4553:
1271:
972:
47:, 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
1266:{\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}}}
381:
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'
2224:
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.
31:
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
62: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.
1490:
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
1483:, 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
977:
232:
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.
819:
461:
529:
202:). 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
684:
192:
2342:
2285:
2397:
1477:
1424:
199:
138:
2932:
1572:
2218:
1897:
1617:
731:
576:
1716:
2124:
2041:
1977:
3607:
832:
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,
1524:
1851:
1824:
1797:
1770:
1743:
2172:
2144:
2100:
2062:
2017:
1997:
1953:
1933:
3690:
2831:
141:
4004:
4162:
2449:
2950:
58:
An alternative way to express this is to assume one or more solutions or examples exists, from which a smallest solution or example—a
4017:
3340:
245:
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
1670:
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:
3602:
4022:
4012:
3749:
2955:
2584:
1368:
3500:
2946:
1338:
that have the same ratio. But infinite descent on the natural numbers is impossible, so this disproves the original assumption that
4587:
4158:
2569:
761:
4255:
3999:
2824:
373:
discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is
262:
3560:
3253:
2994:
2402:
cannot have non-trivial solutions, since non-trivial solutions would give
Pythagorean triangles with two sides being squares.
4577:
4516:
4218:
3981:
3976:
3801:
3222:
2906:
4511:
4294:
4211:
3924:
3855:
3732:
2974:
2703:
870:
were rational, no "smallest" representation as a fraction could exist, as any attempt to find a "smallest" representation
3582:
4582:
4436:
4262:
3948:
3181:
3587:
3919:
3658:
2916:
2817:
954:
4314:
4309:
4243:
3833:
3227:
3195:
2886:
1487:
cannot have any two of its sides each either a square or twice a square, since there is no smallest such triangle:
2960:
4533:
4482:
4379:
3877:
3838:
3315:
4374:
2989:
429:
328:
4304:
3843:
3695:
3678:
3401:
2881:
484:
1480:
618:
43:
Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more
4206:
4183:
4144:
4030:
3971:
3617:
3537:
3381:
3325:
2938:
839:
246:
33:
76:
proves that every composite integer is divided (in Euclid's terminology "measured") by some prime number.
4496:
4223:
4201:
4168:
4061:
3907:
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3295:
3126:
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203:
155:
59:
48:
28:
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4071:
3789:
3525:
3431:
3290:
3275:
3156:
3131:
2588:
2291:
2234:
405:
presented another geometric proof by infinite descent that may be more accessible). The following is an
4552:
2349:
1429:
1376:
1289:)âwhich is positive but less than 1âand then simplified independently. So, the resulting products, say
90:
4399:
4361:
4238:
4042:
3882:
3806:
3784:
3612:
3570:
3469:
3436:
3300:
3088:
2999:
2742:
2561:
2154:) each of which is a square or twice a square, with a smaller hypotenuse than the original triangle (
1484:
84:
66:
51:, the original premise—that any solution exists—is incorrect: its correctness produces a
37:
1528:
377:. Little is known with certainty about the time or circumstances of this discovery, but the name of
4528:
4419:
4404:
4384:
4341:
4228:
4178:
4104:
4049:
3986:
3779:
3774:
3722:
3490:
3479:
3151:
3051:
2979:
2970:
2966:
2901:
2896:
2177:
1856:
1576:
690:
535:
2471:
1694:
4557:
4326:
4289:
4274:
4267:
4250:
4036:
3902:
3828:
3811:
3764:
3577:
3486:
3320:
3305:
3265:
3217:
3202:
3190:
3146:
3121:
2891:
2840:
2720:
2105:
2022:
1958:
402:
210:
195:
4054:
3510:
4492:
4299:
4109:
4099:
3991:
3872:
3707:
3683:
3464:
3448:
3353:
3330:
3207:
3176:
3141:
3036:
2871:
2565:
2445:
594:
374:
362:
320:
307:) is finite, which is certainly a necessary condition for the finite generation of the group
209:
In some cases, to the modern eye, his "method of infinite descent" is an exploitation of the
4506:
4501:
4394:
4351:
4173:
4134:
4129:
4114:
3940:
3897:
3794:
3592:
3542:
3116:
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2712:
1494:
398:
351:
2526:
1829:
1802:
1775:
1748:
1721:
4487:
4477:
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4414:
4369:
4331:
4233:
4153:
3960:
3887:
3860:
3848:
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3642:
3597:
3565:
3366:
3168:
3111:
3061:
3026:
2984:
420:
288:
277:
323:. In this way, abstractly-defined cohomology groups in the theory become identified with
4472:
4451:
4409:
4389:
4284:
4139:
3737:
3727:
3717:
3712:
3646:
3520:
3396:
3285:
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2674:
2631:
2627:
2157:
2129:
2085:
2047:
2002:
1982:
1938:
1918:
370:
287:
had to make more explicit the way of quantifying the size of a solution, by means of a
218:
214:
44:
4571:
4446:
4124:
3631:
3416:
3406:
3376:
3361:
3031:
2799:
2553:
2418:
390:
366:
284:
242:
149:
52:
365:(i.e. cannot be expressed as a fraction of two whole numbers) was discovered by the
4346:
4193:
4094:
4086:
3966:
3914:
3823:
3759:
3742:
3673:
3532:
3391:
3093:
2876:
2649:
834:
586:
87:. Two typical examples are showing the non-solvability of the Diophantine equation
2439:
2409:= 4 case of Fermat's Theorem, see the articles by Grant and Perella and Barbara.
4456:
4336:
3515:
3505:
3452:
3136:
3056:
3041:
2921:
2866:
850:
20:
3386:
3241:
3212:
3018:
2803:
2793:
2456:
a special case of proof by contradiction called the method of infinite descent
250:
4538:
4441:
3494:
3411:
3371:
3335:
3271:
3083:
3073:
3046:
369:, and is perhaps the earliest known example of a proof by infinite descent.
878:
would imply that a smaller one existed, which is a similar contradiction.)
853:, which can be either rational or irrational, the only option left is for
4523:
4321:
3769:
3474:
3068:
2789:
2499:"Fermat's Method of Infinite Descent | Brilliant Math & Science Wiki"
2019:) each of which is a square or twice a square, and a smaller hypotenuse (
378:
4119:
2911:
2724:
1627:
406:
394:
254:
1276:
The numerator and denominator were each multiplied by the expression (
2809:
2558:
Number Theory: An approach through history from
Hammurapi to Legendre
80:
73:
2716:
2614:
27:, also known as Fermat's method of descent, is a particular kind of
2755:
Grant, Mike, and
Perella, Malcolm, "Descending to the irrational",
3663:
3009:
2854:
2701:
Sagher, Yoram (February 1988), "What
Pythagoras could have done",
224:. The context is of a hypothetical non-trivial rational point on
2813:
2615:"The discovery of incommensurability by Hippasus of Metapontum"
331:
was at the start of what later became a very extensive theory.
