5152:
1660:
1647:
1081:
1071:
1041:
1031:
1009:
999:
964:
942:
932:
897:
865:
855:
830:
798:
783:
763:
731:
721:
711:
691:
659:
639:
629:
619:
587:
567:
523:
503:
493:
456:
431:
421:
389:
364:
327:
297:
265:
203:
168:
3253:. In the context of the natural numbers, this means that the relation <, which is well-founded, is used instead of the relation â€, which is not. In some texts, the definition of a well-founded relation is changed from the definition above to include these conventions.
2481:
1391:
1320:
3184:
in the domain of the relation. Every reflexive relation on a nonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example, in the natural numbers with their usual order â€, we have
2674:
1635:
1249:
1520:
1476:
3189:. To avoid these trivial descending sequences, when working with a partial order â€, it is common to apply the definition of well foundedness (perhaps implicitly) to the alternate relation < defined such that
1432:
1657:
indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by
1590:
1487:
1443:
1402:
1331:
1260:
1204:
2333:
2029:
1193:
1579:
3531:
2587:
1326:
4206:
1733:
1255:
1814:
1785:
1759:
1547:
1164:
1694:
4289:
3430:
5839:
68:
1585:
5822:
1199:
3045:
is an infinite descending chain. This relation fails to be well-founded even though the entire set has a minimum element, namely the empty string.
5352:
4603:
5188:
2807:
There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all
1482:
4761:
3549:
1438:
5669:
4616:
3939:
1397:
4201:
4621:
4611:
4348:
3554:
4099:
3545:
5805:
5664:
4757:
3403:
3378:
3332:
5659:
4854:
4598:
3423:
4159:
3852:
3593:
3120:
implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation
2158:
5295:
1919:
5377:
5115:
4817:
4580:
4575:
4400:
3821:
3505:
3280:
61:
5696:
5616:
5110:
4893:
4810:
4523:
4454:
4331:
3573:
3371:
2773:
5290:
4181:
5481:
5410:
5035:
4861:
4547:
3780:
3085:
are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let
31:
4186:
5384:
5372:
5335:
5310:
5285:
5239:
5208:
4518:
4257:
3515:
3416:
5315:
5305:
4913:
4908:
3296:
3093:
is a well-founded set, but there are descending chains starting at Ï of arbitrary great (finite) length; the chain
5681:
5181:
4842:
4432:
3826:
3794:
3485:
2789:
3559:
5654:
5320:
5132:
5081:
4978:
4476:
4437:
3914:
2878:
2202:
2047:
2043:
54:
4973:
3588:
5872:
5586:
5213:
4903:
4442:
4294:
4277:
4000:
3480:
3322:
3056:) under the standard ordering, since, for example, the subset of positive rationals (or reals) lacks a minimum.
2146:
5834:
5817:
4805:
4782:
4743:
4629:
4570:
4216:
4136:
3980:
3924:
3537:
2801:
5746:
5362:
5095:
4822:
4800:
4767:
4660:
4506:
4491:
4464:
4415:
4299:
4234:
4059:
4025:
4020:
3894:
3725:
3702:
3038:
2995:
2765:
1098:
91:
1169:
5724:
5559:
5419:
5254:
5218:
5174:
5025:
4878:
4670:
4388:
4124:
4030:
3889:
3874:
3755:
3730:
2812:
2493:
2222:
2142:
5300:
5151:
1553:
5812:
5771:
5761:
5751:
5496:
5459:
5449:
5429:
5414:
4998:
4960:
4837:
4641:
4481:
4405:
4383:
4211:
4169:
4068:
4035:
3899:
3687:
3598:
2819:. When the well-founded relation is set membership on the universal class, the technique is known as
2816:
2815:. When the well-founded set is a set of recursively-defined data structures, the technique is called
1674:
885:
160:
5739:
5650:
5596:
5555:
5545:
5434:
5367:
5330:
5127:
5018:
5003:
4983:
4940:
4827:
4777:
4703:
4648:
4585:
4378:
4373:
4321:
4089:
4078:
3750:
3650:
3578:
3569:
3565:
3500:
3495:
2988:
2476:{\displaystyle (\forall x\in X)\;\implies P(x)]\quad {\text{implies}}\quad (\forall x\in X)\,P(x).}
2154:
1697:
1133:
544:
126:
43:
3384:
3089:
be the union of the positive integers with a new element Ï that is bigger than any integer. Then
1386:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
5778:
5631:
5540:
5530:
5471:
5389:
5156:
4925:
4888:
4873:
4866:
4849:
4635:
4501:
4427:
4410:
4363:
4176:
4085:
3919:
3904:
3864:
3816:
3801:
3789:
3745:
3720:
3490:
3439:
3165:
3117:
2785:
1854:
1128:
1123:
1103:
1093:
1019:
121:
116:
96:
86:
5851:
5691:
5325:
4653:
4109:
1315:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
5788:
5766:
5626:
5611:
5591:
5394:
5091:
4898:
4708:
4698:
4590:
4471:
4306:
4282:
4063:
4047:
3952:
3929:
3806:
3775:
3740:
3635:
3470:
3399:
3374:
3328:
3276:
2913:
2185:
2036:
1850:
1653:
indicates that the column's property is always true the row's term (at the very left), while
5601:
5454:
5105:
5100:
4993:
4950:
4772:
4733:
4728:
4713:
4539:
4496:
4393:
4191:
4141:
3715:
3677:
1703:
818:
751:
2987:
Every class whose elements are sets, with the relation â ("is an element of"). This is the
2820:
5783:
5566:
5444:
5439:
5424:
5249:
5234:
5086:
5076:
5030:
5013:
4968:
4930:
4832:
4752:
4559:
4486:
4459:
4447:
4353:
4267:
4241:
4196:
4164:
3965:
3767:
3710:
3660:
3625:
3583:
3049:
2808:
2742:
2546:
2509:
1877:
1831:
1790:
679:
476:
47:
5340:
1817:
A term's definition may require additional properties that are not listed in this table.
1764:
1738:
1526:
1143:
2221:
An important reason that well-founded relations are interesting is because a version of
5701:
5686:
5676:
5535:
5513:
5491:
5071:
5050:
5008:
4988:
4883:
4738:
4336:
4326:
4316:
4311:
4245:
4119:
3995:
3884:
3879:
3857:
3458:
3037:
The set of strings over a finite alphabet with more than one element, under the usual (
2917:
2852:
1679:
347:
2492:
On par with induction, well-founded relations also support construction of objects by
5866:
5800:
5756:
5734:
5606:
5476:
5464:
5269:
5045:
4723:
4230:
4015:
4005:
3975:
3960:
3630:
3396:
2111:
1118:
1113:
952:
285:
111:
106:
5621:
5503:
5486:
5404:
5244:
5197:
4945:
4792:
4693:
4685:
4565:
4513:
4422:
4358:
4341:
4272:
4131:
3990:
3692:
3475:
3053:
2486:
2115:
2107:
2669:{\displaystyle G(x)=F\left(x,G\vert _{\left\{y:\,y\mathrel {R} x\right\}}\right).}
5827:
5520:
5399:
5264:
5055:
4935:
4114:
4104:
4051:
3735:
3655:
3640:
3520:
3465:
3388:
2119:
1827:
409:
1630:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
5795:
5729:
5570:
3985:
3840:
3811:
3617:
2130:
2123:
1244:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
607:
5846:
5719:
5525:
5137:
5040:
4093:
4010:
3970:
3934:
3870:
3682:
3672:
3645:
2206:
1861:
17:
5641:
5508:
5259:
5122:
4920:
4368:
4073:
3667:
3220:
224:
3034:, with the usual order, since any unbounded subset has no least element.
1515:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
4718:
3510:
2858:
2836:
2485:
Well-founded induction is sometimes called
Noetherian induction, after
1471:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
3408:
1864:
1427:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
30:"Noetherian induction" redirects here. For the use in topology, see
4262:
3608:
3453:
2039:, i.e., that the elements less than any given element form a set.
