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244:: the proof algorithm itself should be proved valid, so that its use can then be regarded as a mere "verification". Arguments that computer-assisted proofs are subject to errors in their source programs, compilers, and hardware can be resolved by providing a formal proof of correctness for the computer program (an approach which was successfully applied to the four-color theorem in 2005) as well as replicating the result using different programming languages, different compilers, and different computer hardware.
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148:. In a computer, the result of each elementary operation is rounded off by the computer precision. However, one can construct an interval provided by upper and lower bounds on the result of an elementary operation. Then one proceeds by replacing numbers with intervals and performing elementary operations between such intervals of representable numbers.
251:
program to demonstrate their correctness. Since validating a given proof is much easier than finding a proof, the checker program is simpler than the original assistant program, and it is correspondingly easier to gain confidence into its correctness. However, this approach of using a computer
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by human beings, and that mathematicians are effectively being asked to replace logical deduction from assumed axioms with trust in an empirical computational process, which is potentially affected by errors in the computer program, as well as defects in the runtime environment and hardware.
283:
science like astronomy, rather than an experimental one like physics or chemistry. This controversy within mathematics is occurring at the same time as questions are being asked in the physics community about whether twenty-first century
97:
in order to ensure that the set-valued output of a numerical program encloses the solution of the original mathematical problem. This is done by controlling, enclosing and propagating round-off and truncation errors using for example
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becomes more important than the application of pure reason in the area of abstract mathematical concepts. This directly relates to the argument within mathematics as to whether mathematics is based on ideas, or "merely" an
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program to prove the output of another program correct does not appeal to computer proof skeptics, who see it as adding another layer of complexity without addressing the perceived need for human understanding.
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Inclusion in this list does not imply that a formal computer-checked proof exists, but rather, that a computer program has been involved in some way. See the main articles for details.
42:. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the
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Brakensiek, Joshua; Heule, Marijn; Mackey, John; NarvĂĄez, David (2020). "The
Resolution of Keller's Conjecture". In Peltier, Nicolas; Sofronie-Stokkermans, Viorica (eds.).
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259:âthat they provide no insights or new and useful concepts. In fact, this is an argument that could be advanced against any lengthy proof by exhaustion.
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first to articulate objections. Those who adhere to
Tymoczko's arguments believe that lengthy computer-assisted proofs are not, in some sense, 'real'
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allow mathematicians to develop human-readable proofs which are nonetheless formally verified for correctness. Since these proofs are generally
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Another possible way of verifying computer-aided proofs is to generate their reasoning steps in a machine-readable form, and then use a
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A Backtracking
Framework for Beowulf Clusters with an Extension to Multi-Cluster Computation and Sat Benchmark Problem Implementation
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1503:(Technical report). Department of Computer Studies, University of Glamorgan. CS-90-4. Archived from the original on 2012-07-17
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Subercaseaux, Bernardo; Heule, Marijn J. H. (2023-01-23). "The
Packing Chromatic Number of the Infinite Square Grid is 15".
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or rigorous numerics. This means computing numerically yet with mathematical rigour. One uses set-valued arithmetic and
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is confronting this debate head-on by focusing on numerical experiments as its main tool for mathematical exploration.
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323:'s universality conjecture in non-linear dynamics. Proven by O. E. Lanford using rigorous computer arithmetic, 1982
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view, all possible mathematical objects in some sense "already exist", whether computer-aided mathematics is an
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An additional philosophical issue raised by computer-aided proofs is whether they make mathematics into a
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have proved a number of new results and found new proofs for known theorems. Additionally, interactive
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1405:(PhD). AI Lab., Stanford University. STAN-CS-76-570, Heuristic Programming Project Report HPP-76-8.
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research to create smaller, explicit, new proofs of mathematical theorems from the bottom up using
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Ahmed, Tanbir (2009). "Some new van der
Waerden numbers and some van der Waerden-type numbers".
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Numerical
Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
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Computer-assisted proofs are the subject of some controversy in the mathematical world, with
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81:) they do not share the controversial implications of computer-aided proofs-by-exhaustion.
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102:. More precisely, one reduces the computation to a sequence of elementary operations, say
8:
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Other mathematicians believe that lengthy computer-assisted proofs should be regarded as
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in formal symbol manipulation. It also raises the question whether, if according to the
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2016:
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1953:
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AM: An artificial intelligence approach to discovery in mathematics as heuristic search
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1103:
1084:
1035:
842:
796:
551:
351:
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503:
The packing chromatic number of the infinite square grid is 15, by
Subercaseaux and
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47:
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1812:
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873:
830:
813:
Ahmed, Tanbir (2010). "Two new van der
Waerden numbers w(2;3,17) and w(2;3,18)".
784:
734:
642:
581:
221:
70:
1255:
1250:. Lecture Notes in Computer Science. Vol. 12166. Springer. pp. 48â65.
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2009:
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One method for using computers in mathematical proofs is by means of so-called
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1416:. IMA volumes in mathematics and its applications. Vol. 28. Springer.
