5141:
1649:
1636:
1070:
1060:
1030:
1020:
998:
988:
953:
931:
921:
886:
854:
844:
819:
787:
772:
752:
720:
710:
700:
680:
648:
628:
618:
608:
576:
556:
512:
492:
482:
445:
420:
410:
378:
353:
316:
286:
254:
192:
157:
3242:. 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.
2470:
1380:
1309:
3173:
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
2663:
1624:
1238:
1509:
1465:
3178:. 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
1421:
1646:
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
1579:
1476:
1432:
1391:
1320:
1249:
1193:
2322:
2018:
1182:
1568:
2576:
1315:
1722:
1244:
1803:
1774:
1748:
1536:
1153:
1683:
3520:
1574:
1188:
3034:
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.
4195:
2796:
There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all
1471:
1427:
4278:
3419:
1386:
5828:
57:
5811:
5341:
4592:
3109:
implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation
5177:
1908:
4750:
3538:
5658:
4605:
3928:
3074:
are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let
4610:
4600:
4337:
4190:
3543:
3534:
5794:
5653:
4746:
3392:
3367:
3321:
3285:
4088:
3082:
is a well-founded set, but there are descending chains starting at Ï of arbitrary great (finite) length; the chain
5648:
4843:
4587:
3412:
5284:
4148:
3841:
2147:
3582:
5366:
5104:
4806:
4569:
4564:
4389:
3810:
3494:
3311:
3269:
3045:) under the standard ordering, since, for example, the subset of positive rationals (or reals) lacks a minimum.
50:
5685:
5605:
5099:
4882:
4799:
4512:
4443:
4320:
3562:
3360:
2762:
5470:
5399:
5279:
5024:
4850:
4536:
4170:
3769:
20:
5373:
5361:
5324:
5299:
5274:
5228:
5197:
4902:
4897:
4507:
4246:
4175:
3504:
3405:
1158:
5670:
5304:
5294:
5170:
4831:
4421:
3815:
3783:
3474:
2778:
2498:
1542:
5643:
5309:
5121:
5070:
4967:
4465:
4426:
3903:
3548:
2867:
2808:. When the well-founded relation is set membership on the universal class, the technique is known as
2804:. When the well-founded set is a set of recursively-defined data structures, the technique is called
2191:
2036:
2032:
43:
3577:
5861:
5575:
5202:
4962:
4892:
4431:
4283:
4266:
3989:
3469:
2465:{\displaystyle (\forall x\in X)\;\implies P(x)]\quad {\text{implies}}\quad (\forall x\in X)\,P(x).}
2135:
1117:
941:
110:
3078:
be the union of the positive integers with a new element Ï that is bigger than any integer. Then
1375:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
5823:
5806:
4794:
4771:
4732:
4618:
4559:
4205:
4125:
3969:
3913:
3526:
2790:
5735:
5351:
5084:
4811:
4789:
4756:
4649:
4495:
4480:
4453:
4404:
4288:
4223:
4048:
4014:
4009:
3883:
3714:
3691:
3027:
2984:
2754:
1304:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
1087:
80:
5713:
5548:
5408:
5289:
5243:
5207:
5163:
5014:
4867:
4659:
4377:
4113:
4019:
3878:
3863:
3744:
3719:
2801:
2482:
2211:
2131:
1642:
indicates that the column's property is always true the row's term (at the very left), while
1692:
5801:
5760:
5750:
5740:
5485:
5448:
5438:
5418:
5403:
4987:
4949:
4826:
4630:
4470:
4394:
4372:
4200:
4158:
4057:
4024:
3888:
3676:
3587:
2976:
Every class whose elements are sets, with the relation â ("is an element of"). This is the
2805:
1663:
874:
149:
1779:
8:
5728:
5639:
5585:
5544:
5534:
5423:
5356:
5319:
5116:
5007:
4992:
4972:
4929:
4816:
4766:
4692:
4637:
4574:
4367:
4362:
4310:
4078:
4067:
3739:
3639:
3567:
3558:
3554:
3489:
3484:
2977:
2143:
1806:
A term's definition may require additional properties that are not listed in this table.
1753:
1727:
1686:
1515:
1132:
1122:
533:
115:
32:
2210:
An important reason that well-founded relations are interesting is because a version of
5840:
5767:
5620:
5529:
5519:
5460:
5378:
5314:
5145:
4914:
4877:
4862:
4855:
4838:
4642:
4624:
4490:
4416:
4399:
4352:
4165:
4074:
3908:
3893:
3853:
3805:
3790:
3778:
3734:
3709:
3479:
3428:
3373:
3154:
3106:
3026:
The set of strings over a finite alphabet with more than one element, under the usual (
2774:
1843:
1668:
1112:
1092:
1082:
1008:
105:
85:
75:
5680:
4098:
2481:
On par with induction, well-founded relations also support construction of objects by
5777:
5755:
5615:
5600:
5580:
5383:
5140:
5080:
4887:
4697:
4687:
4579:
4460:
4295:
4271:
4052:
4036:
3941:
3918:
3795:
3764:
3729:
3624:
3459:
3388:
3363:
3317:
3265:
2902:
2174:
2025:
1839:
5590:
5443:
5094:
5089:
4982:
4939:
4761:
4722:
4717:
4702:
4528:
4485:
4382:
4180:
4130:
3704:
3666:
807:
740:
2658:{\displaystyle G(x)=F\left(x,G\vert _{\left\{y:\,y\mathrel {R} x\right\}}\right).}
5772:
5555:
5433:
5428:
5413:
5329:
5238:
5223:
5075:
5065:
5019:
5002:
4957:
4919:
4821:
4741:
4548:
4475:
4448:
4436:
4342:
4256:
4230:
4185:
4153:
3954:
3756:
3699:
3649:
3614:
3572:
3038:
2809:
2797:
2731:
2535:
1866:
1820:
668:
465:
36:
5690:
5675:
5665:
5524:
5502:
5480:
5060:
5039:
4997:
4977:
4872:
4727:
4325:
4315:
4305:
4300:
4234:
4108:
3984:
3873:
3868:
3846:
3447:
2906:
2841:
1619:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
336:
1233:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
5855:
5789:
5745:
5723:
5595:
5465:
5453:
5258:
5034:
4712:
4219:
4004:
3994:
3964:
3949:
3619:
3385:
2100:
1107:
1102:
274:
100:
95:
5610:
5492:
5475:
5393:
5233:
5186:
4934:
4781:
4682:
4674:
4554:
4502:
4411:
4347:
4330:
4261:
4120:
3979:
3681:
3464:
3042:
2475:
2104:
2096:
5816:
5509:
5388:
5253:
5044:
4924:
4103:
4093:
4040:
3724:
3644:
3629:
3509:
3454:
3377:
2108:
1816:
398:
3023:, with the usual order, since any unbounded subset has no least element.
1504:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
5784:
5718:
5559:
3974:
3829:
3800:
3606:
2474:
Well-founded induction is sometimes called
Noetherian induction, after
2119:
2112:
596:
1460:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
5835:
5708:
5514:
5126:
5029:
4082:
3999:
3959:
3923:
3859:
3671:
3661:
3634:
3397:
2195:
1850:
1416:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
19:"Noetherian induction" redirects here. For the use in topology, see
5630:
5497:
5248:
5111:
4909:
4357:
4062:
3656:
3209:
213:
2028:, i.e., that the elements less than any given element form a set.
4707:
3499:
2847:
2825:
5155:
1853:
3286:"Infinite Sequence Property of Strictly Well-Founded Relation"
4251:
3597:
3442:
2820:
Well-founded relations that are not totally ordered include:
3212:â€, it is common to use the relation < defined such that
3015:
Examples of relations that are not well-founded include:
2039:, which can be proved when there is no infinite sequence
3384:, 3rd edition, "Well-founded relations", pages 251â5,
2579:
2325:
1911:
1782:
1756:
1730:
1695:
1671:
1577:
1545:
1518:
1474:
1430:
1389:
1318:
1247:
1191:
1161:
1135:
2712:As an example, consider the well-founded relation
2657:
2464:
2012:
1797:
1768:
1742:
1716:
1677:
1618:
1562:
1530:
1503:
1459:
1415:
1374:
1303:
1232:
1176:
1147:
3264:(2nd, rev. ed.). New York: Springer-Verlag.