5170:
5166:
3412:
3297:"Infinite Sequence Property of Strictly Well-Founded Relation"
2831:
Well-founded relations that are not totally ordered include:
3223:â€, it is common to use the relation < defined such that
3026:
Examples of relations that are not well-founded include:
2050:, which can be proved when there is no infinite sequence
3395:, 3rd edition, "Well-founded relations", pages 251â5,
2590:
2336:
1922:
1793:
1767:
1741:
1706:
1682:
1588:
1556:
1529:
1485:
1441:
1400:
1329:
1258:
1202:
1172:
1146:
5712:
5640:
5579:
5349:
5278:
5227:
5064:
4959:
4791:
4684:
4536:
4229:
4152:
4046:
3950:
3839:
3766:
3701:
3616:
3607:
3529:
3446:
2723:As an example, consider the well-founded relation
2668:
2475:
2023:
1808:
1779:
1753:
1727:
1688:
1629:
1573:
1541:
1514:
1470:
1426:
1385:
1314:
1243:
1187:
1158:
3275:(2nd, rev. ed.). New York: Springer-Verlag.
2881:over a fixed alphabet, with the order defined by
2046:, a relation is well-founded when it contains no
1916:. In other words, a relation is well founded if:
1405:
2209:systems, a Noetherian relation is also called
2118:is a well-founded relation. If the order is a
5182:
3424:
3324:Theory of Relations, Volume 145 - 1st Edition
3043:"B" > "AB" > "AAB" > "AAAB" > ...
2031:Some authors include an extra condition that
1670:in the "Antisymmetric" column, respectively.
62:
8:
3356:Elements of mathematics. Commutative algebra
2678:That is, if we want to construct a function
2624:
2114:is called well-founded if the corresponding
5840:Positive cone of a partially ordered group
5189:
5175:
5167:
4250:
3845:
3613:
3431:
3417:
3409:
3128:that is extensional, there exists a class
2416:
2412:
2396:
2392:
2377:
2355:
2161:, asserts that all sets are well-founded.
1958:
1954:
1941:
69:
55:
38:
3081:, then the descending chains starting at
2643:
2639:
2627:
2589:
2457:
2433:
2384:
2335:
2024:{\displaystyle (\forall S\subseteq X)\;.}
2004:
1921:
1792:
1766:
1740:
1705:
1681:
1609:
1589:
1587:
1557:
1555:
1528:
1503:
1486:
1484:
1459:
1442:
1440:
1415:
1401:
1399:
1363:
1340:
1330:
1328:
1301:
1272:
1259:
1257:
1224:
1203:
1201:
1171:
1145:
5823:Positive cone of an ordered vector space
3366:Just, Winfried and Weese, Martin (1998)
3262:
2823:. See those articles for more details.
2749:is the graph of the successor function
1951:
1179:
52:
3327:(1st ed.). Elsevier. p. 46.
3219:. More generally, when working with a
3014:if and only if there is an edge from
2800:is well-founded is also known as the
7:
3273:Introduction to axiomatic set theory
2776:. If we consider the order relation
1673:All definitions tacitly require the
1188:{\displaystyle S\neq \varnothing :}
5350:Properties & Types (
2570:. Then there is a unique function
2516:a function that assigns an object
2442:
2362:
2340:
1995:
1980:
1962:
1926:
25:
5806:Positive cone of an ordered field
5660:Ordered topological vector space
5150:
3368:Discovering Modern Set Theory. I
3321:Fraisse, R. (15 December 2000).
3271:Zaring W.M., G. Takeuti (1971).
2246:is some property of elements of
2157:, which is one of the axioms of
2145:relation is well-founded on the
1658:
1645:
1574:{\displaystyle {\text{not }}aRa}
1079:
1069:
1039:
1029:
1007:
997:
962:
940:
930:
895:
863:
853:
828:
796:
781:
761:
729:
719:
709:
689:
657:
637:
627:
617:
585:
565:
521:
501:
491:
454:
429:
419:
387:
362:
325:
295:
263:
201:
166:
3073:is a well-founded relation and
2438:
2432:
2600:
2594:
2467:
2461:
2454:
2439:
2429:
2426:
2420:
2413:
2409:
2406:
2400:
2393:
2378:
2374:
2359:
2356:
2352:
2337:
2235:) is a well-founded relation,
2015:
2012:
1998:
1992:
1977:
1974:
1959:
1955:
1942:
1938:
1923:
1666:in the "Symmetric" column and
1602:
1347:
1292:
1221:
1:
5617:Series-parallel partial order
5111:History of mathematical logic
3372:American Mathematical Society
3041:) order, since the sequence
1667:
1654:
1064:
1059:
1054:
1049:
1024:
992:
987:
982:
977:
972:
957:
925:
920:
915:
910:
905:
890:
878:
873:
848:
843:
838:
823:
811:
806:
791:
776:
771:
756:
744:
739:
704:
699:
684:
672:
667:
652:
647:
612:
600:
595:
580:
575:
560:
555:
550:
536:
531:
516:
511:
486:
481:
469:
464:
449:
444:
439:
414:
402:
397:
382:
377:
372:
357:
352:
340:
335:
320:
315:
310:
305:
290:
278:
273:
258:
253:
248:
243:
238:
233:
216:
211:
196:
191:
186:
181:
176:
5296:Cantor's isomorphism theorem
5036:Primitive recursive function
2842:, with the order defined by
2201:is also said to satisfy the
32:Noetherian topological space
5336:Szpilrajn extension theorem
5311:Hausdorff maximal principle
5286:Boolean prime ideal theorem
2531:to each pair of an element
2250:, and we want to show that
2159:ZermeloâFraenkel set theory
2042:Equivalently, assuming the
1884:; that is, there exists an
5889:
5682:Topological vector lattice
4100:SchröderâBernstein theorem
3827:Monadic predicate calculus
3486:Foundations of mathematics
3393:Introduction to Set Theory
2811:, the technique is called
2790:course-of-values recursion
2512:well-founded relation and
2276:it suffices to show that:
2048:infinite descending chains
29:
5204:
5146:
5133:Philosophy of mathematics
5082:Automated theorem proving
4253:
4207:Von NeumannâBernaysâGödel
3848:
2895:is a proper substring of
2225:can be used on them: if (
2203:ascending chain condition
2099:for every natural number
2044:axiom of dependent choice
5291:CantorâBernstein theorem
3118:Mostowski collapse lemma
3048:The set of non-negative
2994:The nodes of any finite
5835:Partially ordered group
5655:Specialization preorder
4783:Self-verifying theories
4604:Tarski's axiomatization
3555:Tarski's undefinability
3550:incompleteness theorems
3269:See Definition 6.21 in
2802:well-ordering principle
2264:holds for all elements
2217:Induction and recursion
27:Type of binary relation
5321:Kruskal's tree theorem
5316:KnasterâTarski theorem
5306:DushnikâMiller theorem
5157:Mathematics portal
4768:Proof of impossibility
4416:propositional variable
3726:Propositional calculus
3030:The negative integers
2996:directed acyclic graph
2877:The set of all finite
2766:mathematical induction
2670:
2477:
2025:
1853:or, more generally, a
1810:
1781:
1755:
1729:
1728:{\displaystyle a,b,c,}
1690:
1631:
1575:
1543:
1516:
1472:
1428:
1387:
1316:
1245:
1189:
1160:
5026:Kolmogorov complexity
4979:Computably enumerable
4879:Model complete theory
4671:Principia Mathematica
3731:Propositional formula
3560:BanachâTarski paradox
2813:transfinite induction
2792:. The statement that
2671:
2494:transfinite recursion
2478:
2223:transfinite induction
2170:converse well-founded
2026:
1894:such that, for every
1811:
1782:
1756:
1730:
1691:
1632:
1576:
1544:
1517:
1473:
1429:
1388:
1317:
1246:
1190:
1161:
1140:Definitions, for all
5813:Ordered vector space
4974:ChurchâTuring thesis
4961:Computability theory
4170:continuum hypothesis
3688:Square of opposition
3546:Gödel's completeness
3354:Bourbaki, N. (1972)
2998:, with the relation
2817:structural induction
2760:. Then induction on
2697:using the values of
2588:
2574:such that for every
2334:
2205:. In the context of
2174:upwards well-founded
2122:then it is called a
1920:
1904:, one does not have
1809:{\displaystyle aRc.}
1791:
1765:
1739:
1704:
1680:
1675:homogeneous relation
1586:
1554:
1527:
1483:
1439:
1398:
1327:
1256:
1200:
1170:
1144:
886:Strict partial order
161:Equivalence relation
5651:Alexandrov topology
5597:Lexicographic order
5556:Well-quasi-ordering
5128:Mathematical object
5019:P versus NP problem
4984:Computable function
4778:Reverse mathematics
4704:Logical consequence
4581:primitive recursive
4576:elementary function
4349:Free/bound variable
4202:TarskiâGrothendieck
3721:Logical connectives
3651:Logical equivalence
3501:Logical consequence
2989:axiom of regularity
2774:primitive recursion
2768:, and recursion on
2193:is well-founded on
2155:axiom of regularity
1780:{\displaystyle bRc}
1754:{\displaystyle aRb}
1542:{\displaystyle aRa}
1159:{\displaystyle a,b}
545:Well-quasi-ordering
5632:Transitive closure
5592:Converse/Transpose
5301:Dilworth's theorem
4926:Transfer principle
4889:Semantics of logic
4874:Categorical theory
4850:Non-standard model
4364:Logical connective
3491:Information theory
3440:Mathematical logic
3002:defined such that
2786:complete induction
2741:is the set of all
2666:
2473:
2326:must also be true.