1222:
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1039:
869:
826:
780:
504:
483:
326:
725:
Kouril, Michal (2012). "Computing the van der
Waerden number W(3,4)=293".
3496:
3374:
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3087:
2266:
2256:
2203:
1887:
1807:
1792:
1672:
1617:
1143:"Rigorous computer-assisted application of KAM theory: a modern approach"
856:
Ahmed, Tanbir (2012). "On computation of exact van der
Waerden numbers".
447:
288:
is becoming too mathematical, and leaving behind its experimental roots.
228:
because they involve so many logical steps that they are not practically
3348:
752:
Kouril, Michal (2015). "Leveraging FPGA clusters for SAT computations".
2137:
1992:
1963:
1769:
1176:
607:
443:
419:
390:
192: in this section. Unsourced material may be challenged and removed.
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673:
Electronic Research Announcements of the American Mathematical Society
34:
Most computer-aided proofs to date have been implementations of large
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2022:
1834:
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457:
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599:
584:(1979), "The Four-Color Problem and its Mathematical Significance",
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2225:
1819:
1320:
1207:
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541:
497:
28:
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1662:
514:
Existence of commuting functions with no common fixed point, 1967
396:
255:
Another argument against computer-aided proofs is that they lack
39:
3521:
432:
1337:"The Number 15 Describes the Secret Limit of an Infinite Grid"
2414:
1760:
1605:
1245:
3676:
3670:
3485:
899:
Ahmed, Tanbir (2013). "Some More Van der Waerden numbers".
386:
16:
Mathematical proof at least partially generated by computer
1437:. Springer Series in Computational Mathematics. Springer.
496:
in dimension 7 the only remaining case in 2020 with a 200
1461:"A computer-assisted proof of the Feigenbaum conjectures"
979:
926:
Ahmed, Tanbir; Kullmann, Oliver; Snevily, Hunter (2014).
422:
to prove w(2; 3, 17) = 279 and w(2; 3, 18) = 312 in 2010.
1104:"Rigorous estimates for a computer-assisted KAM theory"
354:, 1998 â the problem of optimal sphere packing in a box
666:
482:, the proof that S(5) = 161 was announced in 2017 by
108:
3350:
Numerical methods for partial differential equations
925:
1371:. Transactions of the American Mathematical Society
46:was the first major theorem to be verified using a
1291:"Computer Search Settles 90-Year-Old Math Problem"
1055:"Two-hundred-terabyte maths proof is largest ever"
299:Theorems proved with the help of computer programs
140:
77:(albeit with difficulty, as with the proof of the
1430:
1313:
1140:
1101:
3806:
1520:: CS1 maint: bot: original URL status unknown (
1362:"Commuting Functions with No Common Fixed Point"
381:Kouril (between 2006 and 2016) computed several
305:Proof assistant § Notable formalized proofs
406:Ahmed (between 2009 and 2014) computed several
667:Hass, J.; Hutchings, M.; Schlafly, R. (1995).
428:can be obtained in at most 20 face moves, 2010
27:that has been at least partially generated by
3334:
1576:
1409:
1141:Figueras, J.L.; Haro, A.; Luque, A. (2017).
634:Notices of the American Mathematical Society
53:Attempts have also been made in the area of
808:
806:
754:Parallel Computing: On the Road to Exascale
151:
3341:
3327:
1768:
1583:
1569:
1531:"Number proofs done by computer might err"
1431:Nakao, M.; Plum, M.; Watanabe, Y. (2019).
446:. The full conjecture was later solved by
1479:
1410:Meyer, K.R.; Schmidt, D.S., eds. (2012).
1319:
1273:
1206:
1158:
1078:
1029:
955:
945:
928:"On the van der Waerden numbers w(2;3,t)"
745:
718:
714:(Ph.D. thesis). University of Cincinnati.
701:
684:
208:Learn how and when to remove this message
1334:
1288:
1147:Foundations of Computational Mathematics
892:
849:
803:
760:
620:
580:
1458:
919:
574:
431:Minimum number of clues for a solvable
418:. Ahmed first used cluster-distributed
3807:
1590:
1528:
1006:"Maths whizz solves a master's riddle"
1003:
751:
724:
707:
511:for the chromatic number of the plane)
3322:
1564:
1496:
1392:
1197:
898:
855:
812:
766:
626:"Formal ProofâThe Four-Color Theorem"
3592:Moving particle semi-implicit method
3503:Weighted essentially non-oscillatory
1102:Celletti, A.; Chierchia, L. (1987).
1052:
190:adding citations to reliable sources
161:
454:Boolean Pythagorean triples problem
13:
3441:Finite-difference frequency-domain
1386:
474:automorphism group of a free group
426:Optimal solutions for Rubik's Cube
14:
3851:
1547:"A Special Issue on Formal Proof"
1452:
1413:Computer aided proofs in analysis
1359:
3302:
1004:Cesare, Chris (1 October 2015).
