660:
845:
597:
98:. Two typical examples are showing the non-solvability of the Diophantine equation
2450:
2420:= 4 case of Fermat's Theorem, see the articles by Grant and Perella and Barbara.
4467:
4347:
3526:
3516:
3463:
3147:
3067:
3052:
2932:
2877:
861:
31:
17:
3397:
3252:
3223:
3029:
2814:
2804:
2467:
a special case of proof by contradiction called the method of infinite descent
261:
4549:
4452:
3505:
3422:
3382:
3346:
3282:
3094:
3084:
3057:
380:, and is perhaps the earliest known example of a proof by infinite descent.
889:
would imply that a smaller one existed, which is a similar contradiction.)
864:, which can be either rational or irrational, the only option left is for
4534:
4332:
3780:
3485:
3079:
2800:
2510:"Fermat's Method of Infinite Descent | Brilliant Math & Science Wiki"
2030:) each of which is a square or twice a square, and a smaller hypotenuse (
389:
4130:
2922:
2735:
1638:
417:
405:
265:
1287:
The numerator and denominator were each multiplied by the expression (
2820:
2569:
Number Theory: An approach through history from
Hammurapi to Legendre
91:
84:
2727:
2625:
38:, also known as Fermat's method of descent, is a particular kind of
2766:
Grant, Mike, and
Perella, Malcolm, "Descending to the irrational",
3674:
3020:
2865:
2712:
Sagher, Yoram (February 1988), "What
Pythagoras could have done",
235:. The context is of a hypothetical non-trivial rational point on
2824:
2626:"The discovery of incommensurability by Hippasus of Metapontum"
342:
was at the start of what later became a very extensive theory.
2612:
825:{\displaystyle {\frac {p}{q}}={\frac {2r}{2s}}={\frac {r}{s}}}
76:
The earliest uses of the method of infinite descent appear in
47:, and is often used to show that a given equation, such as a
919:
is not an integer, but is rational and can be expressed as
330:, one must do calculations in what later was recognised as
83:. A typical example is Proposition 31 of Book 7, in which
1437:
in integers is sufficient to show the non-solvability of
1702:
are both squares. But then the right triangle with legs
1316:
respectively. Therefore, no matter what natural numbers
1783:
also would have integer sides including a square leg (
2416:
For other similar proofs by infinite descent for the
2363:
2305:
2248:
2191:
2171:
2143:
2119:
2099:
2061:
2036:
2016:
1996:
1972:
1952:
1932:
1870:
1843:
1816:
1789:
1762:
1735:
1708:
1590:
1542:
1508:
1443:
1390:
1380:
Fermat's right triangle theorem § Fermat's proof
986:
775:
704:
632:
549:
498:
443:
169:
104:
4476:
4371:
4203:
4096:
3948:
3641:
3564:
3458:
3362:
3251:
3178:
3113:
3028:
3019:
2941:
2858:
302:â a concept that became foundational. To show that
2402:
2347:
2290:
2223:
2177:
2149:
2129:
2105:
2093:is a square. The integer right triangle with legs
2067:
2046:
2022:
2002:
1982:
1958:
1938:
1926:is a square. The integer right triangle with legs
1902:
1856:
1829:
1802:
1775:
1748:
1721:
1622:
1577:
1529:
1482:
1429:
1358:could be expressed as a ratio of natural numbers.
1276:
824:
736:
689:
581:
534:
466:
197:
143:
2779:Barbara, Roy, "Fermat's last theorem in the case
851:this is impossible in the set of natural numbers
268:, that the rational points on an elliptic curve
27:Mathematical proof technique using contradiction
2836:
1308:, are themselves integers, and are less than
276:, used an infinite descent argument based on
8:
2452:The Moment of Proof: Mathematical Epiphanies
1673:is a square or twice a square, then each of
94:, who coined the term and often used it for
3662:
3257:
3025:
2843:
2829:
2821:
2661:"Square root of 2 is irrational (Proof 8)"
467:{\displaystyle {\sqrt {2}}={\frac {p}{q}}}
2394:
2381:
2368:
2362:
2336:
2323:
2310:
2304:
2279:
2266:
2253:
2247:
2215:
2202:
2190:
2170:
2142:
2120:
2118:
2098:
2060:
2037:
2035:
2015:
1995:
1973:
1971:
1951:
1931:
1894:
1881:
1869:
1848:
1842:
1821:
1815:
1794:
1788:
1767:
1761:
1740:
1734:
1709:
1707:
1614:
1601:
1589:
1566:
1553:
1541:
1507:
1474:
1461:
1448:
1442:
1421:
1408:
1395:
1389:
1235:
1199:
1174:
1162:
1151:
1122:
1101:
1095:
1064:
1036:
1025:
1005:
991:
987:
985:
812:
789:
776:
774:
725:
709:
703:
678:
662:
640:
631:
570:
557:
548:
535:{\displaystyle 2={\frac {p^{2}}{q^{2}}},}
521:
511:
505:
497:
454:
444:
442:
394:
179:
168:
135:
122:
109:
103:
1837:), and would have a smaller hypotenuse (
1490:in integers, which is a special case of
2455:. Oxford University Press. p. 43.
2441:
1669:can be twice a square. Furthermore, if
690:{\displaystyle 2q^{2}=(2r)^{2}=4r^{2},}
153:Fermat's theorem on sums of two squares
90:The method was much later developed by
1333:, there exist smaller natural numbers
7:
2531:
2529:
2504:
2502:
2477:
2475:
874:(Alternatively, this proves that if
198:{\displaystyle p\equiv 1{\pmod {4}}}
187:
2348:{\displaystyle r^{4}+s^{2}=t^{4},}
2291:{\displaystyle r^{2}+s^{4}=t^{4},}
25:
2747:Dolan, Stan, "Fermat's method of
2403:{\displaystyle r^{4}+s^{4}=t^{2}}
1661:are each odd means that neither
1483:{\displaystyle q^{4}+s^{4}=t^{4}}
1430:{\displaystyle r^{2}+s^{4}=t^{4}}
951:be the largest integer less than
287:To extend this to the case of an
159:can be expressed as a sum of two
155:, which states that an odd prime
144:{\displaystyle r^{2}+s^{4}=t^{4}}
4562:
2811:Example of Fermat's last theorem
2684:Conrad, Keith (August 6, 2008).
