33:
2951:
955:
941:
803:
In practice, not every proof is traced back to the axioms. At times, it is not even clear which collection of axioms a proof appeals to. For example, a number-theoretic statement might be expressible in the language of arithmetic (i.e. the language of the Peano axioms) and a proof might be given that
172:
if it cannot be proven or disproven from other axioms in the system. A system is called independent if each of its underlying axioms is independent. Unlike consistency, independence is not a necessary requirement for a functioning axiomatic system â though it is usually sought after to minimize the
791:
then tells us that there are certain consistent bodies of propositions with no recursive axiomatization. Typically, the computer can recognize the axioms and logical rules for deriving theorems, and the computer can recognize whether a proof is valid, but to determine whether a proof exists for a
592:). In fact, it has an infinite number of models, one for each cardinality of an infinite set. However, the property distinguishing these models is their cardinality â a property which cannot be defined within the system. Thus the system is not categorial. However it can be shown to be complete.
815:
Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitrary axiomatic system will not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything. Philosophers of mathematics sometimes assert that
629:
attempted to show that all mathematical theory could be reduced to some collection of axioms. More generally, the reduction of a body of propositions to a particular collection of axioms underlies the mathematician's research program. This was very prominent in the mathematics of the twentieth
538:
261:). The property of categoriality (categoricity) ensures the completeness of a system, however the converse is not true: Completeness does not ensure the categoriality (categoricity) of a system, since two models can differ in properties that cannot be expressed by the
190:
Beyond consistency, relative consistency is also the mark of a worthwhile axiom system. This describes the scenario where the undefined terms of a first axiom system are provided definitions from a second, such that the axioms of the first are theorems of the second.
816:
mathematicians choose axioms "arbitrarily", but it is possible that although they may appear arbitrary when viewed only from the point of view of the canons of deductive logic, that appearance is due to a limitation on the purposes that deductive logic serves.
245:
Models can also be used to show the independence of an axiom in the system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.
792:
statement is only soluble by "waiting" for the proof or disproof to be generated. The result is that one will not know which propositions are theorems and the axiomatic method breaks down. An example of such a body of propositions is the theory of the
159:. That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (
367:
124:
is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of
637:
The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that the mathematician would like to work with. For example, mathematicians opted that
376:
253:
if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship. An axiomatic system for which every model is isomorphic to another is called
1330:
600:
Stating definitions and propositions in a way such that each new term can be formally eliminated by the priorly introduced terms requires primitive notions (axioms) to avoid
2005:
182:
if for every statement, either itself or its negation is derivable from the system's axioms (equivalently, every statement is capable of being proven true or false).
568:
230:, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. The existence of a
710:
Mathematical methods developed to some degree of sophistication in ancient Egypt, Babylon, India, and China, apparently without employing the axiomatic method.
2088:
1229:
783:
Not every consistent body of propositions can be captured by a describable collection of axioms. In recursion theory, a collection of axioms is called
886:
If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers ("
668:, a result of the axiomatic method applied to set theory, allowed the "proper" formulation of set-theory problems and helped avoid the paradoxes of
2402:
2560:
974:
909:
is the process of taking a body of knowledge and working backwards towards its axioms. It is the formulation of a system of statements (i.e.
1348:
980:
788:
2415:
1738:
54:
2000:
1014: â Standard system of axiomatic set theory, an axiomatic system for set theory and today's most common foundation for mathematics.
2420:
2410:
2147:
1353:
1898:
1344:
291:
2556:
168:
76:
2653:
2397:
1222:
238:
of a system. A model is called concrete if the meanings assigned are objects and relations from the real world, as opposed to an
214:) in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems.
1958:
1651:
1392:
1011:
685:
665:
533:{\displaystyle \exists x_{1}:\exists x_{2}:\exists x_{3}:\lnot (x_{1}=x_{2})\land \lnot (x_{1}=x_{3})\land \lnot (x_{2}=x_{3})}
2914:
2616:
2379:
2374:
2199:
1620:
1304:
986:
688:
to refer to the axioms of
ZermeloâFraenkel set theory with the axiom of choice excluded. Today ZFC is the standard form of
2909:
2692:
2609:
2322:
2253:
2130:
1372:
1186:
573:
Informally, this infinite set of axioms states that there are infinitely many different items. However, the concept of an
1980:
812:. It might not be immediately clear whether another proof can be found that derives itself solely from the Peano axioms.
