3031:
640:
The last two examples can be combined: given any o-minimal expansion of the real field (such as the real field with restricted analytic functions), one can define its
Pfaffian closure, which is again an o-minimal structure. (The Pfaffian closure of a structure is, in particular, closed under Pfaffian
653:. A major line of current research is based on discovering expansions of the real ordered field that are o-minimal. Despite the generality of application, one can show a great deal about the geometry of set definable in o-minimal structures. There is a cell decomposition theorem,
586:
622:
added (i.e. analytic functions on a neighborhood of , restricted to ; note that the unrestricted sine function has infinitely many roots, and so cannot be definable in an o-minimal structure.)
742:
The condition that the interpretation of < be dense is not strictly necessary, but it is known that discrete orders lead to essentially trivial o-minimal structures, see, for example,
778:
in: Lecture notes on o-minimal structures and real analytic geometry, C. Miller, J.-P. Rolin, and P. Speissegger (eds.), Fields
Institute Communications vol. 62, 2012, pp. 179â218.
671:, a property that has been used to guarantee the convergence of some non-smooth optimization methods, such as the stochastic subgradient method (under some mild assumptions).
1410:
1158:
1113:
1065:
2085:
1002:
2168:
1309:
1209:
466:
O-minimal structures originated in model theory and so have a simpler â but equivalent â definition using the language of model theory. Specifically if
2482:
1202:"Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function"
2640:
1038:
975:
1428:
2495:
1818:
1057:
2080:
2500:
2490:
2227:
1433:
1978:
1424:
2636:
3065:
2733:
2477:
1302:
2038:
1731:
1472:
523:
2994:
2696:
2459:
2454:
2279:
1700:
1384:
36:
2989:
2772:
2689:
2402:
2333:
2210:
1452:
2060:
2914:
2740:
2426:
1659:
661:
2065:
3055:
2397:
2136:
1394:
1295:
994:
182:
110:
2792:
2787:
1201:
148:
O-minimal structures can be defined without recourse to model theory. Here we define a structure on a nonempty set
2721:
2311:
1705:
1673:
1364:
967:
695:
1438:
3011:
2960:
2857:
2355:
2316:
1793:
668:
2852:
1467:
2782:
2321:
2173:
2156:
1879:
1359:
667:
Moreover, continuously differentiable definable functions in a o-minimal structure satisfy a generalization of
2684:
2661:
2622:
2508:
2449:
2095:
2015:
1859:
1803:
1416:
690:
685:
650:
137:
103:
2974:
2701:
2679:
2646:
2539:
2385:
2370:
2343:
2294:
2178:
2113:
1938:
1904:
1899:
1773:
1604:
1581:
79:
72:
3060:
2904:
2757:
2549:
2267:
2003:
1909:
1768:
1753:
1634:
1609:
1238:
3030:
1273:
458:
The "o" stands for "order", since any o-minimal structure requires an ordering on the underlying set.
132:
of an o-minimal structure is an o-minimal theory. This result is remarkable because, in contrast, the
2877:
2839:
2716:
2520:
2360:
2284:
2262:
2090:
2048:
1947:
1914:
1778:
1566:
1477:
626:
3006:
2897:
2882:
2862:
2819:
2706:
2656:
2582:
2527:
2464:
2257:
2252:
2200:
1968:
1957:
1629:
1529:
1457:
1448:
1444:
1379:
1374:
908:(PhD thesis). UniversitĂ© Montpellier; Universidad tĂ©cnica Federico Santa MarĂa (Valparaiso, Chili).