2601:
814:{\displaystyle {\frac {p}{q}}={\frac {2r}{2s}}={\frac {r}{s}}}
65:
The earliest uses of the method of infinite descent appear in
36:, and is often used to show that a given equation, such as a
908:
is not an integer, but is rational and can be expressed as
319:, one must do calculations in what later was recognised as
72:. A typical example is Proposition 31 of Book 7, in which
1426:
in integers is sufficient to show the non-solvability of
1691:
are both squares. But then the right triangle with legs
1305:
respectively. Therefore, no matter what natural numbers
1772:
also would have integer sides including a square leg (
2405:
For other similar proofs by infinite descent for the
2352:
2294:
2237:
2180:
2160:
2132:
2108:
2088:
2050:
2025:
2005:
1985:
1961:
1941:
1921:
1859:
1832:
1805:
1778:
1751:
1724:
1697:
1579:
1531:
1497:
1432:
1379:
1369:
Fermat's right triangle theorem § Fermat's proof
975:
764:
693:
621:
538:
487:
432:
158:
93:
4465:
4360:
4192:
4085:
3937:
3630:
3553:
3447:
3351:
3240:
3167:
3102:
3017:
3008:
2930:
2847:
291:â a concept that became foundational. To show that
2391:
2336:
2279:
2212:
2166:
2138:
2118:
2094:
2082:is a square. The integer right triangle with legs
2056:
2035:
2011:
1991:
1971:
1947:
1927:
1915:is a square. The integer right triangle with legs
1891:
1845:
1818:
1791:
1764:
1737:
1710:
1611:
1566:
1518:
1471:
1418:
1347:could be expressed as a ratio of natural numbers.
1265:
813:
725:
678:
570:
523:
455:
186:
132:
2768:Barbara, Roy, "Fermat's last theorem in the case
840:this is impossible in the set of natural numbers
257:, that the rational points on an elliptic curve
16:Mathematical proof technique using contradiction
2825:
1297:, are themselves integers, and are less than
265:, used an infinite descent argument based on
8:
2441:The Moment of Proof: Mathematical Epiphanies
1662:is a square or twice a square, then each of
83:, who coined the term and often used it for
3651:
3246:
3014:
2832:
2818:
2810:
2650:"Square root of 2 is irrational (Proof 8)"
456:{\displaystyle {\sqrt {2}}={\frac {p}{q}}}
2383:
2370:
2357:
2351:
2325:
2312:
2299:
2293:
2268:
2255:
2242:
2236:
2204:
2191:
2179:
2159:
2131:
2109:
2107:
2087:
2049:
2026:
2024:
2004:
1984:
1962:
1960:
1940:
1920:
1883:
1870:
1858:
1837:
1831:
1810:
1804:
1783:
1777:
1756:
1750:
1729:
1723:
1698:
1696:
1603:
1590:
1578:
1555:
1542:
1530:
1496:
1463:
1450:
1437:
1431:
1410:
1397:
1384:
1378:
1224:
1188:
1163:
1151:
1140:
1111:
1090:
1084:
1053:
1025:
1014:
994:
980:
976:
974:
801:
778:
765:
763:
714:
698:
692:
667:
651:
629:
620:
559:
546:
537:
524:{\displaystyle 2={\frac {p^{2}}{q^{2}}},}
510:
500:
494:
486:
443:
433:
431:
383:
168:
157:
124:
111:
98:
92:
1826:), and would have a smaller hypotenuse (
1479:in integers, which is a special case of
2444:. Oxford University Press. p. 43.
2430:
1658:can be twice a square. Furthermore, if
679:{\displaystyle 2q^{2}=(2r)^{2}=4r^{2},}
142:Fermat's theorem on sums of two squares
79:The method was much later developed by
1322:, there exist smaller natural numbers
7:
2520:
2518:
2493:
2491:
2466:
2464:
863:(Alternatively, this proves that if
187:{\displaystyle p\equiv 1{\pmod {4}}}
176:
2337:{\displaystyle r^{4}+s^{2}=t^{4},}
2280:{\displaystyle r^{2}+s^{4}=t^{4},}
14:
2736:Dolan, Stan, "Fermat's method of
2392:{\displaystyle r^{4}+s^{4}=t^{2}}
1650:are each odd means that neither
1472:{\displaystyle q^{4}+s^{4}=t^{4}}
1419:{\displaystyle r^{2}+s^{4}=t^{4}}
940:be the largest integer less than
276:To extend this to the case of an
148:can be expressed as a sum of two
144:, which states that an odd prime
133:{\displaystyle r^{2}+s^{4}=t^{4}}
4551:
2800:Example of Fermat's last theorem
2673:Conrad, Keith (August 6, 2008).
2228:This implies that the equations
327:in the tradition of Fermat. The
263:finitely-generated abelian group
736:which shows that 2 must divide
169:
2617:, Annals of Mathematics, 1945.
1567:{\displaystyle y=a^{2}-b^{2},}
648:
638:
423:. Then it could be written as
180:
170:
1:
4512:History of mathematical logic
2704:American Mathematical Monthly
2213:{\displaystyle z=a^{2}+b^{2}}
1892:{\displaystyle z=a^{2}+b^{2}}
1612:{\displaystyle z=a^{2}+b^{2}}
726:{\displaystyle q^{2}=2r^{2},}
571:{\displaystyle 2q^{2}=p^{2},}
213:of the doubling function for
4437:Primitive recursive function
2527:"Fermat's Method of Descent"
1711:{\displaystyle {\sqrt {yz}}}
1642:both odd. The property that
2759:83, July 1999, pp. 263â267.
2604:, Nrich.org, November 2004.
2119:{\displaystyle {\sqrt {y}}}
2036:{\displaystyle {\sqrt {z}}}
1979:also would have two sides (
1972:{\displaystyle {\sqrt {z}}}
1799:) and a square hypotenuse (
478:. Then squaring would give
409:proof along similar lines:
253:. The structural result of
4604:
3501:SchröderâBernstein theorem
3228:Monadic predicate calculus
2887:Foundations of mathematics
2472:"What Is Infinite Descent"
2438:Benson, Donald C. (2000).
1366:
4547:
4534:Philosophy of mathematics
4483:Automated theorem proving
3654:
3608:Von NeumannâBernaysâGödel
3249:
2602:"The Dangerous Ratio ..."