2239:This implies that the equations
338:in the tradition of Fermat. The
274:finitely-generated abelian group
747:which shows that 2 must divide
180:
2628:, Annals of Mathematics, 1945.
1578:{\displaystyle y=a^{2}-b^{2},}
659:
649:
434:. Then it could be written as
191:
181:
1:
4523:History of mathematical logic
2715:American Mathematical Monthly
2224:{\displaystyle z=a^{2}+b^{2}}
1903:{\displaystyle z=a^{2}+b^{2}}
1623:{\displaystyle z=a^{2}+b^{2}}
737:{\displaystyle q^{2}=2r^{2},}
582:{\displaystyle 2q^{2}=p^{2},}
224:of the doubling function for
4448:Primitive recursive function
2538:"Fermat's Method of Descent"
1722:{\displaystyle {\sqrt {yz}}}
1653:both odd. The property that
2770:83, July 1999, pp. 263â267.
2615:, Nrich.org, November 2004.
2130:{\displaystyle {\sqrt {y}}}
2047:{\displaystyle {\sqrt {z}}}
1990:also would have two sides (
1983:{\displaystyle {\sqrt {z}}}
1810:) and a square hypotenuse (
489:. Then squaring would give
420:proof along similar lines:
264:. The structural result of
4615:
3512:SchröderâBernstein theorem
3239:Monadic predicate calculus
2898:Foundations of mathematics
2483:"What Is Infinite Descent"
2449:Benson, Donald C. (2000).
1377:
4558:
4545:Philosophy of mathematics
4494:Automated theorem proving
3665:
3619:Von NeumannâBernaysâGödel
3260:
2613:"The Dangerous Ratio ..."
2596:"The Pythagorean Theorem"
477:for two natural numbers,
211:proof by infinite descent
4599:Mathematical terminology
2649:, Copernicus, p. 25
326:) of rational points of
4195:Self-verifying theories
4016:Tarski's axiomatization
2967:Tarski's undefinability
2962:incompleteness theorems
2787:91, July 2007, 260â262.
2757:95, July 2011, 269â271.
1384:The non-solvability of
902:if it is not an integer
395:Conway & Guy (1996)
393:Constant", for example
258:algebraic number theory
45:well-ordering principle
4569:Mathematics portal
4180:Proof of impossibility
3828:propositional variable
3138:Propositional calculus
2693:kconrad.math.uconn.edu
2598:, Dept. of Math. Ed.,
2404:
2349:
2292:
2225:
2179:
2157:would have two sides (
2151:
2131:
2107:
2069:
2048:
2024:
2004:
1984:
1960:
1940:
1904:
1858:
1831:
1804:
1777:
1750:
1723:
1624:
1579:
1531:
1530:{\displaystyle x=2ab,}
1484:
1431:
1278:
826:
738:
691:
600:, it must also divide
583:
536:
468:
239:. Doubling a point on
215:arithmetic progression
199:
145:
71:minimal counterexample
60:mathematical induction
40:proof by contradiction
4589:Diophantine equations
4438:Kolmogorov complexity
4391:Computably enumerable
4291:Model complete theory
4083:Principia Mathematica
3143:Propositional formula
2972:BanachâTarski paradox
2600:University of Georgia
2594:Stephanie J. Morris,
2405:
2350:
2293:
2226:
2180:
2152:
2132:
2108:
2070:
2049:
2025:
2005:
1985:
1961:
1941:
1905:
1859:
1857:{\displaystyle a^{2}}
1832:
1830:{\displaystyle a^{2}}
1805:
1803:{\displaystyle b^{2}}
1778:
1776:{\displaystyle a^{2}}
1751:
1749:{\displaystyle b^{2}}
1724:
1625:
1580:
1532:
1492:Fermat's Last Theorem
1485:
1432:
1279:
906:For positive integer
827:
739:
692:
584:
537:
469:
412:by infinite descent (
200:
146:
96:Diophantine equations
4386:ChurchâTuring thesis
4373:Computability theory
3582:continuum hypothesis
3100:Square of opposition
2958:Gödel's completeness
2785:Mathematical Gazette
2768:Mathematical Gazette
2754:Mathematical Gazette
2665:www.cut-the-knot.org
2487:www.cut-the-knot.org
2361:
2303:
2246:
2189:
2169:
2141:
2117:
2097:
2059:
2034:
2014:
1994:
1970:
1950:
1930:
1868:
1841:
1814:
1787:
1760:
1733:
1706:
1588:
1540:
1506:
1496:Pythagorean triangle
1441:
1388:
1374:and its permutations
1324:are used to express
984:
939:for natural numbers
773:
702:
630:
547:
496:
441:
346:Application examples
340:MordellâWeil theorem
167:
102:
51:, has no solutions.
49:Diophantine equation
4594:Mathematical proofs
4540:Mathematical object
4431:P versus NP problem
4396:Computable function
4190:Reverse mathematics
4116:Logical consequence
3993:primitive recursive
3988:elementary function
3761:Free/bound variable
3614:TarskiâGrothendieck
3133:Logical connectives
3063:Logical equivalence
2913:Logical consequence
2647:The Book of Numbers
1362:Non-solvability of
616:, for some integer
361:The proof that the
284:in Fermat's style.
4338:Transfer principle
4301:Semantics of logic
4286:Categorical theory
4262:Non-standard model
3776:Logical connective
2903:Information theory
2852:Mathematical logic
2686:"Infinite Descent"
2575:, pp. 75â79,
2400:
2345:
2288:
2221:
2175:
2147:
2127:
2103:
2065:
2044:
2020:
2000:
1980:
1956:
1936:
1900:
1854:
1827:
1800:
1773:
1746:
1719:
1620:
1575:
1527:
1480:
1427:
1274:
1272:
871:to be irrational.