2975:
2834:
2660:
2346:
1579:
589:
588:(isomorphic to any other countably infinite set), and another is the real numbers (isomorphic to any other set with the
1985:
2317:
2056:
1314:
1215:
1181:
121:
2712:
2707:
47:
41:
2641:
2231:
1625:
1593:
1284:
693:
223:
1358:
2985:
2931:
2880:
2777:
2275:
2236:
1713:
2772:
1387:
58:
2702:
2241:
2093:
2076:
1799:
1279:
2980:
2604:
2581:
2542:
2428:
2369:
2015:
1935:
1779:
1723:
1336:
918:
2894:
2621:
2599:
2566:
2459:
2305:
2290:
2263:
2214:
2098:
2033:
1858:
1824:
1819:
1693:
1524:
1501:
888:
767:
was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that
740:
705:
622:
160:
2824:
2677:
2469:
2187:
1923:
1829:
1688:
1673:
1554:
1529:
1000:
617:
2950:
954:
1176:
1083:
2797:
2759:
2636:
2440:
2280:
2204:
2182:
2010:
1968:
1867:
1834:
1698:
1486:
1397:
772:
689:
673:
178:
2926:
2817:
2802:
2782:
2739:
2626:
2576:
2502:
2447:
2384:
2177:
2172:
2120:
1888:
1877:
1549:
1449:
1377:
1368:
1364:
1299:
1294:
922:
771:
should be required, for example), the subject could proceed autonomously, without reference to the
631:
2955:
2724:
2687:
2672:
2665:
2648:
2434:
2300:
2226:
2209:
2162:
1975:
1884:
1718:
1703:
1663:
1615:
1600:
1588:
1544:
1519:
1289:
1238:
960:
926:
843:
733:
721:
638:
278:
138:
2452:
1908:
977: â View that mathematics does not necessarily represent reality, but is more akin to a game
2890:
2697:
2507:
2497:
2389:
2270:
2105:
2081:
1862:
1846:
1751:
1728:
1605:
1574:
1539:
1434:
1269:
946:
914:
787:
if a computer program can recognize whether a given proposition in the language is a theorem.
650:
274:
235:
227:
195:
105:
2904:
2899:
2792:
2749:
2571:
2532:
2512:
2295:
2192:
1990:
1940:
1514:
1476:
1058:
995:
834:
0, 1, 2, 3, 4, ... is based on an axiomatic system first devised by the mathematician
809:
669:
654:
642:
626:
601:
211:
207:
109:
1121:
2885:
2875:
2829:
2812:
2767:
2729:
2631:
2551:
2358:
2285:
2258:
2246:
2152:
2066:
2040:
1995:
1963:
1764:
1566:
1509:
1459:
1424:
1382:
768:
677:
658:
203:
547:
2870:
2849:
2807:
2787:
2682:
2537:
2135:
2125:
2115:
2110:
2044:
1918:
1794:
1683:
1678:
1656:
1257:
1146:
835:
831:
793:
585:
2969:
2844:
2522:
2029:
1814:
1804:
1774:
1759:
1429:
784:
760:
756:
744:
725:
156:
126:
1198:
2744:
2591:
2492:
2484:
2364:
2312:
2221:
2157:
2140:
2071:
1930:
1789:
1491:
1274:
968:
825:
797:
764:
748:
646:
574:
282:
134:
130:
2854:
2734:
1913:
1903:
1850:
1534:
1454:
1439:
1319:
1264:
1202:
902:
838:
in 1889. He chose the axioms, in the language of a single unary function symbol
578:
250:
199:
151:
133:) that describes a set of sentences that is closed under logical implication. A
93:
17:
1784:
1639:
1610:
1416:
936:
752:
732:. Then, using these axioms, he established the truth of other propositions by
717:
2936:
2839:
1892:
1809:
1769:
1733:
1669:
1481:
1471:
1444:
929:
of any proposition should be, in principle, traceable back to these axioms.
262:
1033:
739:
Many axiomatic systems were developed in the nineteenth century, including
2921:
2719:
2167:
1872:
1466:
1005:
805:
612:
971: â Short notation for a set of statements that are taken to be true
2517:
1309:
117:
1207:
1090:(Winter 2018 ed.), Metaphysics Research Lab, Stanford University
729:
713:
728:. His idea begins with five undeniable geometric assumptions called
129:. A formal theory is an axiomatic system (usually formulated within
676:. ZermeloâFraenkel set theory, with the historically controversial
2061:
1407:
1252:
910:
763:'s 'new' use of axiomatic method as a research tool. For example,
113:
97:
273:
As an example, observe the following axiomatic system, based on
1211:
362:{\displaystyle \exists x_{1}:\exists x_{2}:\lnot (x_{1}=x_{2})}
681:
26:
913:) that relate a number of primitive terms â in order that a
649:'s original formulation. Mathematicians decided to consider
89:
Mathematical term; concerning axioms used to derive theorems
584:
The system has at least two different models â one is the
281:
many axioms added (these can be easily formalized as an
991:
Pages displaying short descriptions of redirect targets
720:
authored the earliest extant axiomatic presentation of
1108:
Set Theory and its
Philosophy, a Critical Introduction
867:
Distinct natural numbers have distinct successors: if
550:
379:
294:
820:
Example: The Peano axiomatization of natural numbers
577:
cannot be defined within the system â let alone the
2863:
2758:
2590:
2483:
2335:
2028:
1951:
1845:
1749:
1638:
1565:
1500:
1415:
1406:
1328:
1245:
1008: â Programme in the philosophy of mathematics
684:, where "C" stands for "choice". Many authors use
611:A common attitude towards the axiomatic method is
562:
532:
361:
1145:Lehman, Eric; Meyer, Albert R; Leighton, F Tom.