2585:"The Pythagorean Theorem"
466:for two natural numbers,
200:proof by infinite descent
4588:Mathematical terminology
2638:, Copernicus, p. 25
315:) of rational points of
4184:Self-verifying theories
4005:Tarski's axiomatization
2956:Tarski's undefinability
2951:incompleteness theorems
2776:91, July 2007, 260â262.
2746:95, July 2011, 269â271.
1373:The non-solvability of
891:if it is not an integer
384:Conway & Guy (1996)
382:Constant", for example
247:algebraic number theory
34:well-ordering principle
4558:Mathematics portal
4169:Proof of impossibility
3817:propositional variable
3127:Propositional calculus
2682:kconrad.math.uconn.edu
2587:, Dept. of Math. Ed.,
2393:
2338:
2281:
2214:
2168:
2146:would have two sides (
2140:
2120:
2096:
2058:
2037:
2013:
1993:
1973:
1949:
1929:
1893:
1847:
1820:
1793:
1766:
1739:
1712:
1613:
1568:
1520:
1519:{\displaystyle x=2ab,}
1473:
1420:
1267:
815:
727:
680:
589:, it must also divide
572:
525:
457:
228:. Doubling a point on
204:arithmetic progression
188:
134:
60:minimal counterexample
49:mathematical induction
29:proof by contradiction
4578:Diophantine equations
4427:Kolmogorov complexity
4380:Computably enumerable
4280:Model complete theory
4072:Principia Mathematica
3132:Propositional formula
2961:BanachâTarski paradox
2589:University of Georgia
2583:Stephanie J. Morris,
2394:
2339:
2282:
2215:
2169:
2141:
2121:
2097:
2059:
2038:
2014:
1994:
1974:
1950:
1930:
1894:
1848:
1846:{\displaystyle a^{2}}
1821:
1819:{\displaystyle a^{2}}
1794:
1792:{\displaystyle b^{2}}
1767:
1765:{\displaystyle a^{2}}
1740:
1738:{\displaystyle b^{2}}
1713:
1614:
1569:
1521:
1481:Fermat's Last Theorem
1474:
1421:
1268:
895:For positive integer
816:
728:
681:
573:
526:
458:
401:by infinite descent (
189:
135:
85:Diophantine equations
4375:ChurchâTuring thesis
4362:Computability theory
3571:continuum hypothesis
3089:Square of opposition
2947:Gödel's completeness
2774:Mathematical Gazette
2757:Mathematical Gazette
2743:Mathematical Gazette
2654:www.cut-the-knot.org
2476:www.cut-the-knot.org
2350:
2292:
2235:
2178:
2158:
2130:
2106:
2086:
2048:
2023:
2003:
1983:
1959:
1939:
1919:
1857:
1830:
1803:
1776:
1749:
1722:
1695:
1577:
1529:
1495:
1485:Pythagorean triangle
1430:
1377:
1363:and its permutations
1313:are used to express
973:
928:for natural numbers
762:
691:
619:
536:
485:
430:
335:Application examples
329:MordellâWeil theorem
156:
91:
40:, has no solutions.
38:Diophantine equation
4583:Mathematical proofs
4529:Mathematical object
4420:P versus NP problem
4385:Computable function
4179:Reverse mathematics
4105:Logical consequence
3982:primitive recursive
3977:elementary function
3750:Free/bound variable
3603:TarskiâGrothendieck
3122:Logical connectives
3052:Logical equivalence
2902:Logical consequence
2636:The Book of Numbers
1351:Non-solvability of
605:, for some integer
350:The proof that the
273:in Fermat's style.
4327:Transfer principle
4290:Semantics of logic
4275:Categorical theory
4251:Non-standard model
3765:Logical connective
2892:Information theory
2841:Mathematical logic
2675:"Infinite Descent"
2564:, pp. 75â79,
2389:
2334:
2277:
2210:
2164:
2136:
2116:
2092:
2054:
2033:
2009:
1989:
1969:
1945:
1925:
1889:
1843:
1816:
1789:
1762:
1735:
1708:
1609:
1564:
1516:
1469:
1416:
1263:
1261:
860:to be irrational.
811:
723:
676:
568:
521:
453:
403:John Horton Conway
196:Modular arithmetic
184:
130:
4565:
4564:
4497:Abstract category
4300:Theories of truth
4110:Rule of inference
4100:Natural deduction
4081:
4080:
3626:
3625:
3331:Cartesian product
3236:
3235:
3142:Many-valued logic
3117:Boolean functions
3000:Russell's paradox
2975:diagonal argument
2872:First-order logic
2772: = 4",
2525:Donaldson, Neil.
2451:978-0-19-513919-8
2167:{\displaystyle a}
2139:{\displaystyle a}
2114:
2095:{\displaystyle b}
2057:{\displaystyle z}
2031:
2012:{\displaystyle b}
1992:{\displaystyle a}
1967:
1948:{\displaystyle b}
1928:{\displaystyle a}
1706:
1257:
1212:
1196:
1168:
1156:
1128:
1116:
1095:
1072:
1058:
1030:
1002:
985:
882:Irrationality of
809:
796:
773:
748:for some integer
585:. Because 2 is a
581:so 2 must divide
516:
451:
438:
340:Irrationality of
321:Galois cohomology
249:and the study of
4595:
4556:
4555:
4507:History of logic
4502:Category of sets
4395:Decision problem
4174:Ordinal analysis
4115:Sequent calculus
4013:Boolean algebras
3953:
3952:
3927:
3898:logical/constant
3652:
3638:
3561:ZermeloâFraenkel
3312:Set operations:
3247:
3184:
3015:
2995:LöwenheimâSkolem
2882:Formal semantics
2834:
2827:
2820:
2811:
2790:Infinite descent
2777:
2766:
2760:
2753:
2747:
2738:descente infinie
2734:
2728:
2727:
2698:
2692:
2691:
2689:
2688:
2679:
2670:
2664:
2663:
2661:
2660:
2646:
2640:
2639:
2624:
2618:
2613:Kurt von Fritz,
2611:
2605:
2598:
2592:
2581:
2575:
2574:
2550:
2544:
2543:
2541:
2540:
2531:
2522:
2513:
2512:
2510:
2509:
2495:
2486:
2485:
2483:
2482:
2468:
2459:
2458:
2435:
2398:
2396:
2395:
2390:
2388:
2387:
2375:
2374:
2362:
2361:
2343:
2341:
2340:
2335:
2330:
2329:
2317:
2316:
2304:
2303:
2286:
2284:
2283:
2278:
2273:
2272:
2260:
2259:
2247:
2246:
2219:
2217:
2216:
2211:
2209:
2208:
2196:
2195:
2173:
2171:
2170:
2165:
2145:
2143:
2142:
2137:
2125:
2123:
2122:
2117:
2115:
2110:
2101:
2099:
2098:
2093:
2065:
2063:
2061:
2060:
2055:
2042:
2040:
2039:
2034:
2032:
2027:
2018:
2016:
2015:
2010:
1998:
1996:
1995:
1990:
1978:
1976:
1975:
1970:
1968:
1963:
1954:
1952:
1951:
1946:
1934:
1932:
1931:
1926:
1898:
1896:
1895:
1890:
1888:
1887:
1875:
1874:
1852:
1850:
1849:
1844:
1842:
1841:
1825:
1823:
1822:
1817:
1815:
1814:
1798:
1796:
1795:
1790:
1788:
1787:
1771:
1769:
1768:
1763:
1761:
1760:
1744:
1742:
1741:
1736:
1734:
1733:
1717:
1715:
1714:
1709:
1707:
1699:
1628:relatively prime
1618:
1616:
1615:
1610:
1608:
1607:
1595:
1594:
1573:
1571:
1570:
1565:
1560:
1559:
1547:
1546:
1525:
1523:
1522:
1517:
1478:
1476:
1475:
1470:
1468:
1467:
1455:
1454:
1442:
1441:
1425:
1423:
1422:
1417:
1415:
1414:
1402:
1401:
1389:
1388:
1346:
1345:
1334: <
1326: <
1321:
1320:
1284:
1283:
1272:
1270:
1269:
1264:
1262:
1258:
1256:
1242:
1225:
1217:
1213:
1211:
1201:
1197:
1189:
1179:
1169:
1164:
1162:
1158:
1157:
1152:
1141:
1133:
1129:
1127:
1117:
1112:
1106:
1096:
1091:
1085:
1077:
1073:
1071:
1070:
1066:
1059:
1054:
1043:
1042:
1038:
1031:
1026:
1015:
1007:
1003:
995:
986:
981:
965:
964:
948:
947:
927:
925:
924:
919:
916:
907:
906:
890:
889:
869:
868:
859:
858:
848:
847:
831:
830:
820:
818:
817:
812:
810:
802:
797:
795:
787:
779:
774:
766:
732:
730:
729:
724:
719:
718:
703:
702:
685:
683:
682:
677:
672:
671:
656:
655:
634:
633:
577:
575:
574:
569:
564:
563:
551:
550:
530:
528:
527:
522:
517:
515:
514:
505:
504:
495:
477:
471:
462:
460:
459:
454:
452:
444:
439:
434:
418:
417:
360:
359:
352:square root of 2
346:
345:
193:
191:
190:
185:
183:
139:
137:
136:
131:
129:
128:
116:
115:
103:
102:
25:infinite descent
4603:
4602:
4598:
4597:
4596:
4594:
4593:
4592:
4568:
4567:
4566:
4561:
4550:
4543:
4488:Category theory
4478:Algebraic logic
4461:
4432:Lambda calculus
4370:Church encoding
4356:
4332:Truth predicate
4188:
4154:Complete theory
4077:
3946:
3942:
3938:
3933:
3925:
3645: and
3641:
3636:
3622:
3598:New Foundations
3566:axiom of choice
3549:
3511:Gödel numbering
3451: and
3443:
3347:
3232:
3182:
3163:
3112:Boolean algebra
3098:
3062:Equiconsistency
3027:Classical logic
3004:
2985:Halting problem
2973: and
2949: and
2937: and
2936:
2931:Theorems (
2926:
2843:
2838:
2786:
2784:Further reading
2781:
2780:
2767:
2763:
2754:
2750:
2735:
2731:
2717:10.2307/2323064
2700:
2699:
2695:
2686:
2684:
2677:
2672:
2671:
2667:
2658:
2656:
2648:
2647:
2643:
2632:Guy, Richard K.
2628:Conway, John H.
2626:
2625:
2621:
2612:
2608:
2599:
2595:
2582:
2578:
2572:
2552:
2551:
2547:
2538:
2536:
2529:
2524:
2523:
2516:
2507:
2505:
2497:
2496:
2489:
2480:
2478:
2470:
2469:
2462:
2452:
2437:
2436:
2432:
2427:
2415:
2379:
2366:
2353:
2348:
2347:
2321:
2308:
2295:
2290:
2289:
2264:
2251:
2238:
2233:
2232:
2200:
2187:
2176:
2175:
2156:
2155:
2128:
2127:
2126:and hypotenuse
2104:
2103:
2084:
2083:
2046:
2045:
2044:
2021:
2020:
2001:
2000:
1981:
1980:
1957:
1956:
1955:and hypotenuse
1937:
1936:
1917:
1916:
1879:
1866:
1855:
1854:
1833:
1828:
1827:
1806:
1801:
1800:
1779:
1774:
1773:
1752:
1747:
1746:
1745:and hypotenuse
1725:
1720:
1719:
1693:
1692:
1683:: In this case
1599:
1586:
1575:
1574:
1551:
1538:
1527:
1526:
1493:
1492:
1459:
1446:
1433:
1428:
1427:
1406:
1393:
1380:
1375:
1374:
1371:
1365:
1341:
1339:
1316:
1314:
1279:
1277:
1260:
1259:
1243:
1226:
1215:
1214:
1184:
1180:
1147:
1143:
1142:
1131:
1130:
1107:
1086:
1075:
1074:
1052:
1048:
1044:
1024:
1020:
1016:
1005:
1004:
987:
971:
970:
960:
958:
943:
941:
920:
917:
912:
911:
909:
902:
900:
899:, suppose that
893:
885:
883:
866:
864:
856:
854:
845:
843:
828:
826:
788:
780:
760:
759:
710:
694:
689:
688:
663:
647:
625:
617:
616:
555:
542:
534:
533:
506:
496:
483:
482:
473:
467:
428:
427:
415:
413:
399:geometric proof
397:, worked out a
357:
355:
348:
343:
341:
337:
289:height function
278:abelian variety
239:
215:rational points
154:
153:
120:
107:
94:
89:
88:
45:natural numbers
17:
12:
11:
5:
4601:
4599:
4591:
4590:
4585:
4580:
4570:
4569:
4563:
4562:
4548:
4545:
4544:
4542:
4541:
4536:
4531:
4526:
4521:
4520:
4519:
4509:
4504:
4499:
4490:
4485:
4480:
4475:
4473:Abstract logic
4469:
4467:
4463:
4462:
4460:
4459:
4454:
4452:Turing machine
4449:
4444:
4439:
4434:
4429:
4424:
4423:
4422:
4417:
4412:
4407:
4402:
4392:
4390:Computable set
4387:
4382:
4377:
4372:
4366:
4364:
4358:
4357:
4355:
4354:
4349:
4344:
4339:
4334:
4329:
4324:
4319:
4318:
4317:
4312:
4307:
4297:
4292:
4287:
4285:Satisfiability
4282:
4277:
4272:
4271:
4270:
4260:
4259:
4258:
4248:
4247:
4246:
4241:
4236:
4231:
4226:
4216:
4215:
4214:
4209:
4202:Interpretation
4198:
4196:
4190:
4189:
4187:
4186:
4181:
4176:
4171:
4166:
4156:
4151:
4150:
4149:
4148:
4147:
4137:
4132:
4122:
4117:
4112:
4107:
4102:
4097:
4091:
4089:
4083:
4082:
4079:
4078:
4076:
4075:
4067:
4066:
4065:
4064:
4059:
4058:
4057:
4052:
4047:
4027:
4026:
4025:
4023:minimal axioms
4020:
4009:
4008:
4007:
3996:
3995:
3994:
3989:
3984:
3979:
3974:
3969:
3956:
3954:
3935:
3934:
3932:
3931:
3930:
3929:
3917:
3912:
3911:
3910:
3905:
3900:
3895:
3885:
3880:
3875:
3870:
3869:
3868:
3863:
3853:
3852:
3851:
3846:
3841:
3836:
3826:
3821:
3820:
3819:
3814:
3809:
3799:
3798:
3797:
3792:
3787:
3782:
3777:
3772:
3762:
3757:
3752:
3747:
3746:
3745:
3740:
3735:
3730:
3720:
3715:
3713:Formation rule
3710:
3705:
3704:
3703:
3698:
3688:
3687:
3686:
3676:
3671:
3666:
3661:
3655:
3649:
3632:Formal systems
3628:
3627:
3624:
3623:
3621:
3620:
3615:
3610:
3605:
3600:
3595:
3590:
3585:
3580:
3575:
3574:
3573:
3568:
3557:
3555:
3551:
3550:
3548:
3547:
3546:
3545:
3535:
3530:
3529:
3528:
3521:Large cardinal
3518:
3513:
3508:
3503:
3498:
3484:
3483:
3482:
3477:
3472:
3457:
3455:
3445:
3444:
3442:
3441:
3440:
3439:
3434:
3429:
3419:
3414:
3409:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3369:
3364:
3358:
3356:
3349:
3348:
3346:
3345:
3344:
3343:
3338:
3333:
3328:
3323:
3318:
3310:
3309:
3308:
3303:
3293:
3288:
3286:Extensionality
3283:
3281:Ordinal number
3278:
3268:
3263:
3262:
3261:
3250:
3244:
3238:
3237:
3234:
3233:
3231:
3230:
3225:
3220:
3215:
3210:
3205:
3200:
3199:
3198:
3188:
3187:
3186:
3173:
3171:
3165:
3164:
3162:
3161:
3160:
3159:
3154:
3149:
3139:
3134:
3129:
3124:
3119:
3114:
3108:
3106:
3100:
3099:
3097:
3096:
3091:
3086:
3081:
3076:
3071:
3066:
3065:
3064:
3054:
3049:
3044:
3039:
3034:
3029:
3023:
3021:
3012:
3006:
3005:
3003:
3002:
2997:
2992:
2987:
2982:
2977:
2965:Cantor's
2963:
2958:
2953:
2943:
2941:
2928:
2927:
2925:
2924:
2919:
2914:
2909:
2904:
2899:
2894:
2889:
2884:
2879:
2874:
2869:
2864:
2863:
2862:
2851:
2849:
2845:
2844:
2839:
2837:
2836:
2829:
2822:
2814:
2808:
2807:
2797:
2785:
2782:
2779:
2778:
2761:
2748:
2729:
2693:
2665:
2641:
2619:
2606:
2593:
2576:
2570:
2545:
2514:
2487:
2460:
2450:
2429:
2428:
2426:
2423:
2422:
2421:
2414:
2411:
2400:
2399:
2386:
2382:
2378:
2373:
2369:
2365:
2360:
2356:
2345:
2333:
2328:
2324:
2320:
2315:
2311:
2307:
2302:
2298:
2287:
2276:
2271:
2267:
2263:
2258:
2254:
2250:
2245:
2241:
2222:
2221:
2207:
2203:
2199:
2194:
2190:
2186:
2183:
2163:
2135:
2113:
2091:
2067:
2053:
2030:
2008:
1988:
1966:
1944:
1924:
1900:
1886:
1882:
1878:
1873:
1869:
1865:
1862:
1840:
1836:
1813:
1809:
1786:
1782:
1759:
1755:
1732:
1728:
1705:
1702:
1634:odd and hence
1606:
1602:
1598:
1593:
1589:
1585:
1582:
1563:
1558:
1554:
1550:
1545:
1541:
1537:
1534:
1515:
1512:
1509:
1506:
1503:
1500:
1466:
1462:
1458:
1453:
1449:
1445:
1440:
1436:
1413:
1409:
1405:
1400:
1396:
1392:
1387:
1383:
1364:
1349:
1274:
1273:
1255:
1252:
1249:
1246:
1241:
1238:
1235:
1232:
1229:
1223:
1220:
1218:
1216:
1210:
1207:
1204:
1200:
1195:
1192:
1187:
1183:
1178:
1175:
1172:
1167:
1161:
1155:
1150:
1146:
1139:
1136:
1134:
1132:
1126:
1123:
1120:
1115:
1110:
1105:
1102:
1099:
1094:
1089:
1083:
1080:
1078:
1076:
1069:
1065:
1062:
1057:
1051:
1047:
1041:
1037:
1034:
1029:
1023:
1019:
1013:
1010:
1008:
1006:
1001:
998:
993:
990:
988:
984:
979:
978:
892:
880:
825:Therefore, if
823:
822:
808:
805:
800:
794:
791:
786:
783:
777:
772:
769:
734:
733:
722:
717:
713:
709:
706:
701:
697:
686:
675:
670:
666:
662:
659:
654:
650:
646:
643:
640:
637:
632:
628:
624:
595:Euclid's lemma
579:
578:
567:
562:
558:
554:
549:
545:
541:
531:
520:
513:
509:
503:
499:
493:
490:
464:
463:
450:
447:
442:
437:
391:ancient Greeks
367:ancient Greeks
347:
338:
336:
333:
238:
235:
219:elliptic curve
182:
179:
175:
172:
167:
164:
161:
127:
123:
119:
114:
110:
106:
101:
97:
15:
13:
10:
9:
6:
4:
3:
2:
4600:
4589:
4586:
4584:
4581:
4579:
4576:
4575:
4573:
4560:
4559:
4554:
4546:
4540:
4537:
4535:
4532:
4530:
4527:
4525:
4522:
4518:
4515:
4514:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4494:
4491:
4489:
4486:
4484:
4481:
4479:
4476:
4474:
4471:
4470:
4468:
4464:
4458:
4455:
4453:
4450:
4448:
4447:Recursive set
4445:
4443:
4440:
4438:
4435:
4433:
4430:
4428:
4425:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4397:
4396:
4393:
4391:
4388:
4386:
4383:
4381:
4378:
4376:
4373:
4371:
4368:
4367:
4365:
4363:
4359:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4320:
4316:
4313:
4311:
4308:
4306:
4303:
4302:
4301:
4298:
4296:
4293:
4291:
4288:
4286:
4283:
4281:
4278:
4276:
4273:
4269:
4266:
4265:
4264:
4261:
4257:
4256:of arithmetic
4254:
4253:
4252:
4249:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4221:
4220:
4217:
4213:
4210:
4208:
4205:
4204:
4203:
4200:
4199:
4197:
4195:
4191:
4185:
4182:
4180:
4177:
4175:
4172:
4170:
4167:
4164:
4163:from ZFC
4160:
4157:
4155:
4152:
4146:
4143:
4142:
4141:
4138:
4136:
4133:
4131:
4128:
4127:
4126:
4123:
4121:
4118:
4116:
4113:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4092:
4090:
4088:
4084:
4074:
4073:
4069:
4068:
4063:
4062:non-Euclidean
4060:
4056:
4053:
4051:
4048:
4046:
4045:
4041:
4040:
4038:
4035:
4034:
4032:
4028:
4024:
4021:
4019:
4016:
4015:
4014:
4010:
4006:
4003:
4002:
4001:
3997:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3964:
3962:
3958:
3957:
3955:
3950:
3944:
3939:Example
3936:
3928:
3923:
3922:
3921:
3918:
3916:
3913:
3909:
3906:
3904:
3901:
3899:
3896:
3894:
3891:
3890:
3889:
3886:
3884:
3881:
3879:
3876:
3874:
3871:
3867:
3864:
3862:
3859:
3858:
3857:
3854:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3831:
3830:
3827:
3825:
3822:
3818:
3815:
3813:
3810:
3808:
3805:
3804:
3803:
3800:
3796:
3793:
3791:
3788:
3786:
3783:
3781:
3778:
3776:
3773:
3771:
3768:
3767:
3766:
3763:
3761:
3758:
3756:
3753:
3751:
3748:
3744:
3741:
3739:
3736:
3734:
3731:
3729:
3726:
3725:
3724:
3721:
3719:
3716:
3714:
3711:
3709:
3706:
3702:
3699:
3697:
3696:by definition
3694:
3693:
3692:
3689:
3685:
3682:
3681:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3656:
3653:
3650:
3648:
3644:
3639:
3633:
3629:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3583:KripkeâPlatek
3581:
3579:
3576:
3572:
3569:
3567:
3564:
3563:
3562:
3559:
3558:
3556:
3552:
3544:
3541:
3540:
3539:
3536:
3534:
3531:
3527:
3524:
3523:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3496:
3492:
3488:
3485:
3481:
3478:
3476:
3473:
3471:
3468:
3467:
3466:
3462:
3459:
3458:
3456:
3454:
3450:
3446:
3438:
3435:
3433:
3430:
3428:
3427:constructible
3425:
3424:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3359:
3357:
3355:
3350:
3342:
3339:
3337:
3334:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3313:
3311:
3307:
3304:
3302:
3299:
3298:
3297:
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3273:
3269:
3267:
3264:
3260:
3257:
3256:
3255:
3252:
3251:
3248:
3245:
3243:
3239:
3229:
3226:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3206:
3204:
3201:
3197:
3194:
3193:
3192:
3189:
3185:
3180:
3179:
3178:
3175:
3174:
3172:
3170:
3166:
3158:
3155:
3153:
3150:
3148:
3145:
3144:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3109:
3107:
3105:
3104:Propositional
3101:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3063:
3060:
3059:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3032:Logical truth
3030:
3028:
3025:
3024:
3022:
3020:
3016:
3013:
3011:
3007:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2972:
2968:
2964:
2962:
2959:
2957:
2954:
2952:
2948:
2945:
2944:
2942:
2940:
2934:
2929:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2888:
2885:
2883:
2880:
2878:
2875:
2873:
2870:
2868:
2865:
2861:
2858:
2857:
2856:
2853:
2852:
2850:
2846:
2842:
2835:
2830:
2828:
2823:
2821:
2816:
2815:
2812:
2805:
2801:
2798:
2795:
2791:
2788:
2787:
2783:
2775:
2771:
2765:
2762:
2758:
2752:
2749:
2745:
2744:
2739:
2733:
2730:
2726:
2722:
2718:
2714:
2710:
2706:
2705:
2697:
2694:
2683:
2676:
2669:
2666:
2655:
2651:
2645:
2642:
2637:
2633:
2629:
2623:
2620:
2616:
2610:
2607:
2603:
2600:Brian Clegg,
2597:
2594:
2590:
2586:
2580:
2577:
2573:
2571:0-8176-3141-0
2567:
2563:
2559:
2555:
2549:
2546:
2535:
2528:
2521:
2519:
2515:
2504:
2503:brilliant.org
2500:
2494:
2492:
2488:
2477:
2473:
2467:
2465:
2461:
2457:
2453:
2447:
2443:
2442:
2434:
2431:
2424:
2420:
2419:Vieta jumping
2417:
2416:
2412:
2410:
2408:
2403:
2384:
2380:
2376:
2371:
2367:
2363:
2358:
2354:
2346:
2331:
2326:
2322:
2318:
2313:
2309:
2305:
2300:
2296:
2288:
2274:
2269:
2265:
2261:
2256:
2252:
2248:
2243:
2239:
2231:
2230:
2229:
2226:
2205:
2201:
2197:
2192:
2188:
2184:
2181:
2161:
2153:
2149:
2133:
2111:
2089:
2081:
2077:
2076:
2072:
2068:
2051:
2028:
2006:
1986:
1964:
1942:
1922:
1914:
1910:
1909:
1905:
1901:
1884:
1880:
1876:
1871:
1867:
1863:
1860:
1838:
1834:
1811:
1807:
1784:
1780:
1757:
1753:
1730:
1726:
1703:
1700:
1690:
1686:
1682:
1681:
1677:
1673:
1672:
1671:
1669:
1665:
1661:
1657:
1653:
1649:
1645:
1641:
1637:
1633:
1629:
1626:
1622:
1604:
1600:
1596:
1591:
1587:
1583:
1580:
1561:
1556:
1552:
1548:
1543:
1539:
1535:
1532:
1513:
1510:
1507:
1504:
1501:
1498:
1488:
1486:
1482:
1464:
1460:
1456:
1451:
1447:
1443:
1438:
1434:
1411:
1407:
1403:
1398:
1394:
1390:
1385:
1381:
1370:
1362:
1358:
1354:
1350:
1348:
1344:
1337:
1333:
1329:
1325:
1319:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1285: â
1282:
1253:
1250:
1247:
1244:
1239:
1236:
1233:
1230:
1227:
1221:
1219:
1208:
1205:
1202:
1198:
1193:
1190:
1185:
1181:
1176:
1173:
1170:
1165:
1159:
1153:
1148:
1144:
1137:
1135:
1124:
1121:
1118:
1113:
1108:
1103:
1100:
1097:
1092:
1087:
1081:
1079:
1067:
1063:
1060:
1055:
1049:
1045:
1039:
1035:
1032:
1027:
1021:
1017:
1011:
1009:
999:
996:
991:
989:
982:
969:
968:
967:
963:
956:
952:
946:
939:
935:
931:
923:
915:
905:
898:
888:
881:
879:
877:
873:
861:
852:
841:
837:
836:
806:
803:
798:
792:
789:
784:
781:
775:
770:
767:
758:
757:
756:
753:
751:
747:
743:
739:
720:
715:
711:
707:
704:
699:
695:
687:
673:
668:
664:
660:
657:
652:
644:
641:
635:
630:
626:
622:
615:
614:
613:
610:
608:
604:
600:
596:
592:
588:
584:
565:
560:
556:
552:
547:
543:
539:
532:
518:
511:
507:
501:
497:
491:
488:
481:
480:
479:
476:
470:
448:
445:
440:
435:
426:
425:
424:
422:
412:Suppose that
410:
408:
404:
400:
396:
393:, not having
392:
387:
385:
380:
376:
372:
368:
364:
353:
339:
334:
332:
330:
326:
322:
318:
314:
310:
306:
302:
298:
294:
290:
286:
282:
279:
274:
272:
268:
264:
260:
256:
252:
248:
244:
243:number theory
237:Number theory
236:
234:
231:
227:
223:
220:
216:
212:
207:
205:
201:
197:
177:
173:
165:
162:
159:
151:
147:
143:
125:
121:
117:
112:
108:
104:
99:
95:
86:
82:
77:
75:
71:
70:
63:
61:
56:
54:
53:contradiction
50:
46:
41:
39:
35:
30:
26:
23:, a proof by
22:
4549:
4347:Ultraproduct
4194:Model theory
4159:Independence
4095:Formal proof
4087:Proof theory
4070:
4043:
4000:real numbers
3972:second-order
3883:Substitution
3760:Metalanguage
3701:conservative
3674:Axiom schema
3618:Constructive
3588:MorseâKelley
3554:Set theories
3533:Aleph number
3526:inaccessible
3432:Grothendieck
3316:intersection
3203:Higher-order
3191:Second-order
3137:Truth tables
3094:Venn diagram
2877:Formal proof
2773:
2769:
2764:
2756:
2751:
2741:
2737:
2732:
2708:
2702:
2696:
2685:. Retrieved
2681:
2668:
2657:. Retrieved
2653:
2644:
2635:
2622:
2609:
2596:
2579:
2557:
2548:
2537:. Retrieved
2534:math.uci.edu
2533:
2506:. Retrieved
2502:
2479:. Retrieved
2475:
2455:
2440:
2433:
2406:
2404:
2401:
2227:
2223:
2174:compared to
2151:
2147:
2079:
2074:
2070:
2069:
2043:compared to
1912:
1907:
1903:
1902:
1853:compared to
1688:
1684:
1679:
1675:
1674:
1667:
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1624:
1620:
1489:
1372:
1360:
1356:
1352:
1342:
1335:
1331:
1327:
1323:
1317:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1280:
1275:
961:
950:
944:
937:
933:
929:
921:
913:
903:
896:
894:
886:
875:
871:
862:
835:ad infinitum
833:
824:
754:
749:
745:
741:
740:as well. So
737:
735:
611:
606:
602:
598:
590:
587:prime number
582:
580:
474:
468:
465:
411:
388:
371:Pythagoreans
349:
324:
316:
312:
308:
304:
300:
296:
292:
280:
275:
270:
266:
258:
240:
229:
225:
221:
208:
145:
140:and proving
78:
68:
64:
57:
42:
24:
18:
4457:Type theory
4405:undecidable
4337:Truth value
4224:equivalence
3903:non-logical
3516:Enumeration
3506:Isomorphism
3453:cardinality
3437:Von Neumann
3402:Ultrafilter
3367:Uncountable
3301:equivalence
3218:Quantifiers
3208:Fixed-point
3177:First-order
3057:Consistency
3042:Proposition
3019:Traditional
2990:Lindström's
2980:Compactness
2922:Type theory
2867:Cardinality
2554:Weil, André
851:real number
755:This gives
251:L-functions
21:mathematics
4572:Categories
4268:elementary
3961:arithmetic
3829:Quantifier
3807:functional
3679:Expression
3397:Transitive
3341:identities
3326:complement
3259:hereditary
3242:Set theory
2804:PlanetMath
2794:PlanetMath
2711:(2): 117,
2687:2019-12-10
2659:2019-12-10
2562:BirkhÀuser
2539:2019-12-10
2508:2019-12-10
2481:2019-12-10
2425:References
1367:See also:
949:(that is,
936:, and let
612:But then,
375:irrational
363:irrational
285:André Weil
4539:Supertask
4442:Recursion
4400:decidable
4234:saturated
4212:of models
4135:deductive
4130:axiomatic
4050:Hilbert's
4037:Euclidean
4018:canonical
3941:axiomatic
3873:Signature
3802:Predicate
3691:Extension
3613:Ackermann
3538:Operation
3417:Universal
3407:Recursive
3382:Singleton
3377:Inhabited
3362:Countable
3352:Types of
3336:power set
3306:partition
3223:Predicate
3169:Predicate
3084:Syllogism
3074:Soundness
3047:Inference
3037:Tautology
2939:paradoxes
1630:and with
1549:−
1248:−
1234:−
1203:−
1171:−
1119:−
1098:−
1061:−
1033:−