822:
734:
687:
579:
532:
464:
414:John Horton Conway
207:Modular arithmetic
195:
141:
4576:
4575:
4508:Abstract category
4311:Theories of truth
4121:Rule of inference
4111:Natural deduction
4092:
4091:
3637:
3636:
3342:Cartesian product
3247:
3246:
3153:Many-valued logic
3128:Boolean functions
3011:Russell's paradox
2986:diagonal argument
2883:First-order logic
2783: = 4",
2536:Donaldson, Neil.
2462:978-0-19-513919-8
2178:{\displaystyle a}
2150:{\displaystyle a}
2125:
2106:{\displaystyle b}
2068:{\displaystyle z}
2042:
2023:{\displaystyle b}
2003:{\displaystyle a}
1978:
1959:{\displaystyle b}
1939:{\displaystyle a}
1717:
1268:
1223:
1207:
1179:
1167:
1139:
1127:
1106:
1083:
1069:
1041:
1013:
996:
893:Irrationality of
820:
807:
784:
759:for some integer
596:. Because 2 is a
592:so 2 must divide
527:
462:
449:
351:Irrationality of
332:Galois cohomology
260:and the study of
16:(Redirected from
4606:
4567:
4566:
4518:History of logic
4513:Category of sets
4406:Decision problem
4185:Ordinal analysis
4126:Sequent calculus
4024:Boolean algebras
3964:
3963:
3938:
3909:logical/constant
3663:
3649:
3572:ZermeloâFraenkel
3323:Set operations:
3258:
3195:
3026:
3006:LöwenheimâSkolem
2893:Formal semantics
2845:
2838:
2831:
2822:
2801:Infinite descent
2788:
2777:
2771:
2764:
2758:
2749:descente infinie
2745:
2739:
2738:
2709:
2703:
2702:
2700:
2699:
2690:
2681:
2675:
2674:
2672:
2671:
2657:
2651:
2650:
2635:
2629:
2624:Kurt von Fritz,
2622:
2616:
2609:
2603:
2592:
2586:
2585:
2561:
2555:
2554:
2552:
2551:
2542:
2533:
2524:
2523:
2521:
2520:
2506:
2497:
2496:
2494:
2493:
2479:
2470:
2469:
2446:
2409:
2407:
2406:
2401:
2399:
2398:
2386:
2385:
2373:
2372:
2354:
2352:
2351:
2346:
2341:
2340:
2328:
2327:
2315:
2314:
2297:
2295:
2294:
2289:
2284:
2283:
2271:
2270:
2258:
2257:
2230:
2228:
2227:
2222:
2220:
2219:
2207:
2206:
2184:
2182:
2181:
2176:
2156:
2154:
2153:
2148:
2136:
2134:
2133:
2128:
2126:
2121:
2112:
2110:
2109:
2104:
2076:
2074:
2072:
2071:
2066:
2053:
2051:
2050:
2045:
2043:
2038:
2029:
2027:
2026:
2021:
2009:
2007:
2006:
2001:
1989:
1987:
1986:
1981:
1979:
1974:
1965:
1963:
1962:
1957:
1945:
1943:
1942:
1937:
1909:
1907:
1906:
1901:
1899:
1898:
1886:
1885:
1863:
1861:
1860:
1855:
1853:
1852:
1836:
1834:
1833:
1828:
1826:
1825:
1809:
1807:
1806:
1801:
1799:
1798:
1782:
1780:
1779:
1774:
1772:
1771:
1755:
1753:
1752:
1747:
1745:
1744:
1728:
1726:
1725:
1720:
1718:
1710:
1639:relatively prime
1629:
1627:
1626:
1621:
1619:
1618:
1606:
1605:
1584:
1582:
1581:
1576:
1571:
1570:
1558:
1557:
1536:
1534:
1533:
1528:
1489:
1487:
1486:
1481:
1479:
1478:
1466:
1465:
1453:
1452:
1436:
1434:
1433:
1428:
1426:
1425:
1413:
1412:
1400:
1399:
1357:
1356:
1345: <
1337: <
1332:
1331:
1295:
1294:
1283:
1281:
1280:
1275:
1273:
1269:
1267:
1253:
1236:
1228:
1224:
1222:
1212:
1208:
1200:
1190:
1180:
1175:
1173:
1169:
1168:
1163:
1152:
1144:
1140:
1138:
1128:
1123:
1117:
1107:
1102:
1096:
1088:
1084:
1082:
1081:
1077:
1070:
1065:
1054:
1053:
1049:
1042:
1037:
1026:
1018:
1014:
1006:
997:
992:
976:
975:
959:
958:
938:
936:
935:
930:
927:
918:
917:
901:
900:
880:
879:
870:
869:
859:
858:
842:
841:
831:
829:
828:
823:
821:
813:
808:
806:
798:
790:
785:
777:
743:
741:
740:
735:
730:
729:
714:
713:
696:
694:
693:
688:
683:
682:
667:
666:
645:
644:
588:
586:
585:
580:
575:
574:
562:
561:
541:
539:
538:
533:
528:
526:
525:
516:
515:
506:
488:
482:
473:
471:
470:
465:
463:
455:
450:
445:
429:
428:
371:
370:
363:square root of 2
357:
356:
204:
202:
201:
196:
194:
150:
148:
147:
142:
140:
139:
127:
126:
114:
113:
36:infinite descent
21:
18:Infinite descent
4614:
4613:
4609:
4608:
4607:
4605:
4604:
4603:
4579:
4578:
4577:
4572:
4561:
4554:
4499:Category theory
4489:Algebraic logic
4472:
4443:Lambda calculus
4381:Church encoding
4367:
4343:Truth predicate
4199:
4165:Complete theory
4088:
3957:
3953:
3949:
3944:
3936:
3656: and
3652:
3647:
3633:
3609:New Foundations
3577:axiom of choice
3560:
3522:Gödel numbering
3462: and
3454:
3358:
3243:
3193:
3174:
3123:Boolean algebra
3109:
3073:Equiconsistency
3038:Classical logic
3015:
2996:Halting problem
2984: and
2960: and
2948: and
2947:
2942:Theorems (
2937:
2854:
2849:
2797:
2795:Further reading
2792:
2791:
2778:
2774:
2765:
2761:
2746:
2742:
2728:10.2307/2323064
2711:
2710:
2706:
2697:
2695:
2688:
2683:
2682:
2678:
2669:
2667:
2659:
2658:
2654:
2643:Guy, Richard K.