2147:transitive closure
2021:
1806:
1777:
1751:
1725:
1686:
1627:
1625:
1571:
1539:
1512:
1510:
1468:
1466:
1424:
1422:
1383:
1381:
1312:
1310:
1241:
1239:
1185:
1156:
1020:Strict total order
5860:
5859:
5818:Partially ordered
5627:Symmetric closure
5612:Reflexive closure
5355:
5164:
5163:
5096:Abstract category
4899:Theories of truth
4709:Rule of inference
4699:Natural deduction
4680:
4679:
4225:
4224:
3930:Cartesian product
3835:
3834:
3741:Many-valued logic
3716:Boolean functions
3599:Russell's paradox
3574:diagonal argument
3471:First-order logic
3358:, Addison-Wesley.
3144:is isomorphic to
3077:is an element of
3032:{â1, â2, â3, ...}
2436:
2284:is an element of
2186:converse relation
1822:
1821:
1689:{\displaystyle R}
1640:
1639:
1612:
1560:
1506:
1462:
1418:
1366:
1275:
953:Strict weak order
139:Total, Semiconnex
16:(Redirected from
5880:
5602:Linear extension
5351:
5331:Mirsky's theorem
5191:
5184:
5177:
5168:
5155:
5154:
5106:History of logic
5101:Category of sets
4994:Decision problem
4773:Ordinal analysis
4714:Sequent calculus
4612:Boolean algebras
4552:
4551:
4526:
4497:logical/constant
4251:
4237:
4160:ZermeloâFraenkel
3911:Set operations:
3846:
3783:
3614:
3594:LöwenheimâSkolem
3481:Formal semantics
3433:
3426:
3419:
3410:
3359:
3352:
3346:
3345:
3343:
3341:
3318:
3312:
3311:
3309:
3307:
3293:
3287:
3286:
3267:
3252:
3242:
3232:
3218:
3208:
3198:
3188:
3183:
3180:holds for every
3179:
3163:
3151:
3143:
3131:
3127:
3123:
3112:
3108:
3104:
3092:
3088:
3084:
3080:
3076:
3072:
3061:Other properties
3050:rational numbers
3044:
3033:
3021:
3017:
3013:
3001:
2983:
2967:
2951:
2911:
2898:
2894:
2890:
2873:
2863:
2857:
2851:
2841:
2799:
2783:
2771:
2763:
2759:
2748:
2740:
2734:
2719:
2707:
2696:
2686:, we may define
2685:
2681:
2675:
2673:
2672:
2667:
2662:
2658:
2657:
2656:
2655:
2651:
2647:
2583:
2573:
2569:
2565:
2544:
2540:
2530:
2515:
2507:
2482:
2480:
2479:
2474:
2437:
2434:
2388:
2325:
2314:
2302:
2299:is true for all
2298:
2287:
2283:
2271:
2267:
2263:
2249:
2245:
2234:
2200:
2196:
2192:
2183:
2167:
2152:
2139:well-founded set
2136:
2102:
2098:
2077:
2073:
2034:
2030:
2028:
2027:
2022:
2008:
1915:
1903:
1893:
1883:
1880:with respect to
1875:
1859:
1836:
1815:
1813:
1812:
1807:
1786:
1784:
1783:
1778:
1760:
1758:
1757:
1752:
1734:
1732:
1731:
1726:
1695:
1693:
1692:
1687:
1669:
1665:
1662:
1661:
1656:
1652:
1649:
1648:
1636:
1634:
1633:
1628:
1626:
1613:
1610:
1580:
1578:
1577:
1572:
1561:
1558:
1548:
1546:
1545:
1540:
1521:
1519:
1518:
1513:
1511:
1507:
1504:
1477:
1475:
1474:
1469:
1467:
1463:
1460:
1433:
1431:
1430:
1425:
1423:
1419:
1416:
1392:
1390:
1389:
1384:
1382:
1367:
1364:
1341:
1321:
1319:
1318:
1313:
1311:
1302:
1276:
1273:
1250:
1248:
1247:
1242:
1240:
1225:
1206:
1194:
1192:
1191:
1186:
1165:
1163:
1162:
1157:
1086:
1083:
1082:
1076:
1073:
1072:
1066:
1061:
1056:
1051:
1046:
1043:
1042:
1036:
1033:
1032:
1026:
1014:
1011:
1010:
1004:
1001:
1000:
994:
989:
984:
979:
974:
969:
966:
965:
959:
947:
944:
943:
937:
934:
933:
927:
922:
917:
912:
907:
902:
899:
898:
892:
880:
875:
870:
867:
866:
860:
857:
856:
850:
845:
840:
835:
832:
831:
825:
819:Meet-semilattice
813:
808:
803:
800:
799:
793:
788:
785:
784:
778:
773:
768:
765:
764:
758:
752:Join-semilattice
746:
741:
736:
733:
732:
726:
723:
722:
716:
713:
712:
706:
701:
696:
693:
692:
686:
674:
669:
664:
661:
660:
654:
649:
644:
641:
640:
634:
631:
630:
624:
621:
620:
614:
602:
597:
592:
589:
588:
582:
577:
572:
569:
568:
562:
557:
552:
547:
538:
533:
528:
525:
524:
518:
513:
508:
505:
504:
498:
495:
494:
488:
483:
471:
466:
461:
458:
457:
451:
446:
441:
436:
433:
432:
426:
423:
422:
416:
404:
399:
394:
391:
390:
384:
379:
374:
369:
366:
365:
359:
354:
342:
337:
332:
329:
328:
322:
317:
312:
307:
302:
299:
298:
292:
280:
275:
270:
267:
266:
260:
255:
250:
245:
240:
235:
230:
228:
218:
213:
208:
205:
204:
198:
193:
188:
183:
178:
173:
170:
169:
163:
81:
80:
71:
64:
57:
50:
48:binary relations
39:
21:
5888:
5887:
5883:
5882:
5881:
5879:
5878:
5877:
5873:Wellfoundedness
5863:
5862:
5861:
5856:
5852:Young's lattice
5708:
5636:
5575:
5425:Heyting algebra
5373:Boolean algebra
5345:
5326:Laver's theorem
5274:
5240:Boolean algebra
5235:Binary relation
5223:
5200:
5195:
5165:
5160:
5149:
5142:
5087:Category theory
5077:Algebraic logic
5060:
5031:Lambda calculus
4969:Church encoding
4955:
4931:Truth predicate
4787:
4753:Complete theory
4676:
4545:
4541:
4537:
4532:
4524:
4244: and
4240:
4235:
4221:
4197:New Foundations
4165:axiom of choice
4148:
4110:Gödel numbering
4050: and
4042:
3946:
3831:
3781:
3762:
3711:Boolean algebra
3697:
3661:Equiconsistency
3626:Classical logic
3603:
3584:Halting problem
3572: and
3548: and
3536: and
3535:
3530:Theorems (
3525:
3442:
3437:
3363:
3362:
3353:
3349:
3339:
3337:
3335:
3320:
3319:
3315:
3305:
3303:
3295:
3294:
3290:
3283:
3270:
3268:
3264:
3259:
3244:
3234:
3233:if and only if
3224:
3210:
3200:
3199:if and only if
3190:
3187:1 â„ 1 â„ 1 â„ ...