2870:over a fixed alphabet, with the order defined by
2035:, a relation is well-founded when it contains no
1905:. In other words, a relation is well founded if:
5853:
1394:
2198:systems, a Noetherian relation is also called
2107:is a well-founded relation. If the order is a
5171:
3413:
3313:Theory of Relations, Volume 145 - 1st Edition
3032:"B" > "AB" > "AAB" > "AAAB" > ...
2020:Some authors include an extra condition that
1659:in the "Antisymmetric" column, respectively.
51:
3345:Elements of mathematics. Commutative algebra
2667:That is, if we want to construct a function
2613:
2103:is called well-founded if the corresponding
3259:
5829:Positive cone of a partially ordered group
5178:
5164:
3605:
3420:
3406:
3117:that is extensional, there exists a class
2405:
2401:
2385:
2381:
2366:
2344:
2205:
2150:, asserts that all sets are well-founded.
1947:
1943:
1930:
58:
44:
3070:, then the descending chains starting at
2628:
2446:
2013:{\displaystyle (\forall S\subseteq X)\;.}
5812:Positive cone of an ordered vector space
3355:Just, Winfried and Weese, Martin (1998)
3309:
2812:. See those articles for more details.
2738:is the graph of the successor function
5854:
3427:
3316:(1st ed.). Elsevier. p. 46.
3208:. More generally, when working with a
5159:
3401:
3003:if and only if there is an edge from
2789:is well-founded is also known as the
3262:Introduction to axiomatic set theory
2765:. If we consider the order relation
1662:All definitions tacitly require the
3049:
1177:{\displaystyle S\neq \varnothing :}
13:
5339:Properties & Types (
2559:. Then there is a unique function
2505:a function that assigns an object
2431:
2351:
2329:
1984:
1969:
1951:
1915:
14:
5873:
5795:Positive cone of an ordered field
1940:
1168:
5649:Ordered topological vector space
5185:
5139:
3357:Discovering Modern Set Theory. I
3310:Fraisse, R. (15 December 2000).
3260:Zaring W.M., G. Takeuti (1971).
2235:is some property of elements of
2146:, which is one of the axioms of
2134:relation is well-founded on the
1647:
1634:
1563:{\displaystyle {\text{not }}aRa}
1068:
1058:
1028:
1018:
996:
986:
951:
929:
919:
884:
852:
842:
817:
785:
770:
750:
718:
708:
698:
678:
646:
626:
616:
606:
574:
554:
510:
490:
480:
443:
418:
408:
376:
351:
314:
284:
252:
190:
155:
3062:is a well-founded relation and
2427:
2421:
3337:
3303:
3278:
3252:
3144:
2589:
2583:
2456:
2450:
2443:
2428:
2418:
2415:
2409:
2402:
2398:
2395:
2389:
2382:
2367:
2363:
2348:
2345:
2341:
2326:
2224:) is a well-founded relation,
2004:
2001:
1987:
1981:
1966:
1963:
1948:
1944:
1931:
1927:
1912:
1655:in the "Symmetric" column and
1591:
1336:
1281:
1210:
1:
5606:Series-parallel partial order
5100:History of mathematical logic
3361:American Mathematical Society
3245:
3030:) order, since the sequence
1656:
1643:
1053:
1048:
1043:
1038:
1013:
981:
976:
971:
966:
961:
946:
914:
909:
904:
899:
894:
879:
867:
862:
837:
832:
827:
812:
800:
795:
780:
765:
760:
745:
733:
728:
693:
688:
673:
661:
656:
641:
636:
601:
589:
584:
569:
564:
549:
544:
539:
525:
520:
505:
500:
475:
470:
458:
453:
438:
433:
428:
403:
391:
386:
371:
366:
361:
346:
341:
329:
324:
309:
304:
299:
294:
279:
267:
262:
247:
242:
237:
232:
227:
222:
205:
200:
185:
180:
175:
170:
165:
5285:Cantor's isomorphism theorem
5025:Primitive recursive function
2831:, with the order defined by
2190:is also said to satisfy the
21:Noetherian topological space
7:
5325:Szpilrajn extension theorem
5300:Hausdorff maximal principle
5275:Boolean prime ideal theorem
2815:
2520:to each pair of an element
2239:, and we want to show that
2148:ZermeloâFraenkel set theory
2031:Equivalently, assuming the
1873:; that is, there exists an
10:
5878:
5671:Topological vector lattice
4089:SchröderâBernstein theorem
3816:Monadic predicate calculus
3475:Foundations of mathematics
3382:Introduction to Set Theory
2800:, the technique is called
2779:course-of-values recursion
2501:well-founded relation and
2265:it suffices to show that:
2037:infinite descending chains
18:
5701:
5629:
5568:
5338:
5267:
5216:
5193:
5135:
5122:Philosophy of mathematics
5071:Automated theorem proving
5053:
4948:
4780:
4673:
4525:
4242:
4218:
4196:Von NeumannâBernaysâGödel
4141:
4035:
3939:
3837:
3828:
3755:
3690:
3596:
3518:
3435:
2884:is a proper substring of
2214:can be used on them: if (
2192:ascending chain condition
2088:for every natural number
2033:axiom of dependent choice
5280:CantorâBernstein theorem
3107:Mostowski collapse lemma
3037:The set of non-negative
2983:The nodes of any finite
5824:Partially ordered group
5644:Specialization preorder
4772:Self-verifying theories
4593:Tarski's axiomatization
3544:Tarski's undefinability
3539:incompleteness theorems
3258:See Definition 6.21 in
2791:well-ordering principle
2253:holds for all elements
2206:Induction and recursion
16:Type of binary relation
5310:Kruskal's tree theorem
5305:KnasterâTarski theorem
5295:DushnikâMiller theorem
5146:Mathematics portal
4757:Proof of impossibility
4405:propositional variable
3715:Propositional calculus
3019:The negative integers
2985:directed acyclic graph
2866:The set of all finite
2755:mathematical induction
2659:
2466:
2014:
1842:or, more generally, a
1799:
1770:
1744:
1718:
1717:{\displaystyle a,b,c,}
1679:
1620:
1564:
1532:
1505:
1461:
1417:
1376:
1305:
1234:
1178:
1149:
5015:Kolmogorov complexity
4968:Computably enumerable
4868:Model complete theory
4660:Principia Mathematica
3720:Propositional formula
3549:BanachâTarski paradox
2802:transfinite induction
2781:. The statement that
2660:
2483:transfinite recursion
2467:
2212:transfinite induction
2159:converse well-founded
2015:
1883:such that, for every
1800:
1771:
1745:
1719:
1680:
1621:
1565:
1533:
1506:
1462:
1418:
1377:
1306:
1235:
1179:
1150:
1129:Definitions, for all
5802:Ordered vector space
4963:ChurchâTuring thesis
4950:Computability theory
4159:continuum hypothesis
3677:Square of opposition
3535:Gödel's completeness
3343:Bourbaki, N. (1972)
2987:, with the relation
2806:structural induction
2749:. Then induction on
2686:using the values of
2577:
2563:such that for every
2323:
2194:. In the context of
2163:upwards well-founded
2111:then it is called a
1909:
1893:, one does not have
1798:{\displaystyle aRc.}
1780:
1754:
1728:
1693:
1669:
1664:homogeneous relation
1575:
1543:
1516:
1472:
1428:
1387:
1316:
1245:
1189:
1159:
1133:
875:Strict partial order
150:Equivalence relation
5640:Alexandrov topology
5586:Lexicographic order
5545:Well-quasi-ordering
5117:Mathematical object
5008:P versus NP problem
4973:Computable function
4767:Reverse mathematics
4693:Logical consequence
4570:primitive recursive
4565:elementary function
4338:Free/bound variable
4191:TarskiâGrothendieck
3710:Logical connectives
3640:Logical equivalence
3490:Logical consequence
2978:axiom of regularity
2763:primitive recursion
2757:, and recursion on
2182:is well-founded on
2144:axiom of regularity
1769:{\displaystyle bRc}
1743:{\displaystyle aRb}
1531:{\displaystyle aRa}
1148:{\displaystyle a,b}
534:Well-quasi-ordering
5621:Transitive closure
5581:Converse/Transpose
5290:Dilworth's theorem
4915:Transfer principle
4878:Semantics of logic
4863:Categorical theory
4839:Non-standard model
4353:Logical connective
3480:Information theory
3429:Mathematical logic
2991:defined such that
2775:complete induction
2730:is the set of all
2655:
2462:
2315:must also be true.