166:
3794:Method of fundamental solutions
3580:Smoothed-particle hydrodynamics
1490:10.1090/S0273-0979-1982-15008-X
1353:
1328:
1307:
1282:
1239:
1215:
1191:
1134:
1108:Journal of Mathematical Physics
1095:
1046:
997:
972:
656:from the original on 2011-08-05
177:needs additional citations for
141:{\displaystyle (+,-,\times ,/)}
3435:Alternating direction-implicit
1335:Hartnett, Kevin (2023-04-20).
1289:Hartnett, Kevin (2020-08-19).
669:"The double bubble conjecture"
660:
614:
465:Kolmogorov-Arnold-Moser theory
438:In 2014 a special case of the
135:
109:
1:
3447:Finite-difference time-domain
3263:History of mathematical logic
1553:American Mathematical Society
1529:Begley, S. (April 16, 2018).
1369:American Mathematical Society
1201:(2017). "Schur Number Five".
695:10.1090/S1079-6762-95-03001-0
568:
3486:Advection upstream-splitting
3188:Primitive recursive function
1500:Why did AM run out of steam?
1053:Lamb, Evelyn (26 May 2016).
933:Discrete Applied Mathematics
901:Journal of Integer Sequences
450:without computer assistance.
401:minimum-weight triangulation
7:
3497:Essentially non-oscillatory
3480:Monotonic upstream-centered
1256:10.1007/978-3-030-51074-9_4
519:
10:
3856:
3757:Infinite difference method
3375:Forward-time central-space
2252:SchröderâBernstein theorem
1979:Monadic predicate calculus
1638:Foundations of mathematics
1459:Lanford, Oscar E. (1982).
332:Non-existence of a finite
302:
155:
84:
3840:Philosophy of mathematics
3820:Automated theorem proving
3691:
3660:PoincarĂ©âSteklov operator
3613:
3570:
3512:
3460:
3427:
3419:Method of characteristics
3389:
3365:
3356:
3298:
3285:Philosophy of mathematics
3234:Automated theorem proving
3216:
3111:
2943:
2836:
2688:
2405:
2381:
2359:Von NeumannâBernaysâGödel
2304:
2198:
2102:
2000:
1991:
1918:
1853:
1759:
1681:
1598:
1169:10.1007/s10208-016-9339-3
1080:10.1038/nature.2016.19990
1031:10.1038/nature.2015.18441
957:10.1016/j.dam.2014.05.007
587:The Journal of Philosophy
440:ErdĆs discrepancy problem
67:automated theorem provers
3825:Computer-assisted proofs
3677:Tearing and interconnect
3671:Balancing by constraints
340:Double bubble conjecture
293:experimental mathematics
152:Philosophical objections
3784:Computer-assisted proof
3762:Infinite element method
3550:Gradient discretisation
2935:Self-verifying theories
2756:Tarski's axiomatization
1707:Tarski's undefinability
1702:incompleteness theorems
1535:Pittsburgh Post-Gazette
708:Kouril, Michal (2006).
509:HadwigerâNelson problem
416:distributed SAT-solvers
414:-based stand-alone and
408:van der Waerden numbers
383:van der Waerden numbers
264:quasi-empirical science
55:artificial intelligence
21:computer-assisted proof
3772:PetrovâGalerkin method
3533:Discontinuous Galerkin
3309:Mathematics portal
2920:Proof of impossibility
2568:propositional variable
1878:Propositional calculus
1497:Furse, Edmund (1990).
870:10.1515/integ.2011.112
827:10.1515/integ.2010.032
781:10.1515/integ.2009.007
470:Kazhdan's property (T)
329:, 1988 â a solved game
291:The emerging field of
142:
3752:Isogeometric analysis
3598:Material point method
3178:Kolmogorov complexity
3131:Computably enumerable
3031:Model complete theory
2823:Principia Mathematica
1883:Propositional formula
1712:BanachâTarski paradox
1468:Bull. Amer. Math. Soc
476:of rank at least five
374:17-point case of the
257:mathematical elegance
143:
3789:Integrable algorithm
3615:Domain decomposition
3126:ChurchâTuring thesis
3113:Computability theory
2322:continuum hypothesis
1840:Square of opposition
1698:Gödel's completeness
980:"God's Number is 20"
557:Symbolic computation
463:Applications to the
460:of data in May 2016.
376:Happy Ending problem
186:improve this article
158:Non-surveyable proof
106:
36:proofs-by-exhaustion
3815:Argument technology
3633:Schwarz alternating
3556:Loubignac iteration
3280:Mathematical object
3171:P versus NP problem
3136:Computable function
2930:Reverse mathematics
2856:Logical consequence
2733:primitive recursive
2728:elementary function
2501:Free/bound variable
2354:TarskiâGrothendieck
1873:Logical connectives
1803:Logical equivalence
1653:Logical consequence
1248:Automated Reasoning
1223:"Schur Number Five"
1199:Heule, Marijn J. H.