3186:
3181:
3169:
3161:
3158:
3145:
3133:
3129:
3125:
3121:
3110:
3106:
3094:
3090:
3086:
3082:
3078:
3074:
3066:
3063:
3042:
3031:
3019:
3015:
3003:
2999:
2982:
2975:
2969:
2966:
2959:
2953:
2952:if and only if
2949:
2942:
2935:
2928:
2921:
2918:natural numbers
2903:
2896:
2892:
2891:if and only if
2882:
2865:
2861:
2855:
2843:
2839:
2829:
2809:ordinal numbers
2793:
2777:
2769:
2761:
2750:
2746:
2743:natural numbers
2736:
2724:
2709:
2698:
2687:
2683:
2679:
2632:
2628:
2623:
2613:
2609:
2586:
2585:
2575:
2571:
2567:
2549:
2547:initial segment
2542:
2541:and a function
2532:
2517:
2513:
2497:
2332:
2331:
2316:
2304:
2300:
2289:
2285:
2281:
2269:
2265:
2254:
2247:
2236:
2226:
2219:
2198:
2197:. In this case
2194:
2188:
2181:
2165:
2150:
2134:
2100:
2097:
2088:
2079:
2075:
2074:of elements of
2071:
2064:
2057:
2051:
2032:
1918:
1917:
1905:
1895:
1885:
1881:
1878:minimal element
1867:
1857:
1834:
1832:binary relation
1824:
1823:
1816:
1789:
1788:
1763:
1762:
1737:
1736:
1702:
1701:
1678:
1677:
1671:
1663:
1659:
1650:
1646:
1624:
1623:
1606:
1605:
1584:
1583:
1552:
1551:
1525:
1524:
1509:
1508:
1500:
1499:
1481:
1480:
1465:
1464:
1456:
1455:
1437:
1436:
1421:
1420:
1412:
1411:
1396:
1395:
1380:
1379:
1368:
1351:
1350:
1342:
1325:
1324:
1309:
1308:
1303:
1289:
1288:
1277:
1274: and
1254:
1253:
1238:
1237:
1226:
1218:
1217:
1198:
1197:
1168:
1167:
1142:
1141:
1084:
1080:
1074:
1070:
1044:
1040:
1034:
1030:
1012:
1008:
1002:
998:
967:
963:
945:
941:
935:
931:
900:
896:
868:
864:
858:
854:
833:
829:
801:
797:
786:
782:
766:
762:
734:
730:
724:
720:
714:
710:
694:
690:
662:
658:
642:
638:
632:
628:
622:
618:
590:
586:
570:
566:
543:
526:
522:
506:
502:
496:
492:
477:Prewellordering
459:
455:
434:
430:
424:
420:
392:
388:
367:
363:
330:
326:
300:
296:
268:
264:
226:
223:
206:
202:
171:
167:
159:
151:
75:
42:
35:
28:
23:
22:
15:
12:
11:
5:
5886:
5884:
5876:
5875:
5865:
5864:
5858:
5857:
5855:
5854:
5849:
5844:
5843:
5842:
5832:
5831:
5830:
5825:
5820:
5810:
5809:
5808:
5798:
5793:
5792:
5791:
5786:
5779:Order morphism
5776:
5775:
5774:
5764:
5759:
5754:
5749:
5744:
5743:
5742:
5732:
5727:
5722:
5716:
5714:
5710:
5709:
5707:
5706:
5705:
5704:
5699:
5697:Locally convex
5694:
5689:
5679:
5677:Order topology
5674:
5673:
5672:
5670:Order topology
5667:
5657:
5647:
5645:
5638:
5637:
5635:
5634:
5629:
5624:
5619:
5614:
5609:
5604:
5599:
5594:
5589:
5583:
5581:
5577:
5576:
5574:
5573:
5563:
5553:
5548:
5543:
5538:
5533:
5528:
5523:
5518:
5517:
5516:
5506:
5501:
5500:
5499:
5494:
5489:
5484:
5482:Chain-complete
5474:
5469:
5468:
5467:
5462:
5457:
5452:
5447:
5437:
5432:
5427:
5422:
5417:
5407:
5402:
5397:
5392:
5387:
5382:
5381:
5380:
5370:
5365:
5359:
5357:
5347:
5346:
5344:
5343:
5338:
5333:
5328:
5323:
5318:
5313:
5308:
5303:
5298:
5293:
5288:
5282:
5280:
5276:
5275:
5273:
5272:
5267:
5262:
5257:
5252:
5247:
5242:
5237:
5231:
5229:
5225:
5224:
5222:
5221:
5216:
5211:
5205:
5202:
5201:
5196:
5194:
5193:
5186:
5179:
5171:
5162:
5161:
5147:
5144:
5143:
5141:
5140:
5135:
5130:
5125:
5120:
5119:
5118:
5108:
5103:
5098:
5089:
5084:
5079:
5074:
5072:Abstract logic
5068:
5066:
5062:
5061:
5059:
5058:
5053:
5051:Turing machine
5048:
5043:
5038:
5033:
5028:
5023:
5022:
5021:
5016:
5011:
5006:
5001:
4991:
4989:Computable set
4986:
4981:
4976:
4971:
4965:
4963:
4957:
4956:
4954:
4953:
4948:
4943:
4938:
4933:
4928:
4923:
4918:
4917:
4916:
4911:
4906:
4896:
4891:
4886:
4884:Satisfiability
4881:
4876:
4871:
4870:
4869:
4859:
4858:
4857:
4847:
4846:
4845:
4840:
4835:
4830:
4825:
4815:
4814:
4813:
4808:
4801:Interpretation
4797:
4795:
4789:
4788:
4786:
4785:
4780:
4775:
4770:
4765:
4755:
4750:
4749:
4748:
4747:
4746:
4736:
4731:
4721:
4716:
4711:
4706:
4701:
4696:
4690:
4688:
4682:
4681:
4678:
4677:
4675:
4674:
4666:
4665:
4664:
4663:
4658:
4657:
4656:
4651:
4646:
4626:
4625:
4624:
4622:minimal axioms
4619:
4608:
4607:
4606:
4595:
4594:
4593:
4588:
4583:
4578:
4573:
4568:
4555:
4553:
4534:
4533:
4531:
4530:
4529:
4528:
4516:
4511:
4510:
4509:
4504:
4499:
4494:
4484:
4479:
4474:
4469:
4468:
4467:
4462:
4452:
4451:
4450:
4445:
4440:
4435:
4425:
4420:
4419:
4418:
4413:
4408:
4398:
4397:
4396:
4391:
4386:
4381:
4376:
4371:
4361:
4356:
4351:
4346:
4345:
4344:
4339:
4334:
4329:
4319:
4314:
4312:Formation rule
4309:
4304:
4303:
4302:
4297:
4287:
4286:
4285:
4275:
4270:
4265:
4260:
4254:
4248:
4231:Formal systems
4227:
4226:
4223:
4222:
4220:
4219:
4214:
4209:
4204:
4199:
4194:
4189:
4184:
4179:
4174:
4173:
4172:
4167:
4156:
4154:
4150:
4149:
4147:
4146:
4145:
4144:
4134:
4129:
4128:
4127:
4120:Large cardinal
4117:
4112:
4107:
4102:
4097:
4083:
4082:
4081:
4076:
4071:
4056:
4054:
4044:
4043:
4041:
4040:
4039:
4038:
4033:
4028:
4018:
4013:
4008:
4003:
3998:
3993:
3988:
3983:
3978:
3973:
3968:
3963:
3957:
3955:
3948:
3947:
3945:
3944:
3943:
3942:
3937:
3932:
3927:
3922:
3917:
3909:
3908:
3907:
3902:
3892:
3887:
3885:Extensionality
3882:
3880:Ordinal number
3877:
3867:
3862:
3861:
3860:
3849:
3843:
3837:
3836:
3833:
3832:
3830:
3829:
3824:
3819:
3814:
3809:
3804:
3799:
3798:
3797:
3787:
3786:
3785:
3772:
3770:
3764:
3763:
3761:
3760:
3759:
3758:
3753:
3748:
3738:
3733:
3728:
3723:
3718:
3713:
3707:
3705:
3699:
3698:
3696:
3695:
3690:
3685:
3680:
3675:
3670:
3665:
3664:
3663:
3653:
3648:
3643:
3638:
3633:
3628:
3622:
3620:
3611:
3605:
3604:
3602:
3601:
3596:
3591:
3586:
3581:
3576:
3564:Cantor's
3562:
3557:
3552:
3542:
3540:
3527:
3526:
3524:
3523:
3518:
3513:
3508:
3503:
3498:
3493:
3488:
3483:
3478:
3473:
3468:
3463:
3462:
3461:
3450:
3448:
3444:
3443:
3438:
3436:
3435:
3428:
3421:
3413:
3407:
3406:
3382:
3361:
3360:
3347:
3333:
3313:
3288:
3281:
3261:
3260:
3258:
3255:
3164:is said to be
3157:
3154:
3103:â 2, ..., 2, 1
3062:
3059:
3058:
3057:
3046:
3035:
3024:
3023:
2992:
2985:
2980:
2973:
2964:
2957:
2947:
2940:
2933:
2926:
2900:
2875:
2853:if and only if
2840:{1, 2, 3, ...}
2828:
2825:
2665:
2661:
2654:
2650:
2646:
2642:
2638:
2635:
2631:
2626:
2622:
2619:
2616:
2612:
2608:
2605:
2602:
2599:
2596:
2593:
2472:
2469:
2466:
2463:
2460:
2456:
2453:
2450:
2447:
2444:
2441:
2431:
2428:
2425:
2422:
2419:
2415:
2411:
2408:
2405:
2402:
2399:
2395:
2391:
2387:
2383:
2380:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2354:
2351:
2348:
2345:
2342:
2339:
2328:
2327:
2274:
2273:
2218:
2215:
2143:set membership
2095:
2083:
2069:
2062:
2055:
2020:
2017:
2014:
2011:
2007:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1957:
1953:
1950:
1947:
1944:
1940:
1937:
1934:
1931:
1928:
1925:
1820:
1819:
1805:
1802:
1799:
1796:
1776:
1773:
1770:
1750:
1747:
1744:
1724:
1721:
1718:
1715:
1712:
1709:
1685:
1642:
1641:
1638:
1637:
1622:
1619:
1616:
1608:
1607:
1604:
1601:
1598:
1595:
1592:
1591:
1581:
1570:
1567:
1564:
1549:
1538:
1535:
1532:
1522:
1502:
1501:
1498:
1495:
1492:
1489:
1488:
1478:
1458:
1457:
1454:
1451:
1448:
1445:
1444:
1434:
1414:
1413:
1410:
1407:
1404:
1403:
1393:
1378:
1375:
1372:
1369:
1365: or
1362:
1359:
1356:
1353:
1352:
1349:
1346:
1343:
1339:
1336:
1333:
1332:
1322:
1307:
1304:
1300:
1297:
1294:
1291:
1290:
1287:
1284:
1281:
1278:
1271:
1268:
1265:
1262:
1261:
1251:
1236:
1233:
1230:
1227:
1223:
1220:
1219:
1216:
1213:
1210:
1207:
1205:
1195:
1184:
1181:
1178:
1175:
1155:
1152:
1149:
1137:
1136:
1131:
1126:
1121:
1116:
1111:
1106:
1101:
1096:
1091:
1088:
1087:
1077:
1067:
1062:
1057:
1052:
1047:
1037:
1027:
1022:
1016:
1015:
1005:
995:
990:
985:
980:
975:
970:
960:
955:
949:
948:
938:
928:
923:
918:
913:
908:
903:
893:
888:
882:
881:
876:
871:
861:
851:
846:
841:
836:
826:
821:
815:
814:
809:
804:
794:
789:
779:
774:
769:
759:
754:
748:
747:
742:
737:
727:
717:
707:
702:
697:
687:
682:
676:
675:
670:
665:
655:
650:
645:
635:
625:
615:
610:
604:
603:
598:
593:
583:
578:
573:
563:
558:
553:
548:
540:
539:
534:
529:
519:
514:
509:
499:
489:
484:
479:
473:
472:
467:
462:
452:
447:
442:
437:
427:
417:
412:
406:
405:
400:
395:
385:
380:
375:
370:
360:
355:
350:
348:Total preorder
344:
343:
338:
333:
323:
318:
313:
308:
303:
293:
288:
282:
281:
276:
271:
261:
256:
251:
246:
241:
236:
231:
220:
219:
214:
209:
199:
194:
189:
184:
179:
174:
164:
156:
155:
153:
148:
146:
144:
142:
140:
137:
135:
133:
130:
129:
124:
119:
114:
109:
104:
99:
94:
89:
84:
77:
76:
74:
73:
66:
59:
51:
37:
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5885:
5874:
5871:
5870:
5868:
5853:
5850:
5848:
5845:
5841:
5838:
5837:
5836:
5833:
5829:
5826:
5824:
5821:
5819:
5816:
5815:
5814:
5811:
5807:
5804:
5803:
5802:
5801:Ordered field
5799:
5797:
5794:
5790:
5787:
5785:
5782:
5781:
5780:
5777:
5773:
5770:
5769:
5768:
5765:
5763:
5760:
5758:
5757:Hasse diagram
5755:
5753:
5750:
5748:
5745:
5741:
5738:
5737:
5736:
5735:Comparability
5733:
5731:
5728:
5726:
5723:
5721:
5718:
5717:
5715:
5711:
5703:
5700:
5698:
5695:
5693:
5690:
5688:
5685:
5684:
5683:
5680:
5678:
5675:
5671:
5668:
5666:
5663:
5662:
5661:
5658:
5656:
5652:
5649:
5648:
5646:
5643:
5639:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5607:Product order
5605:
5603:
5600:
5598:
5595:
5593:
5590:
5588:
5585:
5584:
5582:
5580:Constructions
5578:
5572:
5568:
5564:
5561:
5557:
5554:
5552:
5549:
5547:
5544:
5542:
5539:
5537:
5534:
5532:
5529:
5527:
5524:
5522:
5519:
5515:
5512:
5511:
5510:
5507:
5505:
5502:
5498:
5495:
5493:
5490:
5488:
5485:
5483:
5480:
5479:
5478:
5477:Partial order
5475:
5473:
5470:
5466:
5465:Join and meet
5463:
5461:
5458:
5456:
5453:
5451:
5448:
5446:
5443:
5442:
5441:
5438:
5436:
5433:
5431:
5428:
5426:
5423:
5421:
5418:
5416:
5412:
5408:
5406:
5403:
5401:
5398:
5396:
5393:
5391:
5388:
5386:
5383:
5379:
5376:
5375:
5374:
5371:
5369:
5366:
5364:
5363:Antisymmetric
5361:
5360:
5358:
5354:
5348:
5342:
5339:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5317:
5314:
5312:
5309:
5307:
5304:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5283:
5281:
5277:
5271:
5270:Weak ordering
5268:
5266:
5263:
5261:
5258:
5256:
5255:Partial order
5253:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5233:
5232:
5230:
5226:
5220:
5217:
5215:
5212:
5210:
5207:
5206:
5203:
5199:
5192:
5187:
5185:
5180:
5178:
5173:
5172:
5169:
5159:
5158:
5153:
5145:
5139:
5136:
5134:
5131:
5129:
5126:
5124:
5121:
5117:
5114:
5113:
5112:
5109:
5107:
5104:
5102:
5099:
5097:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5070:
5069:
5067:
5063:
5057:
5054:
5052:
5049:
5047:
5046:Recursive set
5044:
5042:
5039:
5037:
5034:
5032:
5029:
5027:
5024:
5020:
5017:
5015:
5012:
5010:
5007:
5005:
5002:
5000:
4997:
4996:
4995:
4992:
4990:
4987:
4985:
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4966:
4964:
4962:
4958:
4952:
4949:
4947:
4944:
4942:
4939:
4937:
4934:
4932:
4929:
4927:
4924:
4922:
4919:
4915:
4912:
4910:
4907:
4905:
4902:
4901:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4872:
4868:
4865:
4864:
4863:
4860:
4856:
4855:of arithmetic
4853:
4852:
4851:
4848:
4844:
4841:
4839:
4836:
4834:
4831:
4829:
4826:
4824:
4821:
4820:
4819:
4816:
4812:
4809:
4807:
4804:
4803:
4802:
4799:
4798:
4796:
4794:
4790:
4784:
4781:
4779:
4776:
4774:
4771:
4769:
4766:
4763:
4762:from ZFC
4759:
4756:
4754:
4751:
4745:
4742:
4741:
4740:
4737:
4735:
4732:
4730:
4727:
4726:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4695:
4692:
4691:
4689:
4687:
4683:
4673:
4672:
4668:
4667:
4662:
4661:non-Euclidean
4659:
4655:
4652:
4650:
4647:
4645:
4644:
4640:
4639:
4637:
4634:
4633:
4631:
4627:
4623:
4620:
4618:
4615:
4614:
4613:
4609:
4605:
4602:
4601:
4600:
4596:
4592:
4589:
4587:
4584:
4582:
4579:
4577:
4574:
4572:
4569:
4567:
4564:
4563:
4561:
4557:
4556:
4554:
4549:
4543:
4538:Example
4535:
4527:
4522:
4521:
4520:
4517:
4515:
4512:
4508:
4505:
4503:
4500:
4498:
4495:
4493:
4490:
4489:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4466:
4463:
4461:
4458:
4457:
4456:
4453:
4449:
4446:
4444:
4441:
4439:
4436:
4434:
4431:
4430:
4429:
4426:
4424:
4421:
4417:
4414:
4412:
4409:
4407:
4404:
4403:
4402:
4399:
4395:
4392:
4390:
4387:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4366:
4365:
4362:
4360:
4357:
4355:
4352:
4350:
4347:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4324:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4301:
4298:
4296:
4295:by definition
4293:
4292:
4291:
4288:
4284:
4281:
4280:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4255:
4252:
4249:
4247:
4243:
4238:
4232:
4228:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4182:KripkeâPlatek
4180:
4178:
4175:
4171:
4168:
4166:
4163:
4162:
4161:
4158:
4157:
4155:
4151:
4143:
4140:
4139:
4138:
4135:
4133:
4130:
4126:
4123:
4122:
4121:
4118:
4116:
4113:
4111:
4108:
4106:
4103:
4101:
4098:
4095:
4091:
4087:
4084:
4080:
4077:
4075:
4072:
4070:
4067:
4066:
4065:
4061:
4058:
4057:
4055:
4053:
4049:
4045:
4037:
4034:
4032:
4029:
4027:
4026:constructible
4024:
4023:
4022:
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3987:
3984:
3982:
3979:
3977:
3974:
3972:
3969:
3967:
3964:
3962:
3959:
3958:
3956:
3954:
3949:
3941:
3938:
3936:
3933:
3931:
3928:
3926:
3923:
3921:
3918:
3916:
3913:
3912:
3910:
3906:
3903:
3901:
3898:
3897:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3872:
3868:
3866:
3863:
3859:
3856:
3855:
3854:
3851:
3850:
3847:
3844:
3842:
3838:
3828:
3825:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3805:
3803:
3800:
3796:
3793:
3792:
3791:
3788:
3784:
3779:
3778:
3777:
3774:
3773:
3771:
3769:
3765:
3757:
3754:
3752:
3749:
3747:
3744:
3743:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3709:
3708:
3706:
3704:
3703:Propositional
3700:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3662:
3659:
3658:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3631:Logical truth
3629:
3627:
3624:
3623:
3621:
3619:
3615:
3612:
3610:
3606:
3600:
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3571:
3567:
3563:
3561:
3558:
3556:
3553:
3551:
3547:
3544:
3543:
3541:
3539:
3533:
3528:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3464:
3460:
3457:
3456:
3455:
3452:
3451:
3449:
3445:
3441:
3434:
3429:
3427:
3422:
3420:
3415:
3414:
3411:
3405:
3404:0-8247-7915-0
3401:
3398:
3397:Marcel Dekker
3394:
3390:
3386:
3385:Karel HrbĂĄÄek
3383:
3380:
3379:0-8218-0266-6
3376:
3373:
3369:
3365:
3364:
3357:
3351:
3348:
3336:
3334:9780444505422
3330:
3326:
3325:
3317:
3314:
3302:
3298:
3292:
3289:
3284:
3278:
3274:
3266:
3263:
3256:
3254:
3251:
3247:
3241:
3237:
3231:
3227:
3222:
3217:
3213:
3207:
3203:
3197:
3193:
3178:
3175:
3172:
3167:
3155:
3153:
3149:
3141:
3137:
3119:
3114:
3102:
3098:
3070:
3060:
3055:
3051:
3047:
3040:
3039:lexicographic
3036:
3029:
3028:
3027:
3012:
3009:
3006:
2997:
2993:
2990:
2986:
2979:
2972:
2963:
2956:
2946:
2939:
2932:
2925:
2920:, ordered by
2919:
2915:
2910:
2906:
2901:
2889:
2885:
2880:
2876:
2872:
2868:
2860:
2854:
2850:
2846:
2838:
2835:The positive
2834:
2833:
2832:
2826:
2824:
2822:
2818:
2814:
2810:
2805:
2803:
2797:
2791:
2787:
2781:
2775:
2767:
2764:is the usual
2757:
2753:
2744:
2739:
2732:
2728:
2721:
2718:
2715:
2712:
2705:
2701:
2694:
2690:
2676:
2663:
2659:
2652:
2648:
2644:
2640:
2636:
2633:
2629:
2620:
2617:
2614:
2610:
2606:
2603:
2597:
2591:
2582:
2578:
2563:
2560:
2557:
2553:
2548:
2539:
2535:
2528:
2524:
2520:
2511:
2505:
2501:
2495:
2490:
2488:
2483:
2470:
2464:
2458:
2451:
2448:
2445:
2423:
2417:
2403:
2397:
2389:
2385:
2381:
2371:
2368:
2365:
2349:
2346:
2343:
2323:
2319:
2313:
2310:
2307:
2296:
2292:
2279:
2278:
2277:
2261:
2257:
2253:
2252:
2251:
2243:
2239:
2233:
2229:
2224:
2216:
2214:
2212:
2208:
2204:
2191:
2187:
2179:
2175:
2171:
2162:
2160:
2156:
2148:
2144:
2140:
2132:
2127:
2125:
2121:
2117:
2113:
2112:partial order
2109:
2104:
2094:
2091:
2086:
2082:
2068:
2061:
2054:
2049:
2045:
2040:
2038:
2018:
2009:
2005:
2001:
1989:
1986:
1983:
1971:
1968:
1965:
1948:
1945:
1935:
1932:
1929:
1914:
1911:
1908:
1902:
1898:
1892:
1888:
1879:
1874:
1870:
1866:
1863:
1856:
1852:
1848:
1844:
1840:
1833:
1829:
1818:
1803:
1800:
1797:
1794:
1774:
1771:
1768:
1748:
1745:
1742:
1722:
1719:
1716:
1713:
1710:
1707:
1699:
1683:
1676:
1644:
1643:
1620:
1617:
1614:
1599:
1596:
1593:
1582:
1568:
1565:
1562:
1550:
1536:
1533:
1530:
1523:
1496:
1493:
1490:
1479:
1452:
1449:
1446:
1435:
1408:
1394:
1376:
1373:
1370:
1360:
1357:
1354:
1344:
1337:
1334:
1323:
1305:
1298:
1295:
1285:
1282:
1279:
1269:
1266:
1263:
1252:
1234:
1231:
1228:
1214:
1211:
1208:
1196:
1182:
1176:
1173:
1153:
1150:
1147:
1139:
1138:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1099:Antisymmetric
1097:
1095:
1092:
1090:
1089:
1078:
1068:
1063:
1058:
1053:
1048:
1038:
1028:
1023:
1021:
1018:
1017:
1006:
996:
991:
986:
981:
976:
971:
961:
956:
954:
951:
950:
939:
929:
924:
919:
914:
909:
904:
894:
889:
887:
884:
883:
877:
872:
862:
852:
847:
842:
837:
827:
822:
820:
817:
816:
810:
805:
795:
790:
780:
775:
770:
760:
755:
753:
750:
749:
743:
738:
728:
718:
708:
703:
698:
688:
683:
681:
678:
677:
671:
666:
656:
651:
646:
636:
626:
616:
611:
609:
608:Well-ordering
606:
605:
599:
594:
584:
579:
574:
564:
559:
554:
549:
546:
542:
541:
535:
530:
520:
515:
510:
500:
490:
485:
480:
478:
475:
474:
468:
463:
453:
448:
443:
438:
428:
418:
413:
411:
408:
407:
401:
396:
386:
381:
376:
371:
361:
356:
351:
349:
346:
345:
339:
334:
324:
319:
314:
309:
304:
294:
289:
287:
286:Partial order
284:
283:
277:
272:
262:
257:
252:
247:
242:
237:
232:
229:
222:
221:
215:
210:
200:
195:
190:
185:
180:
175:
165:
162:
158:
157:
154:
149:
147:
145:
143:
141:
138:
136:
134:
132:
131:
128:
125:
123:
120:
118:
115:
113:
110:
108:
105:
103:
100:
98:
95:
93:
92:Antisymmetric
90:
88:
85:
83:
82:
79:
78:
72:
67:
65:
60:
58:
53:
49:
45:
41:
40:
33:
19:
5644:& Orders
5622:Star product
5551:Well-founded
5550:
5504:Prefix order
5460:Distributive
5450:Complemented
5420:Foundational
5385:Completeness
5341:Zorn's lemma
5245:Cyclic order
5228:Key concepts
5198:Order theory
5148:
4946:Ultraproduct
4793:Model theory
4758:Independence
4694:Formal proof
4686:Proof theory
4669:
4642:
4599:real numbers
4571:second-order
4482:Substitution
4359:Metalanguage
4300:conservative
4273:Axiom schema
4217:Constructive
4187:MorseâKelley
4153:Set theories
4132:Aleph number
4125:inaccessible
4031:Grothendieck
3915:intersection