2136:transitive closure
2010:
1795:
1766:
1740:
1714:
1675:
1616:
1614:
1560:
1528:
1501:
1499:
1457:
1455:
1413:
1411:
1372:
1370:
1301:
1299:
1230:
1228:
1174:
1145:
1009:Strict total order
5849:
5848:
5807:Partially ordered
5616:Symmetric closure
5601:Reflexive closure
5344:
5153:
5152:
5085:Abstract category
4888:Theories of truth
4698:Rule of inference
4688:Natural deduction
4669:
4668:
4214:
4213:
3919:Cartesian product
3824:
3823:
3730:Many-valued logic
3705:Boolean functions
3588:Russell's paradox
3563:diagonal argument
3460:First-order logic
3347:, Addison-Wesley.
3133:is isomorphic to
3066:is an element of
3021:{â1, â2, â3, ...}
2425:
2273:is an element of
2175:converse relation
1811:
1810:
1678:{\displaystyle R}
1629:
1628:
1601:
1549:
1495:
1451:
1407:
1355:
1264:
942:Strict weak order
128:Total, Semiconnex
5869:
5591:Linear extension
5340:
5320:Mirsky's theorem
5180:
5173:
5166:
5157:
5156:
5144:
5143:
5095:History of logic
5090:Category of sets
4983:Decision problem
4762:Ordinal analysis
4703:Sequent calculus
4601:Boolean algebras
4541:
4540:
4515:
4486:logical/constant
4240:
4239:
4226:
4149:ZermeloâFraenkel
3900:Set operations:
3835:
3834:
3772:
3603:
3602:
3583:LöwenheimâSkolem
3470:Formal semantics
3422:
3415:
3408:
3399:
3398:
3348:
3341:
3335:
3334:
3332:
3330:
3307:
3301:
3300:
3298:
3296:
3282:
3276:
3275:
3256:
3241:
3231:
3221:
3207:
3197:
3187:
3177:
3172:
3169:holds for every
3168:
3152:
3140:
3132:
3120:
3116:
3112:
3101:
3097:
3093:
3081:
3077:
3073:
3069:
3065:
3061:
3050:Other properties
3039:rational numbers
3033:
3022:
3010:
3006:
3002:
2990:
2972:
2956:
2940:
2900:
2887:
2883:
2879:
2862:
2852:
2846:
2840:
2830:
2788:
2772:
2760:
2752:
2748:
2737:
2729:
2723:
2708:
2696:
2685:
2675:, we may define
2674:
2670:
2664:
2662:
2661:
2656:
2651:
2647:
2646:
2645:
2644:
2640:
2636:
2572:
2562:
2558:
2554:
2533:
2529:
2519:
2504:
2496:
2471:
2469:
2468:
2463:
2426:
2423:
2377:
2314:
2303:
2291:
2288:is true for all
2287:
2276:
2272:
2260:
2256:
2252:
2238:
2234:
2223:
2189:
2185:
2181:
2172:
2156:
2141:
2128:well-founded set
2125:
2091:
2087:
2066:
2062:
2023:
2019:
2017:
2016:
2011:
1997:
1904:
1892:
1882:
1872:
1869:with respect to
1864:
1848:
1825:
1804:
1802:
1801:
1796:
1775:
1773:
1772:
1767:
1749:
1747:
1746:
1741:
1723:
1721:
1720:
1715:
1684:
1682:
1681:
1676:
1658:
1654:
1651:
1650:
1645:
1641:
1638:
1637:
1625:
1623:
1622:
1617:
1615:
1602:
1599:
1569:
1567:
1566:
1561:
1550:
1547:
1537:
1535:
1534:
1529:
1510:
1508:
1507:
1502:
1500:
1496:
1493:
1466:
1464:
1463:
1458:
1456:
1452:
1449:
1422:
1420:
1419:
1414:
1412:
1408:
1405:
1381:
1379:
1378:
1373:
1371:
1356:
1353:
1330:
1310:
1308:
1307:
1302:
1300:
1291:
1265:
1262:
1239:
1237:
1236:
1231:
1229:
1214:
1195:
1183:
1181:
1180:
1175:
1154:
1152:
1151:
1146:
1075:
1072:
1071:
1065:
1062:
1061:
1055:
1050:
1045:
1040:
1035:
1032:
1031:
1025:
1022:
1021:
1015:
1003:
1000:
999:
993:
990:
989:
983:
978:
973:
968:
963:
958:
955:
954:
948:
936:
933:
932:
926:
923:
922:
916:
911:
906:
901:
896:
891:
888:
887:
881:
869:
864:
859:
856:
855:
849:
846:
845:
839:
834:
829:
824:
821:
820:
814:
808:Meet-semilattice
802:
797:
792:
789:
788:
782:
777:
774:
773:
767:
762:
757:
754:
753:
747:
741:Join-semilattice
735:
730:
725:
722:
721:
715:
712:
711:
705:
702:
701:
695:
690:
685:
682:
681:
675:
663:
658:
653:
650:
649:
643:
638:
633:
630:
629:
623:
620:
619:
613:
610:
609:
603:
591:
586:
581:
578:
577:
571:
566:
561:
558:
557:
551:
546:
541:
536:
527:
522:
517:
514:
513:
507:
502:
497:
494:
493:
487:
484:
483:
477:
472:
460:
455:
450:
447:
446:
440:
435:
430:
425:
422:
421:
415:
412:
411:
405:
393:
388:
383:
380:
379:
373:
368:
363:
358:
355:
354:
348:
343:
331:
326:
321:
318:
317:
311:
306:
301:
296:
291:
288:
287:
281:
269:
264:
259:
256:
255:
249:
244:
239:
234:
229:
224:
219:
217:
207:
202:
197:
194:
193:
187:
182:
177:
172:
167:
162:
159:
158:
152:
70:
69:
60:
53:
46:
39:
37:binary relations
28:
27:
5877:
5876:
5872:
5871:
5870:
5868:
5867:
5866:
5862:Wellfoundedness
5852:
5851:
5850:
5845:
5841:Young's lattice
5697:
5625:
5564:
5414:Heyting algebra
5362:Boolean algebra
5334:
5315:Laver's theorem
5263:
5229:Boolean algebra
5224:Binary relation
5212:
5189:
5184:
5154:
5149:
5138:
5131:
5076:Category theory
5066:Algebraic logic
5049:
5020:Lambda calculus
4958:Church encoding
4944:
4920:Truth predicate
4776:
4742:Complete theory
4665:
4534:
4530:
4526:
4521:
4513:
4233: and
4229:
4224:
4210:
4186:New Foundations
4154:axiom of choice
4137:
4099:Gödel numbering
4039: and
4031:
3935:
3820:
3770:
3751:
3700:Boolean algebra
3686:
3650:Equiconsistency
3615:Classical logic
3592:
3573:Halting problem
3561: and
3537: and
3525: and
3524:
3519:Theorems (
3514:
3431:
3426:
3352:
3351:
3342:
3338:
3328:
3326:
3324:
3308:
3304:
3294:
3292:
3284:
3283:
3279:
3272:
3257:
3253:
3248:
3233:
3223:
3222:if and only if
3213:
3199:
3189:
3188:if and only if
3179:
3176:1 â„ 1 â„ 1 â„ ...