1120:1987JMP....28.2078C
1071:2016Natur.534...17L
1022:2015Natur.526...19C
527:Formal verification
507:in 2023 (See also:
494:Keller's conjecture
442:was solved using a
370:interval arithmetic
321:Mitchell Feigenbaum
286:theoretical physics
226:mathematical proofs
100:interval arithmetic
95:inclusion principle
61:techniques such as
59:automated reasoning
3835:Numerical analysis
3779:Validated numerics
3078:Transfer principle
3041:Semantics of logic
3026:Categorical theory
3002:Non-standard model
2516:Logical connective
1643:Information theory
1592:Mathematical logic
1360:Boyce, William M.
562:Validated numerics
537:Mathematical proof
346:Robbins conjecture
315:Four color theorem
138:
91:validated numerics
79:Robbins conjecture
44:four color theorem
38:of a mathematical
25:mathematical proof
3802:
3801:
3742:Immersed boundary
3735:Method of moments
3650:NeumannâDirichlet
3643:abstract additive
3628:Fictitious domain
3572:Meshless/Meshfree
3456:
3455:
3358:Finite difference
3316:
3315:
3248:Abstract category
3051:Theories of truth
2861:Rule of inference
2851:Natural deduction
2832:
2831:
2377:
2376:
2082:Cartesian product
1987:
1986:
1893:Many-valued logic
1868:Boolean functions
1751:Russell's paradox
1726:diagonal argument
1623:First-order logic
1423:978-1-4613-9092-3
1265:978-3-030-51074-9
1227:www.cs.utexas.edu
641:(11): 1382â1393,
622:Gonthier, Georges
552:Seventeen or Bust
480:Schur number five
456:solved using 200
360:, 2002 â 14th of
352:Kepler conjecture
336:of order 10, 1989
268:scientific method
218:
217:
210:
3847:
3747:Analytic element
3730:Boundary element
3623:Schur complement
3604:Particle-in-cell
3539:Spectral element
3363:
3362:
3343:
3336:
3329:
3320:
3319:
3307:
3306:
3258:History of logic
3253:Category of sets
3146:Decision problem
2925:Ordinal analysis
2866:Sequent calculus
2764:Boolean algebras
2704:
2703:
2678:
2649:logical/constant
2403:
2402:
2389:
2312:ZermeloâFraenkel
2063:Set operations:
1998:
1997:
1935:
1766:
1765:
1746:LöwenheimâSkolem
1633:Formal semantics
1585:
1578:
1571:
1562:
1561:
1557:
1556:. December 2008.
1542:
1537:. Archived from
1525:
1519:
1516:cite tech report
1511:
1509:
1508:
1493:
1483:
1465:
1448:
1427:
1406:
1404:
1381:
1380:
1378:
1376:
1366:
1357:
1351:
1350:
1348:
1347:
1332:
1326:
1325:
1323:
1311:
1305:
1304:
1302:
1301:
1286:
1280:
1279:
1277:
1243:
1237:
1236:
1234:
1233:
1219:
1213:
1212:
1210:
1195:
1189:
1188:
1162:
1138:
1132:
1131:
1128:10.1063/1.527418
1099:
1093:
1092:
1082:
1050:
1044:
1043:
1033:
1001:
995:
994:
992:
991:
976:
970:
969:
959:
949:
923:
917:
916:
896:
890:
889:
853:
847:
846:
810:
801:
800:
764:
758:
757:
749:
743:
742:
722:
716:
715:
705:
699:
698:
688:
664:
658:
657:
655:
630:
618:
612:
610:
582:Tymoczko, Thomas
578:
362:Smale's problems
358:Lorenz attractor
334:projective plane
213:
206:
202:
199:
193:
170:
162:
147:
145:
144:
139:
134:
96:
75:human-surveyable
71:proof assistants
48:computer program
3855:
3854:
3850:
3849:
3848:
3846:
3845:
3844:
3805:
3804:
3803:
3798:
3767:Galerkin method
3710:Method of lines
3687:
3655:NeumannâNeumann
3609:
3566:
3508:
3475:High-resolution
3452:
3423:
3385:
3352:
3347:
3317:
3312:
3301:
3294:
3239:Category theory
3229:Algebraic logic
3212:
3183:Lambda calculus
3121:Church encoding
3107:
3083:Truth predicate
2939:
2905:Complete theory
2828:
2697:
2693:
2689:
2684:
2676:
2396: and
2392:
2387:
2373:
2349:New Foundations
2317:axiom of choice
2300:
2262:Gödel numbering
2202: and
2194:
2098:
1983:
1933:
1914:
1863:Boolean algebra
1849:
1813:Equiconsistency
1778:Classical logic
1755:
1736:Halting problem
1724: and
1700: and
1688: and
1687:
1682:Theorems (
1677:
1594:
1589:
1551:Notices of the
1545:
1513:
1512:
1506:
1504:
1481:10.1.1.434.8389
1463:
1455:
1445:
1424:
1402:
1389:
1387:Further reading
1384:
1374:
1372:
1364:
1358:
1354:
1345:
1343:
1341:Quanta Magazine
1333:
1329:
1312:
1308:
1299:
1297:
1295:Quanta Magazine
1287:
1283:
1266:
1244:
1240:
1231:
1229:
1221:
1220:
1216:
1196:
1192:
1139:
1135:
1100:
1096:
1065:(7605): 17â18.