983: â Limitative results in mathematical logic
864:There is no natural number whose successor is 0.
630:century, in particular in subjects based around
540:(informally, there exist three different items).
604:. This way of doing mathematics is called the
369:(informally, there exist two different items).
194:A good example is the relative consistency of
1223:
796:, which is only partially axiomatized by the
8:
989: â System of formal deduction in logic
277:with additional semantics of the following
242:which is based on other axiomatic systems.
166:In an axiomatic system, an axiom is called
2049:
1644:
1412:
1230:
1216:
1208:
846:"), for the set of natural numbers to be:
226:for an axiomatic system is a well-defined
1197:, From MathWorldâA Wolfram Web Resource.
1082:Hodges, Wilfrid; Scanlon, Thomas (2018),
549:
521:
508:
486:
473:
451:
438:
419:
403:
387:
378:
350:
337:
318:
302:
293:
77:Learn how and when to remove this message
40:This article includes a list of general
1088:The Stanford Encyclopedia of Philosophy
1024:
925:from these statements. Thereafter, the
7:
789:Gödel's first incompleteness theorem
680:included, is commonly abbreviated
498:
463:
428:
412:
396:
380:
327:
311:
295:
198:with respect to the theory of the
149:An axiomatic system is said to be
46:it lacks sufficient corresponding
25:
1110:S.6; Michael Potter, Oxford, 2004
210:are undefined terms (also called
2949:
1148:Mathematics for Computer Science
953:
939:
173:number of axioms in the system.
31:
981:Gödel's incompleteness theorems
692:and as such is the most common
987:Hilbert-style deduction system
736:, hence the axiomatic method.
527:
501:
492:
466:
457:
431:
356:
330:
176:An axiomatic system is called
1:
2910:History of mathematical logic
1086:, in Zalta, Edward N. (ed.),
137:is a complete rendition of a
2835:Primitive recursive function
857:has a successor, denoted by
850:There is a natural number 0.
759:'s work on foundations, and
590:cardinality of the continuum
1182:Encyclopedia of Mathematics
1059:"Complete Axiomatic Theory"
1012:ZermeloâFraenkel set theory
830:The mathematical system of
672:. One such problem was the
666:Zermelo-Fraenkel set theory
653:more generally without the
3002:
1899:SchröderâBernstein theorem
1626:Monadic predicate calculus
1285:Foundations of mathematics
1084:"First-order Model Theory"
823:
775:origins of those studies.
703:
249:Two models are said to be
2945:
2932:Philosophy of mathematics
2881:Automated theorem proving
2052:
2006:Von NeumannâBernaysâGödel
1647:
1122:"Zermelo-Fraenkel Axioms"
694:foundation of mathematics
141:within a formal system.
2582:Self-verifying theories
2403:Tarski's axiomatization
1354:Tarski's undefinability
1349:incompleteness theorems
661:originally formulated.
61:more precise citations.
2956:Mathematics portal
2567:Proof of impossibility
2215:propositional variable
1525:Propositional calculus
741:non-Euclidean geometry
706:History of Mathematics
645:, which differed from
623:Alfred North Whitehead
564:
534:
363:
161:principle of explosion
2825:Kolmogorov complexity
2778:Computably enumerable
2678:Model complete theory
2470:Principia Mathematica
1530:Propositional formula
1359:BanachâTarski paradox
1199:Mathworld.wolfram.com
1126:mathworld.wolfram.com
1063:mathworld.wolfram.com
1038:mathworld.wolfram.com
1001:List of logic systems
853:Every natural number
743:, the foundations of
704:Further information:
618:Principia Mathematica
565:
535:
364:
2773:ChurchâTuring thesis
2760:Computability theory
1969:continuum hypothesis
1487:Square of opposition
1345:Gödel's completeness
773:transformation group
690:axiomatic set theory
674:continuum hypothesis
548:
377:
292:
279:countably infinitely
186:Relative consistency
116:to logically derive
2976:Mathematical axioms
2927:Mathematical object
2818:P versus NP problem
2783:Computable function
2577:Reverse mathematics
2503:Logical consequence
2380:primitive recursive
2375:elementary function
2148:Free/bound variable
2001:TarskiâGrothendieck
1520:Logical connectives
1450:Logical equivalence
1300:Logical consequence
1193:Eric W. Weisstein,
1120:Weisstein, Eric W.