2639:Conway, John H.
2637:
2636:
2632:
2623:
2619:
2610:
2606:
2593:
2589:
2583:
2563:
2562:
2558:
2549:
2547:
2540:
2535:
2534:
2527:
2518:
2516:
2508:
2507:
2500:
2491:
2489:
2481:
2480:
2473:
2463:
2448:
2447:
2443:
2438:
2426:
2390:
2377:
2364:
2359:
2358:
2332:
2319:
2306:
2301:
2300:
2275:
2262:
2249:
2244:
2243:
2211:
2198:
2187:
2186:
2167:
2166:
2139:
2138:
2137:and hypotenuse
2115:
2114:
2095:
2094:
2057:
2056:
2055:
2032:
2031:
2012:
2011:
1992:
1991:
1968:
1967:
1966:and hypotenuse
1948:
1947:
1928:
1927:
1890:
1877:
1866:
1865:
1844:
1839:
1838:
1817:
1812:
1811:
1790:
1785:
1784:
1763:
1758:
1757:
1756:and hypotenuse
1736:
1731:
1730:
1704:
1703:
1694:: In this case
1610:
1597:
1586:
1585:
1562:
1549:
1538:
1537:
1504:
1503:
1470:
1457:
1444:
1439:
1438:
1417:
1404:
1391:
1386:
1385:
1382:
1376:
1352:
1350:
1327:
1325:
1290:
1288:
1271:
1270:
1254:
1237:
1226:
1225:
1195:
1191:
1158:
1154:
1153:
1142:
1141:
1118:
1097:
1086:
1085:
1063:
1059:
1055:
1035:
1031:
1027:
1016:
1015:
998:
982:
981:
971:
969:
954:
952:
931:
928:
923:
922:
920:
913:
911:
910:, suppose that
904:
896:
894:
877:
875:
867:
865:
856:
854:
839:
837:
799:
791:
771:
770:
721:
705:
700:
699:
674:
658:
636:
628:
627:
566:
553:
545:
544:
517:
507:
494:
493:
484:
478:
439:
438:
426:
424:
410:geometric proof
408:, worked out a
368:
366:
359:
354:
352:
348:
300:height function
289:abelian variety
250:
226:rational points
165:
164:
131:
118:
105:
100:
99:
56:natural numbers
28:
23:
22:
15:
12:
11:
5:
4612:
4610:
4602:
4601:
4596:
4591:
4581:
4580:
4574:
4573:
4559:
4556:
4555:
4553:
4552:
4547:
4542:
4537:
4532:
4531:
4530:
4520:
4515:
4510:
4501:
4496:
4491:
4486:
4484:Abstract logic
4480:
4478:
4474:
4473:
4471:
4470:
4465:
4463:Turing machine
4460:
4455:
4450:
4445:
4440:
4435:
4434:
4433:
4428:
4423:
4418:
4413:
4403:
4401:Computable set
4398:
4393:
4388:
4383:
4377:
4375:
4369:
4368:
4366:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4330:
4329:
4328:
4323:
4318:
4308:
4303:
4298:
4296:Satisfiability
4293:
4288:
4283:
4282:
4281:
4271:
4270:
4269:
4259:
4258:
4257:
4252:
4247:
4242:
4237:
4227:
4226:
4225:
4220:
4213:Interpretation
4209:
4207:
4201:
4200:
4198:
4197:
4192:
4187:
4182:
4177:
4167:
4162:
4161:
4160:
4159:
4158:
4148:
4143:
4133:
4128:
4123:
4118:
4113:
4108:
4102:
4100:
4094:
4093:
4090:
4089:
4087:
4086:
4078:
4077:
4076:
4075:
4070:
4069:
4068:
4063:
4058:
4038:
4037:
4036:
4034:minimal axioms
4031:
4020:
4019:
4018:
4007:
4006:
4005:
4000:
3995:
3990:
3985:
3980:
3967:
3965:
3946:
3945:
3943:
3942:
3941:
3940:
3928:
3923:
3922:
3921:
3916:
3911:
3906:
3896:
3891:
3886:
3881:
3880:
3879:
3874:
3864:
3863:
3862:
3857:
3852:
3847:
3837:
3832:
3831:
3830:
3825:
3820:
3810:
3809:
3808:
3803:
3798:
3793:
3788:
3783:
3773:
3768:
3763:
3758:
3757:
3756:
3751:
3746:
3741:
3731:
3726:
3724:Formation rule
3721:
3716:
3715:
3714:
3709:
3699:
3698:
3697:
3687:
3682:
3677:
3672:
3666:
3660:
3643:Formal systems
3639:
3638:
3635:
3634:
3632:
3631:
3626:
3621:
3616:
3611:
3606:
3601:
3596:
3591:
3586:
3585:
3584:
3579:
3568:
3566:
3562:
3561:
3559:
3558:
3557:
3556:
3546:
3541:
3540:
3539:
3532:Large cardinal
3529:
3524:
3519:
3514:
3509:
3495:
3494:
3493:
3488:
3483:
3468:
3466:
3456:
3455:
3453:
3452:
3451:
3450:
3445:
3440:
3430:
3425:
3420:
3415:
3410:
3405:
3400:
3395:
3390:
3385:
3380:
3375:
3369:
3367:
3360:
3359:
3357:
3356:
3355:
3354:
3349:
3344:
3339:
3334:
3329:
3321:
3320:
3319:
3314:
3304:
3299:
3297:Extensionality
3294:
3292:Ordinal number
3289:
3279:
3274:
3273:
3272:
3261:
3255:
3249:
3248:
3245:
3244:
3242:
3241:
3236:
3231:
3226:
3221:
3216:
3211:
3210:
3209:
3199:
3198:
3197:
3184:
3182:
3176:
3175:
3173:
3172:
3171:
3170:
3165:
3160:
3150:
3145:
3140:
3135:
3130:
3125:
3119:
3117:
3111:
3110:
3108:
3107:
3102:
3097:
3092:
3087:
3082:
3077:
3076:
3075:
3065:
3060:
3055:
3050:
3045:
3040:
3034:
3032:
3023:
3017:
3016:
3014:
3013:
3008:
3003:
2998:
2993:
2988:
2976:Cantor's
2974:
2969:
2964:
2954:
2952:
2939:
2938:
2936:
2935:
2930:
2925:
2920:
2915:
2910:
2905:
2900:
2895:
2890:
2885:
2880:
2875:
2874:
2873:
2862:
2860:
2856:
2855:
2850:
2848:
2847:
2840:
2833:
2825:
2819:
2818:
2808:
2796:
2793:
2790:
2789:
2772:
2759:
2740:
2704:
2676:
2652:
2630:
2617:
2604:
2587:
2581:
2556:
2525:
2498:
2471:
2461:
2440:
2439:
2437:
2434:
2433:
2432:
2425:
2422:
2411:
2410:
2397:
2393:
2389:
2384:
2380:
2376:
2371:
2367:
2356:
2344:
2339:
2335:
2331:
2326:
2322:
2318:
2313:
2309:
2298:
2287:
2282:
2278:
2274:
2269:
2265:
2261:
2256:
2252:
2233:
2232:
2218:
2214:
2210:
2205:
2201:
2197:
2194:
2174:
2146:
2124:
2102:
2078:
2064:
2041:
2019:
1999:
1977:
1955:
1935:
1911:
1897:
1893:
1889:
1884:
1880:
1876:
1873:
1851:
1847:
1824:
1820:
1797:
1793:
1770:
1766:
1743:
1739:
1716:
1713:
1645:odd and hence
1617:
1613:
1609:
1604:
1600:
1596:
1593:
1574:
1569:
1565:
1561:
1556:
1552:
1548:
1545:
1526:
1523:
1520:
1517:
1514:
1511:
1477:
1473:
1469:
1464:
1460:
1456:
1451:
1447:
1424:
1420:
1416:
1411:
1407:
1403:
1398:
1394:
1375:
1360:
1285:
1284:
1266:
1263:
1260:
1257:
1252:
1249:
1246:
1243:
1240:
1234:
1231:
1229:
1227:
1221:
1218:
1215:
1211:
1206:
1203:
1198:
1194:
1189:
1186:
1183:
1178:
1172:
1166:
1161:
1157:
1150:
1147:
1145:
1143:
1137:
1134:
1131:
1126:
1121:
1116:
1113:
1110:
1105:
1100:
1094:
1091:
1089:
1087:
1080:
1076:
1073:
1068:
1062:
1058:
1052:
1048:
1045:
1040:
1034:
1030:
1024:
1021:
1019:
1017:
1012:
1009:
1004:
1001:
999:
995:
990:
989:
903:
891:
836:Therefore, if
834:
833:
819:
816:
811:
805:
802:
797:
794:
788:
783:
780:
745:
744:
733:
728:
724:
720:
717:
712:
708:
697:
686:
681:
677:
673:
670:
665:
661:
657:
654:
651:
648:
643:
639:
635:
606:Euclid's lemma
590:
589:
578:
573:
569:
565:
560:
556:
552:
542:
531:
524:
520:
514:
510:
504:
501:
475:
474:
461:
458:
453:
448:
402:ancient Greeks
378:ancient Greeks
358:
349:
347:
344:
249:
246:
230:elliptic curve
193:
190:
186:
183:
178:
175:
172:
138:
134:
130:
125:
121:
117:
112:
108:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4611:
4600:
4597:
4595:
4592:
4590:
4587:
4586:
4584:
4571:
4570:
4565:
4557:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4529:
4526:
4525:
4524:
4521:
4519:
4516:
4514:
4511:
4509:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4482:
4481:
4479:
4475:
4469:
4466:
4464:
4461:
4459:
4458:Recursive set
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4439:
4436:
4432:
4429:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4408:
4407:
4404:
4402:
4399:
4397:
4394:
4392:
4389:
4387:
4384:
4382:
4379:
4378:
4376:
4374:
4370:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4331:
4327:
4324:
4322:
4319:
4317:
4314:
4313:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4292:
4289:
4287:
4284:
4280:
4277:
4276:
4275:
4272:
4268:
4267:of arithmetic
4265:
4264:
4263:
4260:
4256:
4253:
4251:
4248:
4246:
4243:
4241:
4238:
4236:
4233:
4232:
4231:
4228:
4224:
4221:
4219:
4216:
4215:
4214:
4211:
4210:
4208:
4206:
4202:
4196:
4193:
4191:
4188:
4186:
4183:
4181:
4178:
4175:
4174:from ZFC
4171:
4168:
4166:
4163:
4157:
4154:
4153:
4152:
4149:
4147:
4144:
4142:
4139:
4138:
4137:
4134:
4132:
4129:
4127:
4124:
4122:
4119:
4117:
4114:
4112:
4109:
4107:
4104:
4103:
4101:
4099:
4095:
4085:
4084:
4080:
4079:
4074:
4073:non-Euclidean
4071:
4067:
4064:
4062:
4059:
4057:
4056:
4052:
4051:
4049:
4046:
4045:
4043:
4039:
4035:
4032:
4030:
4027:
4026:
4025:
4021:
4017:
4014:
4013:
4012:
4008:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3975:
3973:
3969:
3968:
3966:
3961:
3955:
3950:Example
3947:
3939:
3934:
3933:
3932:
3929:
3927:
3924:
3920:
3917:
3915:
3912:
3910:
3907:
3905:
3902:
3901:
3900:
3897:
3895:
3892:
3890:
3887:
3885:
3882:
3878:
3875:
3873:
3870:
3869:
3868:
3865:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3842:
3841:
3838:
3836:
3833:
3829:
3826:
3824:
3821:
3819:
3816:
3815:
3814:
3811:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3778:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3736:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3713:
3710:
3708:
3707:by definition
3705:
3704:
3703:
3700:
3696:
3693:
3692:
3691:
3688:
3686:
3683:
3681:
3678:
3676:
3673:
3671:
3668:
3667:
3664:
3661:
3659:
3655:
3650:
3644:
3640:
3630:
3627:
3625:
3622:
3620:
3617:
3615:
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3594:KripkeâPlatek
3592:
3590:
3587:
3583:
3580:
3578:
3575:
3574:
3573:
3570:
3569:
3567:
3563:
3555:
3552:
3551:
3550:
3547:
3545:
3542:
3538:
3535:
3534:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3507:
3503:
3499:
3496:
3492:
3489:
3487:
3484:
3482:
3479:
3478:
3477:
3473:
3470:
3469:
3467:
3465:
3461:
3457:
3449:
3446:
3444:
3441:
3439:
3438:constructible
3436:
3435:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3370:
3368:
3366:
3361:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3324:
3322:
3318:
3315:
3313:
3310:
3309:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3284:
3280:
3278:
3275:
3271:
3268:
3267:
3266:
3263:
3262:
3259:
3256:
3254:
3250:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3208:
3205:
3204:
3203:
3200:
3196:
3191:
3190:
3189:
3186:
3185:
3183:
3181:
3177:
3169:
3166:
3164:
3161:
3159:
3156:
3155:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3120:
3118:
3116:
3115:Propositional
3112:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3074:
3071:
3070:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3043:Logical truth
3041:
3039:
3036:
3035:
3033:
3031:
3027:
3024:
3022:
3018:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2983:
2979:
2975:
2973:
2970:
2968:
2965:
2963:
2959:
2956:
2955:
2953:
2951:
2945:
2940:
2934:
2931:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2872:
2869:
2868:
2867:
2864:
2863:
2861:
2857:
2853:
2846:
2841:
2839:
2834:
2832:
2827:
2826:
2823:
2816:
2812:
2809:
2806:
2802:
2799:
2798:
2794:
2786:
2782:
2776:
2773:
2769:
2763:
2760:
2756:
2755:
2750:
2744:
2741:
2737:
2733:
2729:
2725:
2721:
2717:
2716:
2708:
2705:
2694:
2687:
2680:
2677:
2666:
2662:
2656:
2653:
2648:
2644:
2640:
2634:
2631:
2627:
2621:
2618:
2614:
2611:Brian Clegg,
2608:
2605:
2601:
2597:
2591:
2588:
2584:
2582:0-8176-3141-0
2578:
2574:
2570:
2566:
2560:
2557:
2546:
2539:
2532:
2530:
2526:
2515:
2514:brilliant.org
2511:
2505:
2503:
2499:
2488:
2484:
2478:
2476:
2472:
2468:
2464:
2458:
2454:
2453:
2445:
2442:
2435:
2431:
2430:Vieta jumping
2428:
2427:
2423:
2421:
2419:
2414:
2395:
2391:
2387:
2382:
2378:
2374:
2369:
2365:
2357:
2342:
2337:
2333:
2329:
2324:
2320:
2316:
2311:
2307:
2299:
2285:
2280:
2276:
2272:
2267:
2263:
2259:
2254:
2250:
2242:
2241:
2240:
2237:
2216:
2212:
2208:
2203:
2199:
2195:
2192:
2172:
2164:
2160:
2144:
2122:
2100:
2092:
2088:
2087:
2083:
2079:
2062:
2039:
2017:
1997:
1975:
1953:
1933:
1925:
1921:
1920:
1916:
1912:
1895:
1891:
1887:
1882:
1878:
1874:
1871:
1849:
1845:
1822:
1818:
1795:
1791:
1768:
1764:
1741:
1737:
1714:
1711:
1701:
1697:
1693:
1692:
1688:
1684:
1683:
1682:
1680:
1676:
1672:
1668:
1664:
1660:
1656:
1652:
1648:
1644:
1640:
1637:
1633:
1615:
1611:
1607:
1602:
1598:
1594:
1591:
1572:
1567:
1563:
1559:
1554:
1550:
1546:
1543:
1524:
1521:
1518:
1515:
1512:
1509:
1499:
1497:
1493:
1475:
1471:
1467:
1462:
1458:
1454:
1449:
1445:
1422:
1418:
1414:
1409:
1405:
1401:
1396:
1392:
1381:
1373:
1369:
1365:
1361:
1359:
1355:
1348:
1344:
1340:
1336:
1330:
1323:
1319:
1315:
1311:
1307:
1303:
1299:
1296: â
1293:
1264:
1261:
1258:
1255:
1250:
1247:
1244:
1241:
1238:
1232:
1230:
1219:
1216:
1213:
1209:
1204:
1201:
1196:
1192:
1187:
1184:
1181:
1176:
1170:
1164:
1159:
1155:
1148:
1146:
1135:
1132:
1129:
1124:
1119:
1114:
1111:
1108:
1103:
1098:
1092:
1090:
1078:
1074:
1071:
1066:
1060:
1056:
1050:
1046:
1043:
1038:
1032:
1028:
1022:
1020:
1010:
1007:
1002:
1000:
993:
980:
979:
978:
974:
967:
963:
957:
950:
946:
942:
934:
926:
916:
909:
899:
892:
890:
888:
884:
872:
863:
852:
848:
847:
817:
814:
809:
803:
800:
795:
792:
786:
781:
778:
769:
768:
767:
764:
762:
758:
754:
750:
731:
726:
722:
718:
715:
710:
706:
698:
684:
679:
675:
671:
668:
663:
655:
652:
646:
641:
637:
633:
626:
625:
624:
621:
619:
615:
611:
607:
603:
599:
595:
576:
571:
567:
563:
558:
554:
550:
543:
529:
522:
518:
512:
508:
502:
499:
492:
491:
490:
487:
481:
459:
456:
451:
446:
437:
436:
435:
433:
423:Suppose that
421:
419:
415:
411:
407:
404:, not having
403:
398:
396:
391:
387:
383:
379:
375:
364:
350:
345:
343:
341:
337:
333:
329:
325:
321:
317:
313:
309:
305:
301:
297:
293:
290:
285:
283:
279:
275:
271:
267:
263:
259:
255:
254:number theory
248:Number theory
247:
245:
242:
238:
234:
231:
227:
223:
218:
216:
212:
208:
188:
184:
176:
173:
170:
162:
158:
154:
136:
132:
128:
123:
119:
115:
110:
106:
97:
93:
88:
86:
82:
81:
74:
72:
67:
65:
64:contradiction
61:
57:
52:
50:
46:
41:
37:
34:, a proof by
33:
19:
4560:
4358:Ultraproduct
4205:Model theory
4170:Independence
4106:Formal proof
4098:Proof theory
4081:
4054:
4011:real numbers
3983:second-order
3894:Substitution
3771:Metalanguage
3712:conservative
3685:Axiom schema
3629:Constructive
3599:MorseâKelley
3565:Set theories
3544:Aleph number
3537:inaccessible
3443:Grothendieck
3327:intersection
3214:Higher-order
3202:Second-order
3148:Truth tables
3105:Venn diagram
2888:Formal proof
2784:
2780:
2775:
2767:
2762:
2752:
2748:
2743:
2719:
2713:
2707:
2696:. Retrieved
2692:
2679:
2668:. Retrieved
2664:
2655:
2646:
2633:
2620:
2607:
2590:
2568:
2559:
2548:. Retrieved
2545:math.uci.edu
2544:
2517:. Retrieved
2513:
2490:. Retrieved
2486:
2466:
2451:
2444:
2417:
2415:
2412:
2238:
2234:
2185:compared to
2162:
2158:
2090:
2085:
2081:
2080:
2054:compared to
1923:
1918:
1914:
1913:
1864:compared to
1699:
1695:
1690:
1686:
1685:
1678:
1674:
1670:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1635:
1631:
1500:
1383:
1371:
1367:
1363:
1353:
1346:
1342:
1338:
1334:
1328:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1291:
1286:
972:
961:
955:
948:
944:
940:
932:
924:
914:
907:
905:
897:
886:
882:
873:
846:ad infinitum
844:
835:
765:
760:
756:
752:
751:as well. So
748:
746:
622:
617:
613:
609:
601:
598:prime number
593:
591:
485:
479:
476:
422:
399:
382:Pythagoreans
360:
335:
327:
323:
319:
315:
311:
307:
303:
291:
286:
281:
277:
269:
251:
240:
236:
232:
219:
156:
151:and proving
89:
79:
75:
68:
53:
35:
29:
4468:Type theory
4416:undecidable
4348:Truth value
4235:equivalence
3914:non-logical
3527:Enumeration
3517:Isomorphism
3464:cardinality
3448:Von Neumann
3413:Ultrafilter
3378:Uncountable
3312:equivalence
3229:Quantifiers
3219:Fixed-point
3188:First-order
3068:Consistency
3053:Proposition
3030:Traditional
3001:Lindström's
2991:Compactness
2933:Type theory
2878:Cardinality
2565:Weil, André
862:real number
766:This gives
262:L-functions
32:mathematics
4583:Categories
4279:elementary
3972:arithmetic
3840:Quantifier
3818:functional
3690:Expression
3408:Transitive
3352:identities
3337:complement
3270:hereditary
3253:Set theory
2815:PlanetMath
2805:PlanetMath
2722:(2): 117,
2698:2019-12-10
2670:2019-12-10
2573:BirkhÀuser
2550:2019-12-10
2519:2019-12-10
2492:2019-12-10
2436:References
1378:See also:
960:(that is,
947:, and let
623:But then,
386:irrational
374:irrational
296:André Weil
4550:Supertask
4453:Recursion
4411:decidable
4245:saturated
4223:of models
4146:deductive
4141:axiomatic
4061:Hilbert's
4048:Euclidean
4029:canonical
3952:axiomatic
3884:Signature
3813:Predicate
3702:Extension
3624:Ackermann
3549:Operation
3428:Universal
3418:Recursive
3393:Singleton
3388:Inhabited
3373:Countable
3363:Types of
3347:power set
3317:partition
3234:Predicate
3180:Predicate
3095:Syllogism
3085:Soundness
3058:Inference
3048:Tautology
2950:paradoxes
1641:and with
1560:−
1259:−
1245:−
1214:−
1182:−
1130:−
1109:−
1072:−
1044:−
977:). Then
418:algebraic
222:inversion
174:≡
78:Euclid's
4535:Logicism
4528:timeline
4504:Concrete
4363:Validity
4333:T-schema
4326:Kripke's
4321:Tarski's
4316:semantic
4306:Strength
4255:submodel
4250:spectrum
4218:function
4066:Tarski's
4055:Elements
4042:geometry
3998:Robinson
3919:variable
3904:function
3877:spectrum
3867:Sentence
3823:variable
3766:Language
3719:Relation
3680:Automata
3670:Alphabet
3654:language
3508:-jection
3486:codomain
3472:Function
3433:Universe
3403:Infinite
3307:Relation
3090:Validity
3080:Argument
2978:theorem,
2645:(1996),
2567:(1984),
2424:See also
853:. Since
432:rational
390:Hippasus
336:descents
80:Elements
4477:Related
4274:Diagram
4172: (
4151:Hilbert
4136:Systems
4131:Theorem
4009:of the
3954:systems
3734:Formula
3729:Grammar
3645: (
3589:General
3302:Forcing
3287:Element
3207:Monadic
2982:paradox
2923:Theorem
2859:General
2736:2323064
1630:, with
1351:√
1326:√
1289:√
970:√
964:is the
953:√
937:
921:
912:√
895:√
876:√
866:√
855:√
838:√
425:√
406:algebra
367:√
353:√
272:form a
266:Mordell
252:In the
161:squares
4240:finite
4003:Skolem
3956:
3931:Theory
3899:Symbol
3889:String
3872:atomic
3749:ground
3744:closed
3739:atomic
3695:ground
3658:syntax
3554:binary
3481:domain
3398:Finite
3163:finite
3021:Logics
2980:
2928:Theory
2734:
2579:
2459:
849:. But
228:on an
92:Fermat
85:Euclid
4230:Model
3978:Peano
3835:Proof
3675:Arity
3604:Naive
3491:image
3423:Fuzzy
3383:Empty
3332:union
3277:Class
2918:Model
2908:Lemma
2866:Axiom
2732:JSTOR
2689:(PDF)
2541:(PDF)
966:floor
860:is a
608:. So
604:, by
430:were
372:) is
205:(see
163:when
4353:Type
4156:list
3960:list
3937:list
3926:Term
3860:rank
3754:open
3648:list
3460:Maps
3365:sets
3224:Free
3194:list
2944:list
2871:list
2577:ISBN
2457:ISBN
2161:and
2113:and
2084:and
2010:and
1946:and
1917:and
1729:and
1698:and
1689:and
1677:and
1665:nor
1657:and
1649:and
1634:and
1341:and
1320:and
1312:and
1304:and
943:and
483:and
400:The
209:and
4040:of
4022:of
3970:of
3502:Sur
3476:Map
3283:Ur-
3265:Set
2813:at
2803:at
2751:",
2724:doi
2355:and
1643:a+b
968:of
755:= 2
612:= 2
310:)/2
217:).