1051:
1047:
1016:(7571): 19â20.
1002:
998:
989:
987:
978:
977:
973:
940:(2014): 27â51.
924:
920:
897:
893:
854:
850:
811:
804:
765:
761:
750:
746:
723:
719:
706:
702:
686:10.1.1.527.8616
665:
661:
653:
628:
619:
615:
600:10.2307/2025976
579:
575:
571:
566:
522:
517:
307:
301:
222:Thomas Tymoczko
214:
203:
197:
194:
183:
171:
160:
154:
130:
107:
104:
103:
94:
87:
17:
12:
11:
5:
3853:
3843:
3842:
3837:
3832:
3830:Formal methods
3827:
3822:
3817:
3800:
3799:
3797:
3796:
3791:
3786:
3781:
3776:
3775:
3774:
3764:
3759:
3754:
3749:
3744:
3739:
3738:
3737:
3727:
3722:
3717:
3712:
3707:
3704:Pseudospectral
3701:
3695:
3693:
3689:
3688:
3686:
3685:
3680:
3674:
3668:
3662:
3657:
3652:
3647:
3646:
3645:
3640:
3630:
3625:
3619:
3617:
3611:
3610:
3608:
3607:
3601:
3595:
3589:
3583:
3576:
3574:
3568:
3567:
3565:
3564:
3558:
3553:
3547:
3542:
3536:
3530:
3524:
3518:
3516:
3514:Finite element
3510:
3509:
3507:
3506:
3500:
3494:
3492:Riemann solver
3489:
3483:
3477:
3472:
3466:
3464:
3458:
3457:
3454:
3453:
3451:
3450:
3444:
3438:
3431:
3429:
3425:
3424:
3422:
3421:
3416:
3411:
3406:
3401:
3399:LaxâFriedrichs
3395:
3393:
3387:
3386:
3384:
3383:
3381:CrankâNicolson
3378:
3371:
3369:
3360:
3354:
3353:
3346:
3345:
3338:
3331:
3323:
3314:
3313:
3299:
3296:
3295:
3293:
3292:
3287:
3282:
3277:
3272:
3271:
3270:
3260:
3255:
3250:
3241:
3236:
3231:
3226:
3224:Abstract logic
3220:
3218:
3214:
3213:
3211:
3210:
3205:
3203:Turing machine
3200:
3195:
3190:
3185:
3180:
3175:
3174:
3173:
3168:
3163:
3158:
3153:
3143:
3141:Computable set
3138:
3133:
3128:
3123:
3117:
3115:
3109:
3108:
3106:
3105:
3100:
3095:
3090:
3085:
3080:
3075:
3070:
3069:
3068:
3063:
3058:
3048:
3043:
3038:
3036:Satisfiability
3033:
3028:
3023:
3022:
3021:
3011:
3010:
3009:
2999:
2998:
2997:
2992:
2987:
2982:
2977:
2967:
2966:
2965:
2960:
2953:Interpretation
2949:
2947:
2941:
2940:
2938:
2937:
2932:
2927:
2922:
2917:
2907:
2902:
2901:
2900:
2899:
2898:
2888:
2883:
2873:
2868:
2863:
2858:
2853:
2848:
2842:
2840:
2834:
2833:
2830:
2829:
2827:
2826:
2818:
2817:
2816:
2815:
2810:
2809:
2808:
2803:
2798:
2778:
2777:
2776:
2774:minimal axioms
2771:
2760:
2759:
2758:
2747:
2746:
2745:
2740:
2735:
2730:
2725:
2720:
2707:
2705:
2686:
2685:
2683:
2682:
2681:
2680:
2668:
2663:
2662:
2661:
2656:
2651:
2646:
2636:
2631:
2626:
2621:
2620:
2619:
2614:
2604:
2603:
2602:
2597:
2592:
2587:
2577:
2572:
2571:
2570:
2565:
2560:
2550:
2549:
2548:
2543:
2538:
2533:
2528:
2523:
2513:
2508:
2503:
2498:
2497:
2496:
2491:
2486:
2481:
2471:
2466:
2464:Formation rule
2461:
2456:
2455:
2454:
2449:
2439:
2438:
2437:
2427:
2422:
2417:
2412:
2406:
2400:
2383:Formal systems
2379:
2378:
2375:
2374:
2372:
2371:
2366:
2361:
2356:
2351:
2346:
2341:
2336:
2331:
2326:
2325:
2324:
2319:
2308:
2306:
2302:
2301:
2299:
2298:
2297:
2296:
2286:
2281:
2280:
2279:
2272:Large cardinal
2269:
2264:
2259:
2254:
2249:
2235:
2234:
2233:
2228:
2223:
2208:
2206:
2196:
2195:
2193:
2192:
2191:
2190:
2185:
2180:
2170:
2165:
2160:
2155:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2109:
2107:
2100:
2099:
2097:
2096:
2095:
2094:
2089:
2084:
2079:
2074:
2069:
2061:
2060:
2059:
2054:
2044:
2039:
2037:Extensionality
2034:
2032:Ordinal number
2029:
2019:
2014:
2013:
2012:
2001:
1995:
1989:
1988:
1985:
1984:
1982:
1981:
1976:
1971:
1966:
1961:
1956:
1951:
1950:
1949:
1939:
1938:
1937:
1924:
1922:
1916:
1915:
1913:
1912:
1911:
1910:
1905:
1900:
1890:
1885:
1880:
1875:
1870:
1865:
1859:
1857:
1851:
1850:
1848:
1847:
1842:
1837:
1832:
1827:
1822:
1817:
1816:
1815:
1805:
1800:
1795:
1790:
1785:
1780:
1774:
1772:
1763:
1757:
1756:
1754:
1753:
1748:
1743:
1738:
1733:
1728:
1716:Cantor's
1714:
1709:
1704:
1694:
1692:
1679:
1678:
1676:
1675:
1670:
1665:
1660:
1655:
1650:
1645:
1640:
1635:
1630:
1625:
1620:
1615:
1614:
1613:
1602:
1600:
1596:
1595:
1588:
1587:
1580:
1573:
1565:
1559:
1558:
1543:
1541:on 2018-04-16.
1526:
1494:
1474:(3): 427â434.
1454:
1453:External links
1451:
1450:
1449:
1443:
1428:
1422:
1407:
1388:
1385:
1383:
1382:
1352:
1327:
1306:
1281:
1264:
1238:
1214:
1190:
1153:(5): 1123â93.
1133:
1114:(9): 2078â86.
1094:
1045:
996:
971:
918:
891:
864:(3): 417â425.
848:
821:(4): 369â377.
802:
759:
744:
717:
700:
659:
613:
572:
570:
567:
565:
564:
559:
554:
549:
547:Model checking
544:
539:
534:
532:Logic Theorist
529:
523:
521:
518:
516:
515:
512:
501:
491:
486:and took up 2
477:
467:
461:
451:
436:
429:
423:
412:DPLL algorithm
404:
394:
379:
372:
366:Warwick Tucker
355:
349:
343:
337:
330:
324:
318:
311:
300:
297:
240:, rather than
216:
215:
174:
172:
165:
156:Main article:
153:
150:
137:
133:
129:
126:
123:
120:
117:
114:
111:
86:
83:
15:
9:
6:
4:
3:
2:
3852:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3812:
3810:
3795:
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3773:
3770:
3769:
3768:
3765:
3763:
3760:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3736:
3733:
3732:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3705:
3702:
3700:
3697:
3696:
3694:
3690:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3644:
3641:
3639:
3636:
3635:
3634:
3631:
3629:
3626:
3624:
3621:
3620:
3618:
3616:
3612:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3577:
3575:
3573:
3569:
3562:
3559:
3557:
3554:
3551:
3548:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3523:
3520:
3519:
3517:
3515:
3511:
3504:
3501:
3498:
3495:
3493:
3490:
3487:
3484:
3481:
3478:
3476:
3473:
3471:
3468:
3467:
3465:
3463:
3462:Finite volume
3459:
3448:
3445:
3442:
3439:
3436:
3433:
3432:
3430:
3426:
3420:
3417:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3396:
3394:
3392:
3388:
3382:
3379:
3376:
3373:
3372:
3370:
3368:
3364:
3361:
3359:
3355:
3351:
3344:
3339:
3337:
3332:
3330:
3325:
3324:
3321:
3311:
3310:
3305:
3297:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3269:
3266:
3265:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3221:
3219:
3215:
3209:
3206:
3204:
3201:
3199:
3198:Recursive set
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3172:
3169:
3167:
3164:
3162:
3159:
3157:
3154:
3152:
3149:
3148:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3118:
3116:
3114:
3110:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3074:
3071:
3067:
3064:
3062:
3059:
3057:
3054:
3053:
3052:
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3020:
3017:
3016:
3015:
3012:
3008:
3007:of arithmetic
3005:
3004:
3003:
3000:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2972:
2971:
2968:
2964:
2961:
2959:
2956:
2955:
2954:
2951:
2950:
2948:
2946:
2942:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2915:
2914:from ZFC
2911:
2908:
2906:
2903:
2897:
2894:
2893:
2892:
2889:
2887:
2884:
2882:
2879:
2878:
2877:
2874:
2872:
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2847:
2844:
2843:
2841:
2839:
2835:
2825:
2824:
2820:
2819:
2814:
2813:non-Euclidean
2811:
2807:
2804:
2802:
2799:
2797:
2796:
2792:
2791:
2789:
2786:
2785:
2783:
2779:
2775:
2772:
2770:
2767:
2766:
2765:
2761:
2757:
2754:
2753:
2752:
2748:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2715:
2713:
2709:
2708:
2706:
2701:
2695:
2690:Example
2687:
2679:
2674:
2673:
2672:
2669:
2667:
2664:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2641:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2618:
2615:
2613:
2610:
2609:
2608:
2605:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2582:
2581:
2578:
2576:
2573:
2569:
2566:
2564:
2561:
2559:
2556:
2555:
2554:
2551:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2518:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2476:
2475:
2472:
2470:
2467:
2465:
2462:
2460:
2457:
2453:
2450:
2448:
2447:by definition
2445:
2444:
2443:
2440:
2436:
2433:
2432:
2431:
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2411:
2408:
2407:
2404:
2401:
2399:
2395:
2390:
2384:
2380:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2334:KripkeâPlatek
2332:
2330:
2327:
2323:
2320:
2318:
2315:
2314:
2313:
2310:
2309:
2307:
2303:
2295:
2292:
2291:
2290:
2287:
2285:
2282:
2278:
2275:
2274:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2255:
2253:
2250:
2247:
2243:
2239:
2236:
2232:
2229:
2227:
2224:
2222:
2219:
2218:
2217:
2213:
2210:
2209:
2207:
2205:
2201:
2197:
2189:
2186:
2184:
2181:
2179:
2178:constructible
2176:
2175:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2111:
2110:
2108:
2106:
2101:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2064:
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1855:Propositional
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1783:Logical truth
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907:(4): 13.4.4.
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679:(3): 98â102.
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433:Sudoku puzzle
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281:observational
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249:proof checker
245:
243:
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187:
181:
180:
175:This section
173:
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149:
131:
127:
124:
121:
118:
115:
112:
101:
92:
82:
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76:
72:
68:
65:search. Such
64:
60:
56:
51:
49:
45:
41:
37:
32:
30:
26:
22:
3783:
3586:Peridynamics
3404:LaxâWendroff
3300:
3098:Ultraproduct
2945:Model theory
2910:Independence
2846:Formal proof
2838:Proof theory
2821:
2794:
2751:real numbers
2723:second-order
2634:Substitution
2511:Metalanguage
2452:conservative
2425:Axiom schema
2369:Constructive
2339:MorseâKelley
2305:Set theories
2284:Aleph number
2277:inaccessible
2183:Grothendieck
2067:intersection
1954:Higher-order
1942:Second-order
1888:Truth tables
1845:Venn diagram
1628:Formal proof
1550:
1539:the original
1534:
1505:. Retrieved
1499:
1471:
1467:
1433:
1412:
1398:
1373:. Retrieved
1368:
1355:
1344:. Retrieved
1340:
1330:
1309:
1298:. Retrieved
1294:
1284:
1247:
1241:
1230:. Retrieved
1226:
1217:
1193:
1150:
1146:
1136:
1111:
1107:
1097:
1062:
1058:
1048:
1013:
1009:
999:
988:. Retrieved
983:
974:
937:
931:
921:
904:
900:
894:
861:
857:
851:
818:
814:
772:
768:
762:
753:
747:
730:
726:
720:
710:
703:
676:
672:
662:
638:
632:
616:
594:(2): 57â83,
591:
585:
576:
484:Marijn Heule
327:Connect Four
308:
290:
266:, where the
261:
254:
246:
241:
238:calculations
237:
235:
219:
204:
198:October 2020
195:
184:Please help
179:verification
176:
88:
52:
33:
20:
18:
3720:Collocation
3208:Type theory
3156:undecidable
3088:Truth value
2975:equivalence
2654:non-logical
2267:Enumeration
2257:Isomorphism
2204:cardinality
2188:Von Neumann
2153:Ultrafilter
2118:Uncountable
2052:equivalence
1969:Quantifiers
1959:Fixed-point
1928:First-order
1808:Consistency
1793:Proposition
1770:Traditional
1741:Lindström's
1731:Compactness
1673:Type theory
1618:Cardinality
1394:Lenat, D.B.
1177:2445/192693
986:. July 2010
448:Terence Tao
435:is 17, 2012
420:SAT-solvers
397:NP-hardness
3809:Categories
3409:MacCormack
3391:Hyperbolic
3019:elementary
2712:arithmetic
2580:Quantifier
2558:functional
2430:Expression
2148:Transitive
2092:identities
2077:complement
2010:hereditary
1993:Set theory
1507:2016-09-06
1346:2023-04-20
1321:2301.09757
1300:2021-10-08
1232:2021-10-06
1208:1711.08076
1160:1601.00084
990:2023-10-18
984:cube20.org
756:: 525â532.