3175:
3170:
3158:
3150:
3147:
3134:
3122:
3118:
3114:
3110:
3099:
3095:
3083:
3079:
3075:
3071:
3067:
3063:
3055:
3052:
3031:
3020:
3008:
3004:
2992:
2988:
2971:
2964:
2958:
2955:
2948:
2942:
2941:if and only if
2938:
2931:
2924:
2917:
2910:
2907:natural numbers
2892:
2885:
2881:
2880:if and only if
2871:
2854:
2850:
2844:
2832:
2828:
2818:
2798:ordinal numbers
2782:
2766:
2758:
2750:
2739:
2735:
2732:natural numbers
2725:
2713:
2698:
2687:
2676:
2672:
2668:
2632:
2621:
2617:
2616:
2612:
2602:
2598:
2578:
2575:
2574:
2564:
2560:
2556:
2538:
2536:initial segment
2531:
2530:and a function
2521:
2506:
2502:
2486:
2422:
2373:
2324:
2321:
2320:
2305:
2293:
2289:
2278:
2274:
2270:
2258:
2254:
2243:
2236:
2225:
2215:
2208:
2187:
2186:. In this case
2183:
2177:
2170:
2154:
2139:
2123:
2089:
2086:
2077:
2068:
2064:
2063:of elements of
2060:
2053:
2046:
2040:
2021:
1993:
1910:
1907:
1906:
1894:
1884:
1874:
1870:
1867:minimal element
1856:
1846:
1823:
1821:binary relation
1813:
1812:
1805:
1781:
1778:
1777:
1755:
1752:
1751:
1729:
1726:
1725:
1694:
1691:
1690:
1670:
1667:
1666:
1660:
1652:
1648:
1639:
1635:
1613:
1612:
1598:
1595:
1594:
1578:
1576:
1573:
1572:
1546:
1544:
1541:
1540:
1517:
1514:
1513:
1498:
1497:
1492:
1489:
1488:
1475:
1473:
1470:
1469:
1454:
1453:
1448:
1445:
1444:
1431:
1429:
1426:
1425:
1410:
1409:
1404:
1401:
1400:
1390:
1388:
1385:
1384:
1369:
1368:
1357:
1352:
1340:
1339:
1331:
1329:
1319:
1317:
1314:
1313:
1298:
1297:
1292:
1290:
1278:
1277:
1266:
1263: and
1261:
1248:
1246:
1243:
1242:
1227:
1226:
1215:
1213:
1207:
1206:
1192:
1190:
1187:
1186:
1160:
1157:
1156:
1134:
1131:
1130:
1073:
1069:
1063:
1059:
1033:
1029:
1023:
1019:
1001:
997:
991:
987:
956:
952:
934:
930:
924:
920:
889:
885:
857:
853:
847:
843:
822:
818:
790:
786:
775:
771:
755:
751:
723:
719:
713:
709:
703:
699:
683:
679:
651:
647:
631:
627:
621:
617:
611:
607:
579:
575:
559:
555:
532:
515:
511:
495:
491:
485:
481:
466:Prewellordering
448:
444:
423:
419:
413:
409:
381:
377:
356:
352:
319:
315:
289:
285:
257:
253:
215:
212:
195:
191:
160:
156:
148:
140:
64:
31:
24:
17:
12:
11:
5:
5875:
5865:
5864:
5847:
5846:
5844:
5843:
5838:
5833:
5832:
5831:
5821:
5820:
5819:
5814:
5809:
5799:
5798:
5797:
5787:
5782:
5781:
5780:
5775:
5768:Order morphism
5765:
5764:
5763:
5753:
5748:
5743:
5738:
5733:
5732:
5731:
5721:
5716:
5711:
5705:
5703:
5699:
5698:
5696:
5695:
5694:
5693:
5688:
5686:Locally convex
5683:
5678:
5668:
5666:Order topology
5663:
5662:
5661:
5659:Order topology
5656:
5646:
5636:
5634:
5627:
5626:
5624:
5623:
5618:
5613:
5608:
5603:
5598:
5593:
5588:
5583:
5578:
5572:
5570:
5566:
5565:
5563:
5562:
5552:
5542:
5537:
5532:
5527:
5522:
5517:
5512:
5507:
5506:
5505:
5495:
5490:
5489:
5488:
5483:
5478:
5473:
5471:Chain-complete
5463:
5458:
5457:
5456:
5451:
5446:
5441:
5436:
5426:
5421:
5416:
5411:
5406:
5396:
5391:
5386:
5381:
5376:
5371:
5370:
5369:
5359:
5354:
5348:
5346:
5336:
5335:
5333:
5332:
5327:
5322:
5317:
5312:
5307:
5302:
5297:
5292:
5287:
5282:
5277:
5271:
5269:
5265:
5264:
5262:
5261:
5256:
5251:
5246:
5241:
5236:
5231:
5226:
5220:
5218:
5214:
5213:
5211:
5210:
5205:
5200:
5194:
5191:
5190:
5183:
5182:
5175:
5168:
5160:
5151:
5150:
5136:
5133:
5132:
5130:
5129:
5124:
5119:
5114:
5109:
5108:
5107:
5097:
5092:
5087:
5078:
5073:
5068:
5063:
5061:Abstract logic
5057:
5055:
5051:
5050:
5048:
5047:
5042:
5040:Turing machine
5037:
5032:
5027:
5022:
5017:
5012:
5011:
5010:
5005:
5000:
4995:
4990:
4980:
4978:Computable set
4975:
4970:
4965:
4960:
4954:
4952:
4946:
4945:
4943:
4942:
4937:
4932:
4927:
4922:
4917:
4912:
4907:
4906:
4905:
4900:
4895:
4885:
4880:
4875:
4873:Satisfiability
4870:
4865:
4860:
4859:
4858:
4848:
4847:
4846:
4836:
4835:
4834:
4829:
4824:
4819:
4814:
4804:
4803:
4802:
4797:
4790:Interpretation
4786:
4784:
4778:
4777:
4775:
4774:
4769:
4764:
4759:
4754:
4744:
4739:
4738:
4737:
4736:
4735:
4725:
4720:
4710:
4705:
4700:
4695:
4690:
4685:
4679:
4677:
4671:
4670:
4667:
4666:
4664:
4663:
4655:
4654:
4653:
4652:
4647:
4646:
4645:
4640:
4635:
4615:
4614:
4613:
4611:minimal axioms
4608:
4597:
4596:
4595:
4584:
4583:
4582:
4577:
4572:
4567:
4562:
4557:
4544:
4542:
4523:
4522:
4520:
4519:
4518:
4517:
4505:
4500:
4499:
4498:
4493:
4488:
4483:
4473:
4468:
4463:
4458:
4457:
4456:
4451:
4441:
4440:
4439:
4434:
4429:
4424:
4414:
4409:
4408:
4407:
4402:
4397:
4387:
4386:
4385:
4380:
4375:
4370:
4365:
4360:
4350:
4345:
4340:
4335:
4334:
4333:
4328:
4323:
4318:
4308:
4303:
4301:Formation rule
4298:
4293:
4292:
4291:
4286:
4276:
4275:
4274:
4264:
4259:
4254:
4249:
4243:
4237:
4220:Formal systems
4216:
4215:
4212:
4211:
4209:
4208:
4203:
4198:
4193:
4188:
4183:
4178:
4173:
4168:
4163:
4162:
4161:
4156:
4145:
4143:
4139:
4138:
4136:
4135:
4134:
4133:
4123:
4118:
4117:
4116:
4109:Large cardinal
4106:
4101:
4096:
4091:
4086:
4072:
4071:
4070:
4065:
4060:
4045:
4043:
4033:
4032:
4030:
4029:
4028:
4027:
4022:
4017:
4007:
4002:
3997:
3992:
3987:
3982:
3977:
3972:
3967:
3962:
3957:
3952:
3946:
3944:
3937:
3936:
3934:
3933:
3932:
3931:
3926:
3921:
3916:
3911:
3906:
3898:
3897:
3896:
3891:
3881:
3876:
3874:Extensionality
3871:
3869:Ordinal number
3866:
3856:
3851:
3850:
3849:
3838:
3832:
3826:
3825:
3822:
3821:
3819:
3818:
3813:
3808:
3803:
3798:
3793:
3788:
3787:
3786:
3776:
3775:
3774:
3761:
3759:
3753:
3752:
3750:
3749:
3748:
3747:
3742:
3737:
3727:
3722:
3717:
3712:
3707:
3702:
3696:
3694:
3688:
3687:
3685:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3653:
3652:
3642:
3637:
3632:
3627:
3622:
3617:
3611:
3609:
3600:
3594:
3593:
3591:
3590:
3585:
3580:
3575:
3570:
3565:
3553:Cantor's
3551:
3546:
3541:
3531:
3529:
3516:
3515:
3513:
3512:
3507:
3502:
3497:
3492:
3487:
3482:
3477:
3472:
3467:
3462:
3457:
3452:
3451:
3450:
3439:
3437:
3433:
3432:
3425:
3424:
3417:
3410:
3402:
3396:
3395:
3371:
3350:
3349:
3336:
3322:
3302:
3277:
3270:
3250:
3249:
3247:
3244:
3153:is said to be
3146:
3143:
3092:â 2, ..., 2, 1
3051:
3048:
3047:
3046:
3035:
3024:
3013:
3012:
2981:
2974:
2969:
2962:
2953:
2946:
2936:
2929:
2922:
2915:
2889:
2864:
2842:if and only if