185:mod
30:In
4585::
4426:NP
4050::
4044::
3974::
3651:),
3506:Bi
3498:In
2730:,
2720:95
2718:,
2691:.
2663:.
2641:;
2571:,
2543:.
2528:^
2512:.
2501:^
2485:.
2474:^
2465:.
2231:).
2089::
1922::
1910:).
1370:=
1366:+
1343:nâČ
1335:mâČ
1306:nâČ
1302:mâČ
763:.
620:.
397:.
294:,
280:/2
66:.
4506:/
4421:P
4176:)
3962:)
3958:(
3855:â
3850:!
3845:â
3806:=
3801:â
3796:â
3791:â§
3786:âš
3781:ÂŹ
3504:/
3500:/
3474:/
3285:)
3281:(
3168:â
3158:3
2946:)
2844:e
2837:t
2830:v
2817:.
2807:.
2781:n
2726::
2701:.
2673:.
2602:.
2553:.
2522:.
2495:.
2418:n
2396:2
2392:t
2388:=
2383:4
2379:s
2375:+
2370:4
2366:r
2343:,
2338:4
2334:t
2330:=
2325:2
2321:s
2317:+
2312:4
2308:r
2286:,
2281:4
2277:t
2273:=
2268:4
2264:s
2260:+
2255:2
2251:r
2217:2
2213:b
2209:+
2204:2
2200:a
2196:=
2193:z
2173:a
2163:a
2159:b
2145:a
2123:y
2101:b
2091:y
2086:x
2082:y
2077:.
2075:)
2063:z
2040:z
2018:b
1998:a
1976:z
1954:b
1934:a
1924:z
1919:x
1915:z
1896:2
1892:b
1888:+
1883:2
1879:a
1875:=
1872:z
1850:2
1846:a
1823:2
1819:a
1796:2
1792:b
1769:2
1765:a
1742:2
1738:b
1715:z
1712:y
1700:z
1696:y
1691:z
1687:y
1679:b
1675:a
1671:x
1667:z
1663:y
1659:z
1655:y
1651:z
1647:y
1636:b
1632:a
1616:2
1612:b
1608:+
1603:2
1599:a
1595:=
1592:z
1573:,
1568:2
1564:b
1555:2
1551:a
1547:=
1544:y
1525:,
1522:b
1519:a
1516:2
1513:=
1510:x
1476:4
1472:t
1468:=
1463:4
1459:s
1455:+
1450:4
1446:q
1423:4
1419:t
1415:=
1410:4
1406:s
1402:+
1397:2
1393:r
1372:t
1368:s
1364:r
1354:k
1347:n
1339:m
1329:k
1322:n
1318:m
1314:n
1310:m
1298:q
1292:k
1265:q
1262:n
1256:m
1251:q
1248:m
1242:k
1239:n
1233:=
1220:q
1217:n
1210:)
1205:n
1202:m
1197:(
1193:n
1188:q
1185:m
1177:k
1171:)
1165:k
1160:n
1156:(
1149:=
1136:q
1133:n
1125:k
1120:n
1115:q
1112:m
1104:k
1099:m
1093:=
1079:)
1075:q
1067:k
1061:(
1057:n
1051:)
1047:q
1039:k
1033:(
1029:m
1023:=
1011:n
1008:m
1003:=
994:k
973:k
962:q
956:k
949:q
945:n
941:m
933:n
929:/
925:m
915:k
908:k
898:k
887:q
885:/
883:p
878:2
868:2
857:2
840:2
832:.
818:s
815:r
810:=
804:s
801:2
796:r
793:2
787:=
782:q
779:p
761:s
757:s
753:q
749:q
732:,
727:2
723:r
719:2
716:=
711:2
707:q
685:,
680:2
676:r
672:4
669:=
664:2
660:)
656:r
653:2
650:(
647:=
642:2
638:q
634:2
618:r
614:r
610:p
602:p
594:p
577:,
572:2
568:p
564:=
559:2
555:q
551:2
530:,
523:2
519:q
513:2
509:p
503:=
500:2
486:q
480:p
460:q
457:p
452:=
447:2
427:2
369:2
365:(
355:2
328:A
324:Q
322:(
320:A
316:Q
314:(
312:A
308:Q
306:(
304:A
292:A
282:E
278:E
270:E
241:E
237:E
233:E
192:)
189:4
182:(
177:1
171:p
157:p
137:4
133:t
129:=
124:4
120:s
116:+
111:2
107:r
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.