966:). Then
407:algebraic
211:inversion
163:≡
67:Euclid's
4524:Logicism
4517:timeline
4493:Concrete
4352:Validity
4322:T-schema
4315:Kripke's
4310:Tarski's
4305:semantic
4295:Strength
4244:submodel
4239:spectrum
4207:function
4055:Tarski's
4044:Elements
4031:geometry
3987:Robinson
3908:variable
3893:function
3866:spectrum
3856:Sentence
3812:variable
3755:Language
3708:Relation
3669:Automata
3659:Alphabet
3643:language
3497:-jection
3475:codomain
3461:Function
3422:Universe
3392:Infinite
3296:Relation
3079:Validity
3069:Argument
2967:theorem,
2634:(1996),
2556:(1984),
2413:See also
842:. Since
421:rational
379:Hippasus
325:descents
69:Elements
4466:Related
4263:Diagram
4161: (
4140:Hilbert
4125:Systems
4120:Theorem
3998:of the
3943:systems
3723:Formula
3718:Grammar
3634: (
3578:General
3291:Forcing
3276:Element
3196:Monadic
2971:paradox
2912:Theorem
2848:General
2725:2323064
1619:, with
1340:√
1315:√
1278:√
959:√
953:is the
942:√
926:
910:
901:√
884:√
865:√
855:√
844:√
827:√
414:√
395:algebra
356:√
342:√
261:form a
255:Mordell
241:In the
150:squares
4229:finite
3992:Skolem
3945:
3920:Theory
3888:Symbol
3878:String
3861:atomic
3738:ground
3733:closed
3728:atomic
3684:ground
3647:syntax
3543:binary
3470:domain
3387:Finite
3152:finite
3010:Logics
2969:
2917:Theory
2723:
2568:
2448:
838:. But
217:on an
81:Fermat
74:Euclid
4219:Model
3967:Peano
3824:Proof
3664:Arity
3593:Naive
3480:image
3412:Fuzzy
3372:Empty
3321:union
3266:Class
2907:Model
2897:Lemma
2855:Axiom
2721:JSTOR
2678:(PDF)
2530:(PDF)
955:floor
849:is a
597:. So
593:, by
419:were
361:) is
194:(see
152:when
4342:Type
4145:list
3949:list
3926:list
3915:Term
3849:rank
3743:open
3637:list
3449:Maps
3354:sets
3213:Free
3183:list
2933:list
2860:list
2566:ISBN
2446:ISBN
2150:and
2102:and
2073:and
1999:and
1935:and
1906:and
1718:and
1687:and
1678:and
1666:and
1654:nor
1646:and
1638:and
1623:and
1330:and
1309:and
1301:and
1293:and
932:and
472:and
389:The
198:and
4029:of
4011:of
3959:of
3491:Sur
3465:Map
3272:Ur-
3254:Set
2802:at
2792:at
2740:",
2713:doi
2344:and
1632:a+b
957:of
744:= 2
601:= 2
299:)/2
206:).
174:mod
19:In
4574::
4415:NP
4039::
4033::
3963::
3640:),
3495:Bi
3487:In
2719:,
2709:95
2707:,
2680:.
2652:.
2630:;
2560:,
2532:.
2517:^
2501:.
2490:^
2474:.
2463:^
2454:.
2220:).
2078::
1911::
1899:).
1359:=
1355:+
1332:nâČ
1324:mâČ
1295:nâČ
1291:mâČ
752:.
609:.
386:.
283:,
269:/2
55:.
4495:/
4410:P
4165:)
3951:)
3947:(
3844:â
3839:!
3834:â
3795:=
3790:â
3785:â
3780:â§
3775:âš
3770:ÂŹ
3493:/
3489:/
3463:/
3274:)
3270:(
3157:â
3147:3
2935:)
2833:e
2826:t
2819:v
2806:.
2796:.
2770:n
2715::
2690:.
2662:.
2591:.
2542:.
2511:.
2484:.
2407:n
2385:2
2381:t
2377:=
2372:4
2368:s
2364:+
2359:4
2355:r
2332:,
2327:4
2323:t
2319:=
2314:2
2310:s
2306:+
2301:4
2297:r
2275:,
2270:4
2266:t
2262:=
2257:4
2253:s
2249:+
2244:2
2240:r
2206:2
2202:b
2198:+
2193:2
2189:a
2185:=
2182:z
2162:a
2152:a
2148:b
2134:a
2112:y
2090:b
2080:y
2075:x
2071:y
2066:.
2064:)
2052:z
2029:z
2007:b
1987:a
1965:z
1943:b
1923:a
1913:z
1908:x
1904:z
1885:2
1881:b
1877:+
1872:2
1868:a
1864:=
1861:z
1839:2
1835:a
1812:2
1808:a
1785:2
1781:b
1758:2
1754:a
1731:2
1727:b
1704:z
1701:y
1689:z
1685:y
1680:z
1676:y
1668:b
1664:a
1660:x
1656:z
1652:y
1648:z
1644:y
1640:z
1636:y
1625:b
1621:a
1605:2
1601:b
1597:+
1592:2
1588:a
1584:=
1581:z
1562:,
1557:2
1553:b
1544:2
1540:a
1536:=
1533:y
1514:,
1511:b
1508:a
1505:2
1502:=
1499:x
1465:4
1461:t
1457:=
1452:4
1448:s
1444:+
1439:4
1435:q
1412:4
1408:t
1404:=
1399:4
1395:s
1391:+
1386:2
1382:r
1361:t
1357:s
1353:r
1343:k
1336:n
1328:m
1318:k
1311:n
1307:m
1303:n
1299:m
1287:q
1281:k
1254:q
1251:n
1245:m
1240:q
1237:m
1231:k
1228:n
1222:=
1209:q
1206:n
1199:)
1194:n
1191:m
1186:(
1182:n
1177:q
1174:m
1166:k
1160:)
1154:k
1149:n
1145:(
1138:=
1125:q
1122:n
1114:k
1109:n
1104:q
1101:m
1093:k
1088:m
1082:=
1068:)
1064:q
1056:k
1050:(
1046:n
1040:)
1036:q
1028:k
1022:(
1018:m
1012:=
1000:n
997:m
992:=
983:k
962:k
951:q
945:k
938:q
934:n
930:m
922:n
918:/
914:m
904:k
897:k
887:k
876:q
874:/
872:p
867:2
857:2
846:2
829:2
821:.
807:s
804:r
799:=
793:s
790:2
785:r
782:2
776:=
771:q
768:p
750:s
746:s
742:q
738:q
721:,
716:2
712:r
708:2
705:=
700:2
696:q
674:,
669:2
665:r
661:4
658:=
653:2
649:)
645:r
642:2
639:(
636:=
631:2
627:q
623:2
607:r
603:r
599:p
591:p
583:p
566:,
561:2
557:p
553:=
548:2
544:q
540:2
519:,
512:2
508:q
502:2
498:p
492:=
489:2
475:q
469:p
449:q
446:p
441:=
436:2
416:2
358:2
354:(
344:2
317:A
313:Q
311:(
309:A
305:Q
303:(
301:A
297:Q
295:(
293:A
281:A
271:E
267:E
259:E
230:E
226:E
222:E
181:)
178:4
171:(
166:1
160:p
146:p
126:4
122:t
118:=
113:4
109:s
105:+
100:2
96:r
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