2829:{1, 2, 3, ...}
2817:
2814:
2654:
2650:
2643:
2639:
2635:
2631:
2627:
2624:
2620:
2615:
2611:
2608:
2605:
2601:
2597:
2594:
2591:
2588:
2585:
2582:
2461:
2458:
2455:
2452:
2449:
2445:
2442:
2439:
2436:
2433:
2430:
2420:
2417:
2414:
2411:
2408:
2404:
2400:
2397:
2394:
2391:
2388:
2384:
2380:
2376:
2372:
2369:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2343:
2340:
2337:
2334:
2331:
2328:
2317:
2316:
2263:
2262:
2207:
2204:
2132:set membership
2084:
2072:
2058:
2051:
2044:
2009:
2006:
2003:
2000:
1996:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1946:
1942:
1939:
1936:
1933:
1929:
1926:
1923:
1920:
1917:
1914:
1809:
1808:
1794:
1791:
1788:
1785:
1765:
1762:
1759:
1739:
1736:
1733:
1713:
1710:
1707:
1704:
1701:
1698:
1674:
1631:
1630:
1627:
1626:
1611:
1608:
1605:
1597:
1596:
1593:
1590:
1587:
1584:
1581:
1580:
1570:
1559:
1556:
1553:
1538:
1527:
1524:
1521:
1511:
1491:
1490:
1487:
1484:
1481:
1478:
1477:
1467:
1447:
1446:
1443:
1440:
1437:
1434:
1433:
1423:
1403:
1402:
1399:
1396:
1393:
1392:
1382:
1367:
1364:
1361:
1358:
1354: or
1351:
1348:
1345:
1342:
1341:
1338:
1335:
1332:
1328:
1325:
1322:
1321:
1311:
1296:
1293:
1289:
1286:
1283:
1280:
1279:
1276:
1273:
1270:
1267:
1260:
1257:
1254:
1251:
1250:
1240:
1225:
1222:
1219:
1216:
1212:
1209:
1208:
1205:
1202:
1199:
1196:
1194:
1184:
1173:
1170:
1167:
1164:
1144:
1141:
1138:
1126:
1125:
1120:
1115:
1110:
1105:
1100:
1095:
1090:
1085:
1080:
1077:
1076:
1066:
1056:
1051:
1046:
1041:
1036:
1026:
1016:
1011:
1005:
1004:
994:
984:
979:
974:
969:
964:
959:
949:
944:
938:
937:
927:
917:
912:
907:
902:
897:
892:
882:
877:
871:
870:
865:
860:
850:
840:
835:
830:
825:
815:
810:
804:
803:
798:
793:
783:
778:
768:
763:
758:
748:
743:
737:
736:
731:
726:
716:
706:
696:
691:
686:
676:
671:
665:
664:
659:
654:
644:
639:
634:
624:
614:
604:
599:
593:
592:
587:
582:
572:
567:
562:
552:
547:
542:
537:
529:
528:
523:
518:
508:
503:
498:
488:
478:
473:
468:
462:
461:
456:
451:
441:
436:
431:
426:
416:
406:
401:
395:
394:
389:
384:
374:
369:
364:
359:
349:
344:
339:
337:Total preorder
333:
332:
327:
322:
312:
307:
302:
297:
292:
282:
277:
271:
270:
265:
260:
250:
245:
240:
235:
230:
225:
220:
209:
208:
203:
198:
188:
183:
178:
173:
168:
163:
153:
145:
144:
142:
137:
135:
133:
131:
129:
126:
124:
122:
119:
118:
113:
108:
103:
98:
93:
88:
83:
78:
73:
66:
65:
63:
62:
55:
48:
40:
26:
25:
15:
9:
6:
4:
3:
2:
5874:
5863:
5860:
5859:
5857:
5842:
5839:
5837:
5834:
5830:
5827:
5826:
5825:
5822:
5818:
5815:
5813:
5810:
5808:
5805:
5804:
5803:
5800:
5796:
5793:
5792:
5791:
5790:Ordered field
5788:
5786:
5783:
5779:
5776:
5774:
5771:
5770:
5769:
5766:
5762:
5759:
5758:
5757:
5754:
5752:
5749:
5747:
5746:Hasse diagram
5744:
5742:
5739:
5737:
5734:
5730:
5727:
5726:
5725:
5724:Comparability
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5706:
5704:
5700:
5692:
5689:
5687:
5684:
5682:
5679:
5677:
5674:
5673:
5672:
5669:
5667:
5664:
5660:
5657:
5655:
5652:
5651:
5650:
5647:
5645:
5641:
5638:
5637:
5635:
5632:
5628:
5622:
5619:
5617:
5614:
5612:
5609:
5607:
5604:
5602:
5599:
5597:
5596:Product order
5594:
5592:
5589:
5587:
5584:
5582:
5579:
5577:
5574:
5573:
5571:
5569:Constructions
5567:
5561:
5557:
5553:
5550:
5546:
5543:
5541:
5538:
5536:
5533:
5531:
5528:
5526:
5523:
5521:
5518:
5516:
5513:
5511:
5508:
5504:
5501:
5500:
5499:
5496:
5494:
5491:
5487:
5484:
5482:
5479:
5477:
5474:
5472:
5469:
5468:
5467:
5466:Partial order
5464:
5462:
5459:
5455:
5454:Join and meet
5452:
5450:
5447:
5445:
5442:
5440:
5437:
5435:
5432:
5431:
5430:
5427:
5425:
5422:
5420:
5417:
5415:
5412:
5410:
5407:
5405:
5401:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5368:
5365:
5364:
5363:
5360:
5358:
5355:
5353:
5352:Antisymmetric
5350:
5349:
5347:
5343:
5337:
5331:
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5293:
5291:
5288:
5286:
5283:
5281:
5278:
5276:
5273:
5272:
5270:
5266:
5260:
5259:Weak ordering
5257:
5255:
5252:
5250:
5247:
5245:
5244:Partial order
5242:
5240:
5237:
5235:
5232:
5230:
5227:
5225:
5222:
5221:
5219:
5215:
5209:
5206:
5204:
5201:
5199:
5196:
5195:
5192:
5188:
5181:
5176:
5174:
5169:
5167:
5162:
5161:
5158:
5148:
5147:
5142:
5134:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5106:
5103:
5102:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5082:
5079:
5077:
5074:
5072:
5069:
5067:
5064:
5062:
5059:
5058:
5056:
5052:
5046:
5043:
5041:
5038:
5036:
5035:Recursive set
5033:
5031:
5028:
5026:
5023:
5021:
5018:
5016:
5013:
5009:
5006:
5004:
5001:
4999:
4996:
4994:
4991:
4989:
4986:
4985:
4984:
4981:
4979:
4976:
4974:
4971:
4969:
4966:
4964:
4961:
4959:
4956:
4955:
4953:
4951:
4947:
4941:
4938:
4936:
4933:
4931:
4928:
4926:
4923:
4921:
4918:
4916:
4913:
4911:
4908:
4904:
4901:
4899:
4896:
4894:
4891:
4890:
4889:
4886:
4884:
4881:
4879:
4876:
4874:
4871:
4869:
4866:
4864:
4861:
4857:
4854:
4853:
4852:
4849:
4845:
4844:of arithmetic
4842:
4841:
4840:
4837:
4833:
4830:
4828:
4825:
4823:
4820:
4818:
4815:
4813:
4810:
4809:
4808:
4805:
4801:
4798:
4796:
4793:
4792:
4791:
4788:
4787:
4785:
4783:
4779:
4773:
4770:
4768:
4765:
4763:
4760:
4758:
4755:
4752:
4751:from ZFC
4748:
4745:
4743:
4740:
4734:
4731:
4730:
4729:
4726:
4724:
4721:
4719:
4716:
4715:
4714:
4711:
4709:
4706:
4704:
4701:
4699:
4696:
4694:
4691:
4689:
4686:
4684:
4681:
4680:
4678:
4676:
4672:
4662:
4661:
4657:
4656:
4651:
4650:non-Euclidean
4648:
4644:
4641:
4639:
4636:
4634:
4633:
4629:
4628:
4626:
4623:
4622:
4620:
4616:
4612:
4609:
4607:
4604:
4603:
4602:
4598:
4594:
4591:
4590:
4589:
4585:
4581:
4578:
4576:
4573:
4571:
4568:
4566:
4563:
4561:
4558:
4556:
4553:
4552:
4550:
4546:
4545:
4543:
4538:
4532:
4527:Example
4524:
4516:
4511:
4510:
4509:
4506:
4504:
4501:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4478:
4477:
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4455:
4452:
4450:
4447:
4446:
4445:
4442:
4438:
4435:
4433:
4430:
4428:
4425:
4423:
4420:
4419:
4418:
4415:
4413:
4410:
4406:
4403:
4401:
4398:
4396:
4393:
4392:
4391:
4388:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4355:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4332:
4329:
4327:
4324:
4322:
4319:
4317:
4314:
4313:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4290:
4287:
4285:
4284:by definition
4282:
4281:
4280:
4277:
4273:
4270:
4269:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4244:
4241:
4238:
4236:
4232:
4227:
4221:
4217:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4184:
4182:
4179:
4177:
4174:
4172:
4171:KripkeâPlatek
4169:
4167:
4164:
4160:
4157:
4155:
4152:
4151:
4150:
4147:
4146:
4144:
4140:
4132:
4129:
4128:
4127:
4124:
4122:
4119:
4115:
4112:
4111:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4084:
4080:
4076:
4073:
4069:
4066:
4064:
4061:
4059:
4056:
4055:
4054:
4050:
4047:
4046:
4044:
4042:
4038:
4034:
4026:
4023:
4021:
4018:
4016:
4015:constructible
4013:
4012:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3951:
3948:
3947:
3945:
3943:
3938:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3905:
3902:
3901:
3899:
3895:
3892:
3890:
3887:
3886:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3865:
3861:
3857:
3855:
3852:
3848:
3845:
3844:
3843:
3840:
3839:
3836:
3833:
3831:
3827:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3785:
3782:
3781:
3780:
3777:
3773:
3768:
3767:
3766:
3763:
3762:
3760:
3758:
3754:
3746:
3743:
3741:
3738:
3736:
3733:
3732:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3698:
3697:
3695:
3693:
3692:Propositional
3689:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3651:
3648:
3647:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3620:Logical truth
3618:
3616:
3613:
3612:
3610:
3608:
3604:
3601:
3599:
3595:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3560:
3556:
3552:
3550:
3547:
3545:
3542:
3540:
3536:
3533:
3532:
3530:
3528:
3522:
3517:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3481:
3478:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3449:
3446:
3445:
3444:
3441:
3440:
3438:
3434:
3430:
3423:
3418:
3416:
3411:
3409:
3404:
3403:
3400:
3394:
3393:0-8247-7915-0
3390:
3387:
3386:Marcel Dekker
3383:
3379:
3375:
3374:Karel HrbĂĄÄek
3372:
3369:
3368:0-8218-0266-6
3365:
3362:
3358:
3354:
3353:
3346:
3340:
3325:
3323:9780444505422
3319:
3315:
3314:
3306:
3291:
3287:
3281:
3273:
3267:
3263:
3255:
3251:
3243:
3240:
3236:
3230:
3226:
3220:
3216:
3211:
3206:
3202:
3196:
3192:
3186:
3182:
3167:
3164:
3161:
3156:
3142:
3138:
3130:
3126:
3108:
3103:
3091:
3087:
3059:
3044:
3040:
3036:
3029:
3028:lexicographic
3025:
3018:
3017:
3016:
3001:
2998:
2995:
2986:
2982:
2979:
2975:
2968:
2961:
2952:
2945:
2935:
2928:
2921:
2914:
2909:, ordered by
2908:
2904:
2899:
2895:
2890:
2878:
2874:
2869:
2865:
2861:
2857:
2849:
2843:
2839:
2835:
2827:
2824:The positive
2823:
2822:
2821:
2813:
2811:
2807:
2803:
2799:
2794:
2792:
2786:
2780:
2776:
2770:
2764:
2756:
2753:is the usual
2746:
2742:
2733:
2728:
2721:
2717:
2710:
2707:
2704:
2701:
2694:
2690:
2683:
2679:
2665:
2652:
2648:
2641:
2637:
2633:
2629:
2625:
2622:
2618:
2609:
2606:
2603:
2599:
2595:
2592:
2586:
2580:
2571:
2567:
2552:
2549:
2546:
2542:
2537:
2528:
2524:
2517:
2513:
2509:
2500:
2494:
2490:
2484:
2479:
2477:
2472:
2459:
2453:
2447:
2440:
2437:
2434:
2412:
2406:
2392:
2386:
2378:
2374:
2370:
2360:
2357:
2354:
2338:
2335:
2332:
2312:
2308:
2302:
2299:
2296:
2285:
2281:
2268:
2267:
2266:
2250:
2246:
2242:
2241:
2240:
2232:
2228:
2222:
2218:
2213:
2203:
2201:
2197:
2193:
2180:
2176:
2168:
2164:
2160:
2151:
2149:
2145:
2137:
2133:
2129:
2121:
2116:
2114:
2110:
2106:
2102:
2101:partial order
2098:
2093:
2083:
2080:
2075:
2071:
2057:
2050:
2043:
2038:
2034:
2029:
2027:
2007:
1998:
1994:
1990:
1978:
1975:
1972:
1960:
1957:
1954:
1937:
1934:
1924:
1921:
1918:
1903:
1900:
1897:
1891:
1887:
1881:
1877:
1868:
1863:
1859:
1855:
1852:
1845:
1841:
1837:
1833:
1829:
1822:
1818:
1807:
1792:
1789:
1786:
1783:
1763:
1760:
1757:
1737:
1734:
1731:
1711:
1708:
1705:
1702:
1699:
1696:
1688:
1672:
1665:
1633:
1632:
1609:
1606:
1603:
1588:
1585:
1582:
1571:
1557:
1554:
1551:
1539:
1525:
1522:
1519:
1512:
1485:
1482:
1479:
1468:
1441:
1438:
1435:
1424:
1397:
1383:
1365:
1362:
1359:
1349:
1346:
1343:
1333:
1326:
1323:
1312:
1294:
1287:
1284:
1274:
1271:
1268:
1258:
1255:
1252:
1241:
1223:
1220:
1217:
1203:
1200:
1197:
1185:
1171:
1165:
1162:
1142:
1139:
1136:
1128:
1127:
1124:
1121:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1089:
1088:Antisymmetric
1086:
1084:
1081:
1079:
1078:
1067:
1057:
1052:
1047:
1042:
1037:
1027:
1017:
1012:
1010:
1007:
1006:
995:
985:
980:
975:
970:
965:
960:
950:
945:
943:
940:
939:
928:
918:
913:
908:
903:
898:
893:
883:
878:
876:
873:
872:
866:
861:
851:
841:
836:
831:
826:
816:
811:
809:
806:
805:
799:
794:
784:
779:
769:
764:
759:
749:
744:
742:
739:
738:
732:
727:
717:
707:
697:
692:
687:
677:
672:
670:
667:
666:
660:
655:
645:
640:
635:
625:
615:
605:
600:
598:
597:Well-ordering
595:
594:
588:
583:
573:
568:
563:
553:
548:
543:
538:
535:
531:
530:
524:
519:
509:
504:
499:
489:
479:
474:
469:
467:
464:
463:
457:
452:
442:
437:
432:
427:
417:
407:
402:
400:
397:
396:
390:
385:
375:
370:
365:
360:
350:
345:
340:
338:
335:
334:
328:
323:
313:
308:
303:
298:
293:
283:
278:
276:
275:Partial order
273:
272:
266:
261:
251:
246:
241:
236:
231:
226:
221:
218:
211:
210:
204:
199:
189:
184:
179:
174:
169:
164:
154:
151:
147:
146:
143:
138:
136:
134:
132:
130:
127:
125:
123:
121:
120:
117:
114:
112:
109:
107:
104:
102:
99:
97:
94:
92:
89:
87:
84:
82:
81:Antisymmetric
79:
77:
74:
72:
71:
68:
67:
61:
56:
54:
49:
47:
42:
41:
38:
34:
30:
29:
22:
5633:& Orders
5611:Star product
5540:Well-founded
5539:
5493:Prefix order
5449:Distributive
5439:Complemented
5409:Foundational
5374:Completeness
5330:Zorn's lemma
5234:Cyclic order
5217:Key concepts
5187:Order theory
5137:
4935:Ultraproduct
4782:Model theory
4747:Independence
4683:Formal proof
4675:Proof theory
4658:
4631:
4588:real numbers
4560:second-order
4471:Substitution
4348:Metalanguage
4289:conservative
4262:Axiom schema
4206:Constructive
4176:MorseâKelley
4142:Set theories
4121:Aleph number
4114:inaccessible