1057:Weisstein, Eric W.
1032:Weisstein, Eric W.
800:(described below).
632:homological algebra
563:{\displaystyle ...}
2725:Transfer principle
2688:Semantics of logic
2673:Categorical theory
2649:Non-standard model
2163:Logical connective
1290:Information theory
1239:Mathematical logic
1177:"Axiomatic method"
961:Mathematics portal
722:Euclidean geometry
651:topological spaces
560:
530:
359:
200:real number system
139:mathematical proof
2963:
2962:
2895:Abstract category
2698:Theories of truth
2508:Rule of inference
2498:Natural deduction
2479:
2478:
2024:
2023:
1729:Cartesian product
1634:
1633:
1540:Many-valued logic
1515:Boolean functions
1398:Russell's paradox
1373:diagonal argument
1270:First-order logic
947:Philosophy portal
275:first-order logic
212:primitive notions
196:absolute geometry
110:primitive notions
87:
86:
79:
16:(Redirected from
2993:
2986:Methods of proof
2954:
2953:
2905:History of logic
2900:Category of sets
2793:Decision problem
2572:Ordinal analysis
2513:Sequent calculus
2411:Boolean algebras
2351:
2350:
2325:
2296:logical/constant
2050:
2036:
1959:ZermeloâFraenkel
1710:Set operations:
1645:
1582:
1413:
1393:LöwenheimâSkolem
1280:Formal semantics
1232:
1225:
1218:
1209:
1195:Axiomatic System
1190:
1163:
1162:
1160:
1158:
1153:
1142:
1136:
1135:
1133:
1132:
1117:
1111:
1104:
1098:
1097:
1096:
1095:
1079:
1073:
1072:
1070:
1069:
1054:
1048:
1047:
1045:
1044:
1029:
996:History of logic
992:
963:
958:
957:
949:
944:
943:
942:
810:complex analysis
769:inverse elements
670:naĂŻve set theory
655:separation axiom
627:Bertrand Russell
615:. In their book
606:axiomatic method
602:infinite regress
596:Axiomatic method
581:of such as set.
569:
567:
566:
561:
539:
537:
536:
531:
526:
525:
513:
512:
491:
490:
478:
477:
456:
455:
443:
442:
424:
423:
408:
407:
392:
391:
368:
366:
365:
360:
355:
354:
342:
341:
323:
322:
307:
306:
102:axiomatic system
82:
75:
71:
68:
62:
57:this article by
48:inline citations
35:
34:
27:
21:
3001:
3000:
2996:
2995:
2994:
2992:
2991:
2990:
2966:
2965:
2964:
2959:
2948:
2941:
2886:Category theory
2876:Algebraic logic
2859:
2830:Lambda calculus
2768:Church encoding
2754:
2730:Truth predicate
2586:
2552:Complete theory
2475:
2344:
2340:
2336:
2331:
2323:
2043: and
2039:
2034:
2020:
1996:New Foundations
1964:axiom of choice
1947:
1909:Gödel numbering
1849: and
1841:
1745:
1630:
1580:
1561:
1510:Boolean algebra
1496:
1460:Equiconsistency
1425:Classical logic
1402:
1383:Halting problem
1371: and
1347: and
1335: and
1334:
1329:Theorems (
1324:
1241:
1236:
1175:
1172:
1170:Further reading
1167:
1166:
1156:
1154:
1151:
1144:
1143:
1139:
1130:
1128:
1119:
1118:
1114:
1105:
1101:
1093:
1091:
1081:
1080:
1076:
1067:
1065:
1056:
1055:
1051:
1042:
1040:
1031:
1030:
1026:
1021:
990:
959:
952:
945:
940:
938:
935:
921:may be derived
899:
889:Induction axiom
832:natural numbers
828:
822:
794:natural numbers
781:
708:
702:
678:axiom of choice
659:Felix Hausdorff
598:
586:natural numbers
546:
545:
517:
504:
482:
469:
447:
434:
415:
399:
383:
375:
374:
346:
333:
314:
298:
290:
289:
271:
265:of the system.