3802:Higher-order
3790:Second-order
3736:Truth tables
3693:Venn diagram
3476:Formal proof
3392:
3367:
3355:
3350:
3338:. Retrieved
3323:
3316:
3304:. Retrieved
3300:
3291:
3272:
3265:
3249:
3245:
3239:
3235:
3229:
3225:
3215:
3211:
3205:
3201:
3195:
3191:
3176:
3173:
3170:
3159:
3147:
3139:
3135:
3115:
3100:
3096:
3068:
3064:
3025:
3010:
3007:
3004:
2977:
2970:
2961:
2954:
2944:
2937:
2930:
2923:
2908:
2904:
2887:
2883:
2870:
2866:
2848:
2844:
2830:
2806:
2795:
2784:, we obtain
2779:
2755:
2751:
2737:
2730:
2726:
2722:
2716:
2713:
2710:
2703:
2699:
2692:
2688:
2677:
2580:
2576:
2561:
2558:
2555:
2551:
2537:
2533:
2526:
2522:
2518:
2503:
2499:
2491:
2487:Emmy Noether
2484:
2329:
2321:
2317:
2311:
2308:
2305:
2294:
2290:
2275:
2259:
2255:
2241:
2237:
2231:
2227:
2220:
2210:
2189:
2177:
2173:
2169:
2163:
2138:
2137:is called a
2128:
2116:strict order
2108:order theory
2105:
2092:
2089:
2084:
2080:
2066:
2059:
2052:
2041:
1912:
1909:
1906:
1900:
1896:
1890:
1886:
1872:
1868:
1847:foundational
1846:
1842:
1839:well-founded
1838:
1825:
1672:
1109:Well-founded
1108:
227:(Quasiorder)
102:Well-founded
101:
18:Well-founded
5828:Riesz space
5789:Isomorphism
5665:Normal cone
5587:Composition
5521:Semilattice
5430:Homogeneous
5415:Equivalence
5265:Total order
5056:Type theory
5004:undecidable
4936:Truth value
4823:equivalence
4502:non-logical
4115:Enumeration
4105:Isomorphism
4052:cardinality
4036:Von Neumann
4001:Ultrafilter
3966:Uncountable
3900:equivalence
3817:Quantifiers
3807:Fixed-point
3776:First-order
3656:Consistency
3641:Proposition
3618:Traditional
3589:Lindström's
3579:Compactness
3521:Type theory
3466:Cardinality
3389:Thomas Jech
3340:20 February
3160:A relation
3156:Reflexivity
3124:on a class
3105:has length
2821:â-induction
2211:terminating
2164:A relation
2120:total order
1843:wellfounded
1828:mathematics
1129:Irreflexive
410:Total order
122:Irreflexive
5796:Order type
5730:Cofinality
5571:Well-order
5546:Transitive
5435:Idempotent
5368:Asymmetric
4867:elementary
4560:arithmetic
4428:Quantifier
4406:functional
4278:Expression
3996:Transitive
3940:identities
3925:complement
3858:hereditary
3841:Set theory
3282:0387900241
3257:References
3132:such that
2303:such that
2178:Noetherian
2131:set theory
2124:well-order
2078:such that
1837:is called
1700:: for all
1698:transitive
1134:Asymmetric
127:Asymmetric
44:Transitive
5847:Upper set
5784:Embedding
5720:Antichain
5541:Tolerance
5531:Symmetric
5526:Semiorder
5472:Reflexive
5390:Connected
5138:Supertask
5041:Recursion
4999:decidable
4833:saturated
4811:of models
4734:deductive
4729:axiomatic
4649:Hilbert's
4636:Euclidean
4617:canonical
4540:axiomatic
4472:Signature
4401:Predicate
4290:Extension
4212:Ackermann
4137:Operation
4016:Universal
4006:Recursive
3981:Singleton
3976:Inhabited
3961:Countable
3951:Types of
3935:power set
3905:partition
3822:Predicate
3768:Predicate
3683:Syllogism
3673:Soundness
3646:Inference
3636:Tautology
3538:paradoxes
3301:ProofWiki
3166:reflexive
2449:∈
2443:∀
2414:⟹
2394:⟹
2369:∈
2363:∀
2347:∈
2341:∀
2330:That is,
2207:rewriting
2184:, if the
1996:¬
1987:∈
1981:∀
1969:∈
1963:∃
1956:⟹
1952:∅
1949:≠
1933:⊆
1927:∀
1862:non-empty
1860:if every
1611:not
1603:⇒
1559:not
1494:∧
1450:∨
1348:⇒
1338:≠
1293:⇒
1222:⇒
1180:∅
1177:≠
1124:Reflexive
1119:Has meets
1114:Has joins
1104:Connected
1094:Symmetric
225:Preorder
152:reflexive
117:Reflexive
112:Has meets
107:Has joins
97:Connected
87:Symmetric
5867:Category
5642:Topology
5509:Preorder
5492:Eulerian
5455:Complete
5405:Directed
5395:Covering
5260:Preorder
5219:Category
5214:Glossary
5123:Logicism
5116:timeline
5092:Concrete
4951:Validity
4921:T-schema
4914:Kripke's
4909:Tarski's
4904:semantic
4894:Strength
4843:submodel
4838:spectrum
4806:function
4654:Tarski's
4643:Elements
4630:geometry
4586:Robinson
4507:variable
4492:function
4465:spectrum
4455:Sentence
4411:variable
4354:Language
4307:Relation
4268:Automata
4258:Alphabet
4242:language
4096:-jection
4074:codomain
4060:Function
4021:Universe
3991:Infinite
3895:Relation
3678:Validity
3668:Argument
3566:theorem,
3221:preorder
3109:for any
2936:) < (
2902:The set
2837:integers
2827:Examples
2735:, where
2510:set-like
2133:, a set
2037:set-like
1668:✗
1655:✗
1065:✗
1060:✗
1055:✗
1050:✗
1025:✗
993:✗
988:✗
983:✗
978:✗
973:✗
958:✗
926:✗
921:✗
916:✗
911:✗
906:✗
891:✗
879:✗
874:✗
849:✗
844:✗
839:✗
824:✗
812:✗
807:✗
792:✗
777:✗
772:✗
757:✗
745:✗
740:✗
705:✗
700:✗
685:✗
673:✗
668:✗
653:✗
648:✗
613:✗
601:✗
596:✗
581:✗
576:✗
561:✗
556:✗
551:✗
537:✗
532:✗
517:✗
512:✗
487:✗
482:✗
470:✗
465:✗
450:✗
445:✗
440:✗
415:✗
403:✗
398:✗
383:✗
378:✗
373:✗
358:✗
353:✗
341:✗
336:✗
321:✗
316:✗
311:✗
306:✗
291:✗
279:✗
274:✗
259:✗
254:✗
249:✗
244:✗
239:✗
234:✗
217:✗
212:✗
197:✗
192:✗
187:✗
182:✗
177:✗
5747:Duality
5725:Cofinal
5713:Related
5692:Fréchet
5569:)
5445:Bounded
5440:Lattice
5413:)
5411:Partial
5279:Results
5250:Lattice
5065:Related
4862:Diagram
4760: (
4739:Hilbert
4724:Systems
4719:Theorem
4597:of the
4542:systems
4322:Formula
4317:Grammar
4233: (
4177:General
3890:Forcing
3875:Element
3795:Monadic
3570:paradox
3511:Theorem
3447:General
3391:(1999)
3071:, <)
2879:strings
2859:divides
2798:, <)
2782:, <)
2545:on the
2435:implies
2315:, then
2141:if the
1849:) on a
680:Lattice
5772:Subnet
5752:Filter
5702:Normed
5687:Banach
5653:&
5560:Better
5497:Strict
5487:Graded
5378:topics
5209:Topics
4828:finite
4591:Skolem
4544:
4519:Theory
4487:Symbol
4477:String
4460:atomic
4337:ground
4332:closed
4327:atomic
4283:ground
4246:syntax
4142:binary
4069:domain
3986:Finite
3751:finite
3609:Logics
3568:
3516:Theory
3402:
3387:&
3377:
3331:
3306:10 May
3279:
2788:, and
2772:gives
2745:, and
2496:. Let
2153:. The
1876:has a
1865:subset
1505:exists
1461:exists
1417:exists
46:
5762:Ideal
5740:Graph
5536:Total
5514:Total
5400:Dense
4818:Model
4566:Peano
4423:Proof
4263:Arity
4192:Naive
4079:image
4011:Fuzzy
3971:Empty
3920:union
3865:Class
3506:Model
3496:Lemma
3454:Axiom
3228:<
3194:<
3099:â 1,
3054:reals
2976:<
2960:<
2914:pairs
2886:<
2847:<
2508:be a
2072:, ...