4020:Grothendieck
3904:intersection
3791:Higher-order
3779:Second-order
3725:Truth tables
3682:Venn diagram
3465:Formal proof
3381:
3356:
3344:
3339:
3327:. Retrieved
3312:
3305:
3293:. Retrieved
3289:
3280:
3261:
3254:
3238:
3234:
3228:
3224:
3218:
3214:
3204:
3200:
3194:
3190:
3184:
3180:
3165:
3162:
3159:
3148:
3136:
3128:
3124:
3104:
3089:
3085:
3057:
3053:
3014:
2999:
2996:
2993:
2966:
2959:
2950:
2943:
2933:
2926:
2919:
2912:
2897:
2893:
2876:
2872:
2859:
2855:
2837:
2833:
2819:
2795:
2784:
2773:, we obtain
2768:
2744:
2740:
2726:
2719:
2715:
2711:
2705:
2702:
2699:
2692:
2688:
2681:
2677:
2666:
2569:
2565:
2550:
2547:
2544:
2540:
2526:
2522:
2515:
2511:
2507:
2492:
2488:
2480:
2476:Emmy Noether
2473:
2318:
2310:
2306:
2300:
2297:
2294:
2283:
2279:
2264:
2248:
2244:
2230:
2226:
2220:
2216:
2209:
2199:
2178:
2166:
2162:
2158:
2152:
2127:
2126:is called a
2117:
2105:strict order
2097:order theory
2094:
2081:
2078:
2073:
2069:
2055:
2048:
2041:
2030:
1901:
1898:
1895:
1889:
1885:
1879:
1875:
1861:
1857:
1836:foundational
1835:
1831:
1828:well-founded
1827:
1814:
1661:
1098:Well-founded
1097:
216:(Quasiorder)
91:Well-founded
90:
5817:Riesz space
5778:Isomorphism
5654:Normal cone
5576:Composition
5510:Semilattice
5419:Homogeneous
5404:Equivalence
5254:Total order
5045:Type theory
4993:undecidable
4925:Truth value
4812:equivalence
4491:non-logical
4104:Enumeration
4094:Isomorphism
4041:cardinality
4025:Von Neumann
3990:Ultrafilter
3955:Uncountable
3889:equivalence
3806:Quantifiers
3796:Fixed-point
3765:First-order
3645:Consistency
3630:Proposition
3607:Traditional
3578:Lindström's
3568:Compactness
3510:Type theory
3455:Cardinality
3378:Thomas Jech
3329:20 February
3149:A relation
3145:Reflexivity
3113:on a class
3094:has length
2810:â-induction
2200:terminating
2153:A relation
2109:total order
1832:wellfounded
1817:mathematics
1118:Irreflexive
399:Total order
111:Irreflexive
5785:Order type
5719:Cofinality
5560:Well-order
5535:Transitive
5424:Idempotent
5357:Asymmetric
4856:elementary
4549:arithmetic
4417:Quantifier
4395:functional
4267:Expression
3985:Transitive
3929:identities
3914:complement
3847:hereditary
3830:Set theory
3271:0387900241
3246:References
3121:such that
2292:such that
2167:Noetherian
2120:set theory
2113:well-order
2067:such that
1826:is called
1689:: for all
1687:transitive
1123:Asymmetric
116:Asymmetric
33:Transitive
5836:Upper set
5773:Embedding
5709:Antichain
5530:Tolerance
5520:Symmetric
5515:Semiorder
5461:Reflexive
5379:Connected
5127:Supertask
5030:Recursion
4988:decidable
4822:saturated
4800:of models
4723:deductive
4718:axiomatic
4638:Hilbert's
4625:Euclidean
4606:canonical
4529:axiomatic
4461:Signature
4390:Predicate
4279:Extension
4201:Ackermann
4126:Operation
4005:Universal
3995:Recursive
3970:Singleton
3965:Inhabited
3950:Countable
3940:Types of
3924:power set
3894:partition
3811:Predicate
3757:Predicate
3672:Syllogism
3662:Soundness
3635:Inference
3625:Tautology
3527:paradoxes
3290:ProofWiki
3155:reflexive
2438:∈
2432:∀
2403:⟹
2383:⟹
2358:∈
2352:∀
2336:∈
2330:∀
2319:That is,
2196:rewriting
2173:, if the
1985:¬
1976:∈
1970:∀
1958:∈
1952:∃
1945:⟹
1941:∅
1938:≠
1922:⊆
1916:∀
1851:non-empty
1849:if every
1600:not
1592:⇒
1548:not
1483:∧
1439:∨
1337:⇒
1327:≠
1282:⇒
1211:⇒
1169:∅
1166:≠
1113:Reflexive
1108:Has meets
1103:Has joins
1093:Connected
1083:Symmetric
214:Preorder
141:reflexive
106:Reflexive
101:Has meets
96:Has joins
86:Connected
76:Symmetric
5856:Category
5631:Topology
5498:Preorder
5481:Eulerian
5444:Complete
5394:Directed
5384:Covering
5249:Preorder
5208:Category
5203:Glossary
5112:Logicism
5105:timeline
5081:Concrete
4940:Validity
4910:T-schema
4903:Kripke's
4898:Tarski's
4893:semantic
4883:Strength
4832:submodel
4827:spectrum
4795:function
4643:Tarski's
4632:Elements
4619:geometry
4575:Robinson
4496:variable
4481:function
4454:spectrum
4444:Sentence
4400:variable
4343:Language
4296:Relation
4257:Automata
4247:Alphabet
4231:language
4085:-jection
4063:codomain
4049:Function
4010:Universe
3980:Infinite
3884:Relation
3667:Validity
3657:Argument
3555:theorem,
3210:preorder
3098:for any
2925:) < (
2891:The set
2826:integers
2816:Examples
2724:, where
2499:set-like
2122:, a set
2026:set-like
1657:✗
1644:✗
1054:✗
1049:✗
1044:✗
1039:✗
1014:✗
982:✗
977:✗
972:✗
967:✗
962:✗
947:✗
915:✗
910:✗
905:✗
900:✗
895:✗
880:✗
868:✗
863:✗
838:✗
833:✗
828:✗
813:✗
801:✗
796:✗
781:✗
766:✗
761:✗
746:✗
734:✗
729:✗
694:✗
689:✗
674:✗
662:✗
657:✗
642:✗
637:✗
602:✗
590:✗
585:✗
570:✗
565:✗
550:✗
545:✗
540:✗
526:✗
521:✗
506:✗
501:✗
476:✗
471:✗
459:✗
454:✗
439:✗
434:✗
429:✗
404:✗
392:✗
387:✗
372:✗
367:✗
362:✗
347:✗
342:✗
330:✗
325:✗
310:✗
305:✗
300:✗
295:✗
280:✗
268:✗
263:✗
248:✗
243:✗
238:✗
233:✗
228:✗
223:✗
206:✗
201:✗
186:✗
181:✗
176:✗
171:✗
166:✗
5736:Duality
5714:Cofinal
5702:Related
5681:Fréchet
5558:)
5434:Bounded
5429:Lattice
5402:)
5400:Partial
5268:Results
5239:Lattice
5054:Related
4851:Diagram
4749: (
4728:Hilbert
4713:Systems
4708:Theorem
4586:of the
4531:systems
4311:Formula
4306:Grammar
4222: (
4166:General
3879:Forcing
3864:Element
3784:Monadic
3559:paradox
3500:Theorem
3436:General
3380:(1999)
3060:, <)
2868:strings
2848:divides
2787:, <)
2771:, <)
2534:on the
2424:implies
2304:, then
2130:if the
1838:) on a
669:Lattice
5761:Subnet
5741:Filter
5691:Normed
5676:Banach
5642:&
5549:Better
5486:Strict
5476:Graded
5367:topics
5198:Topics
4817:finite
4580:Skolem
4533:
4508:Theory
4476:Symbol
4466:String
4449:atomic
4326:ground
4321:closed
4316:atomic
4272:ground
4235:syntax
4131:binary
4058:domain
3975:Finite
3740:finite
3598:Logics
3557:
3505:Theory
3391:
3376:&
3366:
3320:
3295:10 May
3268:
2777:, and
2761:gives
2734:, and
2485:. Let
2142:. The
1865:has a
1854:subset
1494:exists
1450:exists
1406:exists
35:
5751:Ideal
5729:Graph
5525:Total
5503:Total
5389:Dense
4807:Model
4555:Peano
4412:Proof
4252:Arity
4181:Naive
4068:image
4000:Fuzzy
3960:Empty
3909:union
3854:Class
3495:Model
3485:Lemma
3443:Axiom
3217:<
3183:<
3088:â 1,
3043:reals
2965:<
2949:<
2903:pairs
2875:<
2836:<
2497:be a
2061:, ...