220:
188:
147:
90:
83:
72:
66:
63:
53:Please help to
52:
36:
32:
23:
22:
18:Axiomatic proof
15:
12:
11:
5:
2999:
2997:
2989:
2988:
2983:
2981:Formal systems
2978:
2968:
2967:
2961:
2960:
2946:
2943:
2942:
2940:
2939:
2934:
2929:
2924:
2919:
2918:
2917:
2907:
2902:
2897:
2888:
2883:
2878:
2873:
2871:Abstract logic
2867:
2865:
2861:
2860:
2858:
2857:
2852:
2850:Turing machine
2847:
2842:
2837:
2832:
2827:
2822:
2821:
2820:
2815:
2810:
2805:
2800:
2790:
2788:Computable set
2785:
2780:
2775:
2770:
2764:
2762:
2756:
2755:
2753:
2752:
2747:
2742:
2737:
2732:
2727:
2722:
2717:
2716:
2715:
2710:
2705:
2695:
2690:
2685:
2683:Satisfiability
2680:
2675:
2670:
2669:
2668:
2658:
2657:
2656:
2646:
2645:
2644:
2639:
2634:
2629:
2624:
2614:
2613:
2612:
2607:
2600:Interpretation
2596:
2594:
2588:
2587:
2585:
2584:
2579:
2574:
2569:
2564:
2554:
2549:
2548:
2547:
2546:
2545:
2535:
2530:
2520:
2515:
2510:
2505:
2500:
2495:
2489:
2487:
2481:
2480:
2477:
2476:
2474:
2473:
2465:
2464:
2463:
2462:
2457:
2456:
2455:
2450:
2445:
2425:
2424:
2423:
2421:minimal axioms
2418:
2407:
2406:
2405:
2394:
2393:
2392:
2387:
2382:
2377:
2372:
2367:
2354:
2352:
2333:
2332:
2330:
2329:
2328:
2327:
2315:
2310:
2309:
2308:
2303:
2298:
2293:
2283:
2278:
2273:
2268:
2267:
2266:
2261:
2251:
2250:
2249:
2244:
2239:
2234:
2224:
2219:
2218:
2217:
2212:
2207:
2197:
2196:
2195:
2190:
2185:
2180:
2175:
2170:
2160:
2155:
2150:
2145:
2144:
2143:
2138:
2133:
2128:
2118:
2113:
2111:Formation rule
2108:
2103:
2102:
2101:
2096:
2086:
2085:
2084:
2074:
2069:
2064:
2059:
2053:
2047:
2030:Formal systems
2026:
2025:
2022:
2021:
2019:
2018:
2013:
2008:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1972:
1971:
1966:
1955:
1953:
1949:
1948:
1946:
1945:
1944:
1943:
1933:
1928:
1927:
1926:
1919:Large cardinal
1916:
1911:
1906:
1901:
1896:
1882:
1881:
1880:
1875:
1870:
1855:
1853:
1843:
1842:
1840:
1839:
1838:
1837:
1832:
1827:
1817:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1756:
1754:
1747:
1746:
1744:
1743:
1742:
1741:
1736:
1731:
1726:
1721:
1716:
1708:
1707:
1706:
1701:
1691:
1686:
1684:Extensionality
1681:
1679:Ordinal number
1676:
1666:
1661:
1660:
1659:
1648:
1642:
1636:
1635:
1632:
1631:
1629:
1628:
1623:
1618:
1613:
1608:
1603:
1598:
1597:
1596:
1586:
1585:
1584:
1571:
1569:
1563:
1562:
1560:
1559:
1558:
1557:
1552:
1547:
1537:
1532:
1527:
1522:
1517:
1512:
1506:
1504:
1498:
1497:
1495:
1494:
1489:
1484:
1479:
1474:
1469:
1464:
1463:
1462:
1452:
1447:
1442:
1437:
1432:
1427:
1421:
1419:
1410:
1404:
1403:
1401:
1400:
1395:
1390:
1385:
1380:
1375:
1363:Cantor's
1361:
1356:
1351:
1341:
1339:
1326:
1325:
1323:
1322:
1317:
1312:
1307:
1302:
1297:
1292:
1287:
1282:
1277:
1272:
1267:
1262:
1261:
1260:
1249:
1247:
1243:
1242:
1237:
1235:
1234:
1227:
1220:
1212:
1206:
1205:
1191:
1171:
1168:
1165:
1164:
1137:
1112:
1099:
1074:
1049:
1023:
1022:
1020:
1017:
1016:
1015:
1009:
1003:
998:
993:
984:
978:
972:
965:
964:
950:
934:
931:
907:axiomatization
898:
897:Axiomatization
895:
894:
893:
884:
865:
862:
851:
836:Giuseppe Peano
824:Main article:
821:
818:
780:
777:
701:
698:
597:
594:
571:
570:
559:
556:
553:
542:
541:
529:
524:
520:
516:
511:
507:
503:
500:
497:
494:
489:
485:
481:
476:
472:
468:
465:
462:
459:
454:
450:
446:
441:
437:
433:
430:
427:
422:
418:
414:
411:
406:
402:
398:
395:
390:
386:
382:
371:
370:
358:
353:
349:
345:
340:
336:
332:
329:
326:
321:
317:
313:
310:
305:
301:
297:
270:
267:
260:
256:
241:
240:abstract model
233:
232:concrete model
219:
216:
187:
184:
146:
143:
88:
85:
84:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2998:
2987:
2984:
2982:
2979:
2977:
2974:
2973:
2971:
2958:
2957:
2952:
2944:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2916:
2913:
2912:
2911:
2908:
2906:
2903:
2901:
2898:
2896:
2892:
2889:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2868:
2866:
2862:
2856:
2853:
2851:
2848:
2846:
2845:Recursive set
2843:
2841:
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2795:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2765:
2763:
2761:
2757:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2714:
2711:
2709:
2706:
2704:
2701:
2700:
2699:
2696:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2667:
2664:
2663:
2662:
2659:
2655:
2654:of arithmetic
2652:
2651:
2650:
2647:
2643:
2640:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2619:
2618:
2615:
2611:
2608:
2606:
2603:
2602:
2601:
2598:
2597:
2595:
2593:
2589:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2562:
2561:from ZFC
2558:
2555:
2553:
2550:
2544:
2541:
2540:
2539:
2536:
2534:
2531:
2529:
2526:
2525:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2499:
2496:
2494:
2491:
2490:
2488:
2486:
2482:
2472:
2471:
2467:
2466:
2461:
2460:non-Euclidean
2458:
2454:
2451:
2449:
2446:
2444:
2443:
2439:
2438:
2436:
2433:
2432:
2430:
2426:
2422:
2419:
2417:
2414:
2413:
2412:
2408:
2404:
2401:
2400:
2399:
2395:
2391:
2388:
2386:
2383:
2381:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2362:
2360:
2356:
2355:
2353:
2348:
2342:
2337:Example
2334:
2326:
2321:
2320:
2319:
2316:
2314:
2311:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2288:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2265:
2262:
2260:
2257:
2256:
2255:
2252:
2248:
2245:
2243:
2240:
2238:
2235:
2233:
2230:
2229:
2228:
2225:
2223:
2220:
2216:
2213:
2211:
2208:
2206:
2203:
2202:
2201:
2198:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2165:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2123:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2100:
2097:
2095:
2094:by definition
2092:
2091:
2090:
2087:
2083:
2080:
2079:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2054:
2051:
2048:
2046:
2042:
2037:
2031:
2027:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1981:KripkeâPlatek
1979:
1977:
1974:
1970:
1967:
1965:
1962:
1961:
1960:
1957:
1956:
1954:
1950:
1942:
1939:
1938:
1937:
1934:
1932:
1929:
1925:
1922:
1921:
1920:
1917:
1915:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1894:
1890:
1886:
1883:
1879:
1876:
1874:
1871:
1869:
1866:
1865:
1864:
1860:
1857:
1856:
1854:
1852:
1848:
1844:
1836:
1833:
1831:
1828:
1826:
1825:constructible
1823:
1822:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1757:
1755:
1753:
1748:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1711:
1709:
1705:
1702:
1700:
1697:
1696:
1695:
1692:
1690:
1687:
1685:
1682:
1680:
1677:
1675:
1671:
1667:
1665:
1662:
1658:
1655:
1654:
1653:
1650:
1649:
1646:
1643:
1641:
1637:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1604:
1602:
1599:
1595:
1592:
1591:
1590:
1587:
1583:
1578:
1577:
1576:
1573:
1572:
1570:
1568:
1564:
1556:
1553:
1551:
1548:
1546:
1543:
1542:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1507:
1505:
1503:
1502:Propositional
1499:
1493:
1490:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1468:
1465:
1461:
1458:
1457:
1456:
1453:
1451:
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1430:Logical truth
1428:
1426:
1423:
1422:
1420:
1418:
1414:
1411:
1409:
1405:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1370:
1366:
1362:
1360:
1357:
1355:
1352:
1350:
1346:
1343:
1342:
1340:
1338:
1332:
1327:
1321:
1318:
1316:
1313:
1311:
1308:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1286:
1283:
1281:
1278:
1276:
1273:
1271:
1268:
1266:
1263:
1259:
1256:
1255:
1254:
1251:
1250:
1248:
1244:
1240:
1233:
1228:
1226:
1221:
1219:
1214:
1213:
1210:
1204:
1200:
1196:
1192:
1188:
1184:
1183:
1178:
1174:
1173:
1169:
1150:
1149:
1141:
1138:
1127:
1123:
1116:
1113:
1109:
1103:
1100:
1089:
1085:
1078:
1075:
1064:
1060:
1053:
1050:
1039:
1035:
1028:
1025:
1018:
1013:
1010:
1007:
1004:
1002:
999:
997:
994:
988:
985:
982:
979:
976:
973:
970:
967:
966:
962:
956:
951:
948:
937:
932:
930:
928:
924:
920:
916:
912:
908:
904:
896:
891:
890:
885:
882:
878:
874:
870:
866:
863:
860:
856:
852:
849:
848:
847:
845:
841:
837:
833:
827:
819:
817:
813:
811:
807:
801:
799:
795:
790:
786:
778:
776:
774:
770:
766:
762:
758:
754:
750:
746:
745:real analysis
742:
737:
735:
731:
727:
726:number theory
723:
719:
715:
711:
707:
699:
697:
695:
691:
687:
683:
679:
675:
671:
667:
662:
660:
656:
652:
648:
644:
640:
635:
633:
628:
624:
620:
619:
614:
609:
607:
603:
595:
593:
591:
587:
582:
580:
576:
557:
554:
551:
544:
543:
522:
518:
514:
509:
505:
495:
487:
483:
479:
474:
470:
460:
452:
448:
444:
439:
435:
425:
420:
416:
409:
404:
400:
393:
388:
384:
373:
372:
351:
347:
343:
338:
334:
324:
319:
315:
308:
303:
299:
288:
287:
286:
284:
280:
276:
268:
266:
264:
258:
254:
252:
247:
243:
239:
237:
231:
229:
225:
217:
215:
213:
209:
205:
201:
197:
192:
185:
183:
181:
180:
174:
171:
170:
164:
162:
158:
157:contradiction
154:
153:
144:
142:
140:
136:
132:
128:
127:formal system
123:
119:
115:
111:
107:
103:
99:
95:
81:
78:
70:
60:
56:
50:
49:
43:
38:
29:
28:
19:
2947:
2745:Ultraproduct
2592:Model theory
2557:Independence
2527:
2493:Formal proof
2485:Proof theory
2468:
2441:
2398:real numbers
2370:second-order
2338:
2281:Substitution
2158:Metalanguage
2099:conservative
2072:Axiom schema
2016:Constructive
1986:MorseâKelley
1952:Set theories
1931:Aleph number
1924:inaccessible
1830:Grothendieck
1714:intersection
1601:Higher-order
1589:Second-order
1535:Truth tables
1492:Venn diagram
1275:Formal proof
1194:
1180:
1155:. Retrieved
1147:
1140:
1129:. Retrieved
1125:
1115:
1107:
1102:
1092:, retrieved
1087:
1077:
1066:. Retrieved
1062:
1052:
1041:. Retrieved
1037:
1027:
969:Axiom schema
919:propositions
906:
900:
887:
880:
876:
872:
868:
858:
854:
842:(short for "
839:
829:
826:Peano axioms
814:
802:
798:Peano axioms
782:
765:group theory
738:
712:
709:
663:
647:Emmy Noether
641:need not be
636:
616:
610:
605:
599:
583:
575:infinite set
572:
283:axiom schema
272:
248:
244:
221:
193:
189:
177:
175:
167:
165:
155:if it lacks
150:
148:
135:formal proof
131:model theory
101:
91:
73:
64:
45:
2855:Type theory
2803:undecidable
2735:Truth value
2622:equivalence
2301:non-logical
1914:Enumeration
1904:Isomorphism
1851:cardinality
1835:Von Neumann
1800:Ultrafilter
1765:Uncountable
1699:equivalence
1616:Quantifiers
1606:Fixed-point
1575:First-order
1455:Consistency
1440:Proposition
1417:Traditional
1388:Lindström's
1378:Compactness
1320:Type theory
1265:Cardinality
1203:Answers.com
923:deductively
903:mathematics
804:appeals to
643:commutative
579:cardinality
259:categorical
257:(sometimes
236:consistency
234:proves the
169:independent
94:mathematics
59:introducing
2970:Categories
2666:elementary
2359:arithmetic
2227:Quantifier
2205:functional
2077:Expression
1795:Transitive