569:References
444:SAT-solver
391:SAT-solver
364:proved by
303:See also:
230:verifiable
3725:Level-set
3715:Multigrid
3665:Balancing
3367:Parabolic
3290:Supertask
3193:Recursion
3151:decidable
2985:saturated
2963:of models
2886:deductive
2881:axiomatic
2801:Hilbert's
2788:Euclidean
2769:canonical
2692:axiomatic
2624:Signature
2553:Predicate
2442:Extension
2364:Ackermann
2289:Operation
2168:Universal
2158:Recursive
2133:Singleton
2128:Inhabited
2113:Countable
2103:Types of
2087:power set
2057:partition
1974:Predicate
1920:Predicate
1835:Syllogism
1825:Soundness
1798:Inference
1788:Tautology
1690:paradoxes
1476:CiteSeerX
1375:31 August
947:1102.5433
843:124272560
797:122129059
681:CiteSeerX
488:petabytes
458:terabytes
277:Platonist
125:×
119:−
63:heuristic
3699:Spectral
3638:additive
3561:Smoothed
3527:Extended
3275:Logicism
3268:timeline
3244:Concrete
3103:Validity
3073:T-schema
3066:Kripke's
3061:Tarski's
3056:semantic
3046:Strength
2995:submodel
2990:spectrum
2958:function
2806:Tarski's
2795:Elements
2782:geometry
2738:Robinson
2659:variable
2644:function
2617:spectrum
2607:Sentence
2563:variable
2506:Language
2459:Relation
2420:Automata
2410:Alphabet
2394:language
2248:-jection
2226:codomain
2212:Function
2173:Universe
2143:Infinite
2047:Relation
1830:Validity
1820:Argument
1718:theorem,
1396:(1976).
1185:28258285
1089:27251254
1040:26432222
886:11811448
858:Integers
815:Integers
769:Integers
727:Integers
651:archived
624:(2008),
542:Metamath
520:See also
498:gigabyte
490:of space
472:for the
273:exercise
29:computer
3683:FETI-DP
3563:(S-FEM)
3482:(MUSCL)
3470:Godunov
3217:Related
3014:Diagram
2912: (
2891:Hilbert
2876:Systems
2871:Theorem
2749:of the
2694:systems
2474:Formula
2469:Grammar
2385: (
2329:General
2042:Forcing
2027:Element
1947:Monadic
1722:paradox
1663:Theorem
1599:General
1275:7324133
1116:Bibcode
1067:Bibcode
1018:Bibcode
966:3215454
913:3056628
878:2955523
835:2684128
789:2506138
739:3083419
733:: A46.
647:2463991
608:2025976
389:-based
85:Methods
40:theorem
3692:Others
3679:(FETI)
3673:(BDDC)
3545:Mortar
3529:(XFEM)
3522:hp-FEM
3505:(WENO)
3488:(AUSM)
3449:(FDTD)
3443:(FDFD)
3428:Others
3414:Upwind
3377:(FTCS)
2980:finite
2743:Skolem
2696:
2671:Theory
2639:Symbol
2629:String
2612:atomic
2489:ground
2484:closed
2479:atomic
2435:ground
2398:syntax
2294:binary
2221:domain
2138:Finite
1903:finite
1761:Logics
1720:
1668:Theory
1478:
1441:
1420:
1272:
1262:
1183:
1087:
1059:Nature
1038:
1010:Nature
964:
911:
884:
876:
841:
833:
795:
787:
775:: A6.
737:
683:
645:
606:
410:using
403:, 2008
385:using
378:, 2006
368:using
348:, 1996
342:, 1995
317:, 1976
242:proofs
3706:(DVR)
3667:(BDD)
3606:(PIC)
3600:(MPM)
3594:(MPS)
3582:(SPH)
3552:(GDM)
3541:(SEM)
3499:(ENO)
3437:(ADI)
2970:Model
2718:Peano
2575:Proof
2415:Arity
2344:Naive
2231:image
2163:Fuzzy
2123:Empty
2072:union
2017:Class
1658:Model
1648:Lemma
1606:Axiom
1464:(PDF)
1403:(PDF)
1365:(PDF)
1316:arXiv
1203:arXiv
1181:S2CID
1155:arXiv
942:arXiv
882:S2CID
839:S2CID
793:S2CID
654:(PDF)
629:(PDF)
604:JSTOR
505:Heule
500:proof
23:is a
3588:(PD)
3535:(DG)
3093:Type
2896:list
2700:list
2677:list
2666:Term
2600:rank
2494:open
2388:list
2200:Maps
2105:sets
1964:Free
1934:list
1684:list
1611:list
1522:link
1439:ISBN
1418:ISBN
1377:2024
1260:ISBN
1085:PMID
1036:PMID
387:FPGA
2780:of
2762:of
2710:of
2242:Sur
2216:Map
2023:Ur-
2005:Set
1486:doi
1270:PMC
1252:doi
1173:hdl
1165:doi
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