1844:class
1776:then
139:Anti-
5342:list
4930:Type
4733:list
4537:list
4514:list
4503:Term
4437:rank
4331:open
4225:list
4037:Maps
3942:sets
3801:Free
3771:list
3521:list
3448:list
3389:ISBN
3364:ISBN
3331:2019
3318:ISBN
3297:2021
3266:ISBN
3232:and
3198:and
3139:, â)
3105:The
3041:(or
2957:and
2853:and
2697:for
2277:and
2099:, a
1830:(or
1819:, a
1750:and
1155:and
5756:Net
5556:Pre
4617:of
4599:of
4547:of
4079:Sur
4053:Map
3860:Ur-
3842:Set
3237:â°
3157:if
3084:Ï,
3054:If
3007:to
2905:of
2901:of
2671:on
2555:of
2269:If
2257:of
2169:on
2165:or
2157:is
2138:of
2118:In
2095:In
2024:is
1840:set
1834:or
1815:In
1724:if
1685:be
1395:min
5858::
5003:NP
4627::
4621::
4551::
4228:),
4083:Bi
4075:In
3359:,
3288:.
3227:â€
3203:â
3193:â€
3141:.
3127:,
3102:.
2932:,
2918:,
2896:Ă
2858:â
2793:.
2747:+1
2743:âŠ
2718:,
2709:.
2573:,
2568:â
2543::
2525:â
2514:,
2491:,
2478:.
2219:,
2202:.
2161:,
2115:.
2092:.
2076:+1
2054:,
2047:,
1888:â
1878:â
1860:â
5554:(
5551:)
5547:(
5398:(
5345:)
5179:e
5172:t
5165:v
5083:/
4998:P
4753:)
4539:)
4535:(
4432:â
4427:!
4422:â
4383:=
4378:â
4373:â
4368:â§
4363:âš
4358:ÂŹ
4081:/
4077:/
4051:/
3862:)
3858:(
3745:â
3735:3
3523:)
3421:e
3414:t
3407:v
3370:.
3333:.
3299:.
3274:.
3239:a
3235:b
3229:b
3225:a
3219:b
3215:a
3205:b
3201:a
3195:b
3191:a
3185:b
3181:a
3171:a
3166:a
3163:R
3160:a
3151:R
3137:C
3135:(
3131:)
3129:R
3125:X
3123:(
3119:C
3115:X
3111:R
3100:n
3096:n
3090:n
3086:n
3080:X
3076:X
3072:x
3068:X
3064:x
3058:X
3056:(
3011:.
3009:b
3005:a
3000:b
2997:R
2994:a
2989:R
2980:.
2973:.
2970:2
2967:m
2963:2
2960:n
2954:1
2951:m
2947:1
2944:n
2939:)
2937:2
2934:m
2930:1
2927:m
2923:2
2920:n
2916:1
2913:n
2911:(
2898:N
2894:N
2888:.
2886:t
2882:s
2877:t
2873:s
2863:.
2860:b
2856:a
2851:b
2845:a
2838:b
2834:a
2785:N
2783:(
2769:N
2767:(
2759:S
2751:S
2745:x
2741:x
2736:S
2727:N
2722:)
2720:S
2716:N
2714:(
2706:x
2703:R
2700:y
2695:)
2693:y
2691:(
2689:G
2684:)
2682:x
2680:(
2678:G
2673:X
2669:G
2653:.
2649:)
2642:}
2638:x
2634:R
2630:y
2626::
2623:y
2619:{
2614:|
2610:G
2607:,
2604:x
2600:(
2596:F
2593:=
2590:)
2587:x
2584:(
2581:G
2570:X
2566:x
2561:G
2557:X
2553:}
2551:x
2548:R
2545:y
2541:y
2539:{
2532:g
2527:X
2523:x
2518:)
2516:g
2512:x
2510:(
2508:F
2503:F
2495:)
2493:R
2489:X
2487:(
2460:.
2457:)
2454:x
2451:(
2448:P
2444:)
2441:X
2435:x
2429:(
2419:]
2416:)
2413:x
2410:(
2407:P
2399:]
2396:)
2393:y
2390:(
2387:P
2379:x
2375:R
2371:y
2368:[
2364:)
2361:X
2355:y
2349:(
2346:[
2342:)
2339:X
2333:x
2327:(
2313:)
2311:x
2309:(
2307:P
2301:x
2298:R
2295:y
2290:y
2286:)
2284:y
2282:(
2280:P
2275:X
2271:x
2261:,
2259:X
2255:x
2251:)
2249:x
2247:(
2245:P
2237:X
2233:)
2231:x
2229:(
2227:P
2221:R
2217:X
2188:R
2184:X
2179:R
2171:X
2155:R
2140:x
2124:x
2090:n
2085:n
2082:x
2079:R
2074:n
2070:x
2065:X
2059:2
2056:x
2052:1
2049:x
2045:0
2042:x
2022:R
2008:.
2005:]
2002:)
1999:m
1995:R
1991:s
1988:(
1982:)
1979:S
1973:s
1967:(
1964:)
1961:S
1955:m
1949:(
1935:S
1932:[
1928:)
1925:X
1919:S
1913:(
1902:m
1899:R
1896:s
1890:S
1886:s
1880:S
1876:m
1871:R
1862:X
1858:S
1847:X
1824:R
1793:.
1790:c
1787:R
1784:a
1764:c
1761:R
1758:b
1738:b
1735:R
1732:a
1712:,
1709:c
1706:,
1703:b
1700:,
1697:a
1673:R
1653:Y
1640:Y
1610:a
1607:R
1604:b
1589:b
1586:R
1583:a
1558:a
1555:R
1552:a
1526:a
1523:R
1520:a
1486:b
1480:a
1442:b
1436:a
1398:S
1366:a
1363:R
1360:b
1350:b
1347:R
1344:a
1334:b
1324:a
1295:b
1288:=
1285:a
1275:a
1272:R
1269:b
1259:b
1256:R
1253:a
1224:a
1221:R
1218:b
1204:b
1201:R
1198:a
1172::
1163:S
1143:b
1140:,
1137:a
1074:Y
1064:Y
1034:Y
1024:Y
1002:Y
992:Y
957:Y
935:Y
925:Y
890:Y
858:Y
848:Y
823:Y
791:Y
776:Y
756:Y
724:Y
714:Y
704:Y
684:Y
652:Y
632:Y
622:Y
612:Y
580:Y
560:Y
516:Y
496:Y
486:Y
449:Y
424:Y
414:Y
382:Y
357:Y
320:Y
290:Y
258:Y
196:Y
161:Y
59:e
52:t
45:v
23:.
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