1739:identities
1724:complement
1657:hereditary
1640:Set theory
1131:2019-10-31
1094:2019-10-31
1068:2019-10-31
1043:2019-10-31
1019:References
915:consistent
753:set theory
718:Alexandria
255:categorial
251:isomorphic
152:consistent
145:Properties
67:March 2013
42:references
2937:Supertask
2840:Recursion
2798:decidable
2632:saturated
2610:of models
2533:deductive
2528:axiomatic
2448:Hilbert's
2435:Euclidean
2416:canonical
2339:axiomatic
2271:Signature
2200:Predicate
2089:Extension
2011:Ackermann
1936:Operation
1815:Universal
1805:Recursive
1780:Singleton
1775:Inhabited
1760:Countable
1750:Types of
1734:power set
1704:partition
1621:Predicate
1567:Predicate
1482:Syllogism
1472:Soundness
1445:Inference
1435:Tautology
1337:paradoxes
1187:EMS Press
975:Formalism
844:successor
785:recursive
499:¬
496:∧
464:¬
461:∧
429:¬
413:∃
397:∃
381:∃
328:¬
312:∃
296:∃
263:semantics
2922:Logicism
2915:timeline
2891:Concrete
2750:Validity
2720:T-schema
2713:Kripke's
2708:Tarski's
2703:semantic
2693:Strength
2642:submodel
2637:spectrum
2605:function
2453:Tarski's
2442:Elements
2429:geometry
2385:Robinson
2306:variable
2291:function
2264:spectrum
2254:Sentence
2210:variable
2153:Language
2106:Relation
2067:Automata
2057:Alphabet
2041:language
1895:-jection
1873:codomain
1859:Function
1820:Universe
1790:Infinite
1694:Relation
1477:Validity
1467:Argument
1365:theorem,
1034:"Theory"
1006:Logicism
933:See also
917:body of
806:topology
613:logicism
179:complete
118:theorems
2864:Related
2661:Diagram
2559: (
2538:Hilbert
2523:Systems
2518:Theorem
2396:of the
2341:systems
2121:Formula
2116:Grammar
2032: (
1976:General
1689:Forcing
1674:Element
1594:Monadic
1369:paradox
1310:Theorem
1246:General
1189:, 2001
875:, then
761:Hilbert
700:History
269:Example
104:is any
55:improve
2627:finite
2390:Skolem
2343:
2318:Theory
2286:Symbol
2276:String
2259:atomic
2136:ground
2131:closed
2126:atomic
2082:ground
2045:syntax
1941:binary
1868:domain
1785:Finite
1550:finite
1408:Logics
1367:
1315:Theory
1201:&
911:axioms
779:Issues
749:Cantor
734:proofs
730:axioms
714:Euclid
657:which
218:Models
208:points
122:theory
114:axioms
44:, but
2617:Model
2365:Peano
2222:Proof
2062:Arity
1991:Naive
1878:image
1810:Fuzzy
1770:Empty
1719:union
1664:Class
1305:Model
1295:Lemma
1253:Axiom
1157:2 May
1152:(PDF)
927:proof
757:Frege
639:rings
224:model
204:Lines
100:, an
98:logic
2740:Type
2543:list
2347:list
2324:list
2313:Term
2247:rank
2141:open
2035:list
1847:Maps
1752:sets
1611:Free
1581:list
1331:list
1258:list
1159:2023
724:and
664:The
625:and
206:and
120:. A
112:and
96:and
2427:of
2409:of
2357:of
1889:Sur
1863:Map
1670:Ur-
1652:Set
901:In
892:").
808:or
751:'s
716:of
682:ZFC
285:):
228:set
202:.
163:).
108:of
106:set
92:In
2972::
2813:NP
2437::
2431::
2361::
2038:),
1893:Bi
1885:In
1185:,
1179:,
1124:.
1061:.
1036:.
905:,
881:Sb
879:â
877:Sa
871:â
859:Sa
755:,
747:,
696:.
686:ZF
634:.
621:,
608:.
222:A
2893:/
2808:P
2563:)
2349:)
2345:(
2242:â
2237:!
2232:â
2193:=
2188:â
2183:â
2178:â§
2173:âš
2168:ÂŹ
1891:/
1887:/
1861:/
1672:)
1668:(
1555:â
1545:3
1333:)
1231:e
1224:t
1217:v
1161:.
1134:.
1106:"
1071:.
1046:.
883:.
873:b
869:a
861:.
855:a
840:S
558:.
555:.
552:.
528:)
523:3
519:x
515:=
510:2
506:x
502:(
493:)
488:3
484:x
480:=
475:1
471:x
467:(
458:)
453:2
449:x
445:=
440:1
436:x
432:(
426::
421:3
417:x
410::
405:2
401:x
394::
389:1
385:x
357:)
352:2
348:x
344:=
339:1
335:x
331:(
325::
320:2
316:x
309::
304:1
300:x
80:)
74:(
69:)
65:(
51:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.