1855:class
1787:then
150:Anti-
5353:list
4941:Type
4744:list
4548:list
4525:list
4514:Term
4448:rank
4342:open
4236:list
4048:Maps
3953:sets
3812:Free
3782:list
3532:list
3459:list
3400:ISBN
3375:ISBN
3342:2019
3329:ISBN
3308:2021
3277:ISBN
3243:and
3209:and
3150:, â)
3116:The
3052:(or
2968:and
2864:and
2708:for
2288:and
2110:, a
1841:(or
1830:, a
1761:and
1166:and
5767:Net
5567:Pre
4628:of
4610:of
4558:of
4090:Sur
4064:Map
3871:Ur-
3853:Set
3248:â°
3168:if
3095:Ï,
3065:If
3018:to
2916:of
2912:of
2682:on
2566:of
2280:If
2268:of
2180:on
2176:or
2168:is
2149:of
2129:In
2106:In
2035:is
1851:set
1845:or
1826:In
1735:if
1696:be
1406:min
5869::
5014:NP
4638::
4632::
4562::
4239:),
4094:Bi
4086:In
3370:,
3299:.
3238:â€
3214:â
3204:â€
3152:.
3138:,
3113:.
2943:,
2929:,
2907:Ă
2869:â
2804:.
2758:+1
2754:âŠ
2729:,
2720:.
2584:,
2579:â
2554::
2536:â
2525:,
2502:,
2489:.
2230:,
2213:.
2172:,
2126:.
2103:.
2087:+1
2065:,
2058:,
1899:â
1889:â
1871:â
5565:(
5562:)
5558:(
5409:(
5356:)
5190:e
5183:t
5176:v
5094:/
5009:P
4764:)
4550:)
4546:(
4443:â
4438:!
4433:â
4394:=
4389:â
4384:â
4379:â§
4374:âš
4369:ÂŹ
4092:/
4088:/
4062:/
3873:)
3869:(
3756:â
3746:3
3534:)
3432:e
3425:t
3418:v
3381:.
3344:.
3310:.
3285:.
3250:a
3246:b
3240:b
3236:a
3230:b
3226:a
3216:b
3212:a
3206:b
3202:a
3196:b
3192:a
3182:a
3177:a
3174:R
3171:a
3162:R
3148:C
3146:(
3142:)
3140:R
3136:X
3134:(
3130:C
3126:X
3122:R
3111:n
3107:n
3101:n
3097:n
3091:X
3087:X
3083:x
3079:X
3075:x
3069:X
3067:(
3022:.
3020:b
3016:a
3011:b
3008:R
3005:a
3000:R
2991:.
2984:.
2981:2
2978:m
2974:2
2971:n
2965:1
2962:m
2958:1
2955:n
2950:)
2948:2
2945:m
2941:1
2938:m
2934:2
2931:n
2927:1
2924:n
2922:(
2909:N
2905:N
2899:.
2897:t
2893:s
2888:t
2884:s
2874:.
2871:b
2867:a
2862:b
2856:a
2849:b
2845:a
2796:N
2794:(
2780:N
2778:(
2770:S
2762:S
2756:x
2752:x
2747:S
2738:N
2733:)
2731:S
2727:N
2725:(
2717:x
2714:R
2711:y
2706:)
2704:y
2702:(
2700:G
2695:)
2693:x
2691:(
2689:G
2684:X
2680:G
2664:.
2660:)
2653:}
2649:x
2645:R
2641:y
2637::
2634:y
2630:{
2625:|
2621:G
2618:,
2615:x
2611:(
2607:F
2604:=
2601:)
2598:x
2595:(
2592:G
2581:X
2577:x
2572:G
2568:X
2564:}
2562:x
2559:R
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2552:y
2550:{
2543:g
2538:X
2534:x
2529:)
2527:g
2523:x
2521:(
2519:F
2514:F
2506:)
2504:R
2500:X
2498:(
2471:.
2468:)
2465:x
2462:(
2459:P
2455:)
2452:X
2446:x
2440:(
2430:]
2427:)
2424:x
2421:(
2418:P
2410:]
2407:)
2404:y
2401:(
2398:P
2390:x
2386:R
2382:y
2379:[
2375:)
2372:X
2366:y
2360:(
2357:[
2353:)
2350:X
2344:x
2338:(
2324:)
2322:x
2320:(
2318:P
2312:x
2309:R
2306:y
2301:y
2297:)
2295:y
2293:(
2291:P
2286:X
2282:x
2272:,
2270:X
2266:x
2262:)
2260:x
2258:(
2256:P
2248:X
2244:)
2242:x
2240:(
2238:P
2232:R
2228:X
2199:R
2195:X
2190:R
2182:X
2166:R
2151:x
2135:x
2101:n
2096:n
2093:x
2090:R
2085:n
2081:x
2076:X
2070:2
2067:x
2063:1
2060:x
2056:0
2053:x
2033:R
2019:.
2016:]
2013:)
2010:m
2006:R
2002:s
1999:(
1993:)
1990:S
1984:s
1978:(
1975:)
1972:S
1966:m
1960:(
1946:S
1943:[
1939:)
1936:X
1930:S
1924:(
1913:m
1910:R
1907:s
1901:S
1897:s
1891:S
1887:m
1882:R
1873:X
1869:S
1858:X
1835:R
1804:.
1801:c
1798:R
1795:a
1775:c
1772:R
1769:b
1749:b
1746:R
1743:a
1723:,
1720:c
1717:,
1714:b
1711:,
1708:a
1684:R
1664:Y
1651:Y
1621:a
1618:R
1615:b
1600:b
1597:R
1594:a
1569:a
1566:R
1563:a
1537:a
1534:R
1531:a
1497:b
1491:a
1453:b
1447:a
1409:S
1377:a
1374:R
1371:b
1361:b
1358:R
1355:a
1345:b
1335:a
1306:b
1299:=
1296:a
1286:a
1283:R
1280:b
1270:b
1267:R
1264:a
1235:a
1232:R
1229:b
1215:b
1212:R
1209:a
1183::
1174:S
1154:b
1151:,
1148:a
1085:Y
1075:Y
1045:Y
1035:Y
1013:Y
1003:Y
968:Y
946:Y
936:Y
901:Y
869:Y
859:Y
834:Y
802:Y
787:Y
767:Y
735:Y
725:Y
715:Y
695:Y
663:Y
643:Y
633:Y
623:Y
591:Y
571:Y
527:Y
507:Y
497:Y
460:Y
435:Y
425:Y
393:Y
368:Y
331:Y
301:Y
269:Y
207:Y
172:Y
70:e
63:t
56:v
34:.
20:)
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