630:
3035:
2804:
2767:
2752:
2745:
2728:
2514:
2380:
2306:
2289:
2242:
2055:
1964:
1798:
1783:
1743:
1695:
1680:
1668:
1624:
1599:
1369:
1318:
1255:
1177:
1132:
1084:
884:
858:
658:
68:
28:
2532:
1988:
98:
is equivalent to a quantifier-free formula involving only the ordering, also with parameters in
3070:
2970:
2777:
2587:
2577:
2469:
2350:
2185:
2161:
1942:
1926:
1831:
1808:
1685:
1654:
1619:
1514:
1349:
1034:
971:
940:
876:
827:
680:
646:
634:
619:
608:
805:
2984:
2979:
2872:
2829:
2651:
2612:
2607:
2592:
2418:
2375:
2272:
2070:
2020:
1594:
1556:
1247:
1218:
1185:
1167:
1140:
1122:
1092:
1074:
1044:
1011:
981:
932:
905:
Descent dynamical systems and algorithms for tame optimization, and multi-objective problems
868:
817:
779:
700:
753:
746:
2965:
2955:
2909:
2892:
2847:
2809:
2711:
2631:
2438:
2365:
2338:
2326:
2232:
2146:
2120:
2075:
2043:
1844:
1646:
1589:
1539:
1504:
1462:
1189:
1144:
1104:
1096:
1048:
1030:
985:
750:
743:
654:
478:-structure where < is interpreted to satisfy the axioms of a dense linear order, then (
133:
87:
2950:
2929:
2887:
2867:
2762:
2617:
2215:
2205:
2195:
2190:
2124:
1998:
1874:
1763:
1758:
1736:
1337:
846:
3049:
2924:
2602:
2109:
1894:
1884:
1854:
1839:
1509:
705:
91:
52:
2824:
2671:
2572:
2564:
2444:
2392:
2301:
2237:
2220:
2151:
2010:
1869:
1571:
1354:
888:
121:
32:
1223:
1016:
600:
The complete theory of dense linear orders in the language with just the ordering.
140:, that is, there may be an elementarily equivalent structure that is not minimal.
2934:
2814:
1993:
1983:
1930:
1614:
1534:
1519:
1399:
1344:
1197:
903:
783:
615:
44:
872:
1864:
1719:
1690:
1496:
920:
966:. London Mathematical Society Lecture Note Series. Vol. 248. Cambridge:
944:
880:
831:
641:
chains where arbitrary definable functions are used in place of polynomials.)
409:
has a dense linear order without endpoints on it, say <, then a structure
3016:
2919:
1972:
1889:
1849:
1813:
1749:
1561:
1551:
1524:
1280:
3001:
2799:
2247:
1952:
1546:
845:
Davis, Damek; Drusvyatskiy, Dmitriy; Kakade, Sham; Lee, Jason D. (2020).
17:
2597:
1389:
1259:
1181:
1136:
1088:
417:
is called o-minimal (respect to <) if it satisfies the extra axioms
106:
structures, which are exactly the analogous property down to equality.
936:
1287:
822:
604:
482:,<,...) is called an o-minimal structure if for any definable set
1251:
1172:
1153:
1127:
1108:
1079:
863:
2141:
1487:
1332:
664:
theorems and a good notion of dimension and Euler characteristic.
649:. Thus the study of o-minimal structures and theories generalises
724:
Knight, Pillay and
Steinhorn (1986), Pillay and Steinhorn (1988).
1291:
633:. More generally, the complete theory of the real numbers with
1029:. Graduate Texts in Mathematics. Vol. 217. New York, NY:
806:"On gradients of functions definable in o-minimal structures"
625:
The complete theory of the real field with a symbol for the
847:"Stochastic Subgradient Method Converges on Tame Functions"
454:
are precisely the finite unions of intervals and points.
581:{\displaystyle X=X_{0}\cup I_{1}\cup \ldots \cup I_{r}.}
526:
470:
is a language including a binary relation <, and (
1282:
Real
Algebraic and Analytic Geometry Preprint Server
995:"Review of "Tame Topology and o-minimal Structures""
2943:
2838:
2670:
2563:
2415:
2108:
2031:
1925:
1829:
1718:
1645:
1580:
1495:
1486:
1408:
1325:
128:is o-minimal. It is known that the complete theory
580:
1159:Transactions of the American Mathematical Society
1114:Transactions of the American Mathematical Society
1066:Transactions of the American Mathematical Society
645:In the case of RCF, the definable sets are the
78:O-minimality can be regarded as a weak form of
1303:
1003:Bulletin of the American Mathematical Society
8:
1210:Journal of the American Mathematical Society
1107:; Pillay, Anand; Steinhorn, Charles (1986).
2129:
1724:
1492:
1310:
1296:
1288:
1154:"Definable Sets in Ordered Structures III"
1152:Pillay, Anand; Steinhorn, Charles (1988).
1056:Pillay, Anand; Steinhorn, Charles (1986).
1222:
1171:
1126:
1109:"Definable Sets in Ordered Structures II"
1078:
1015:
862:
821:
569:
550:
537:
525:
152:in a set-theoretic manner, as a sequence
1058:"Definable Sets in Ordered Structures I"
851:Foundations of Computational Mathematics
394:. A subset is called definable if it is
717:
490:there are finitely many open intervals
1232:Denef, J.; van den Dries, L. (1989). "
964:Tame Topology and o-minimal Structures
338:, we consider the smallest structure
136:of a minimal structure need not be a
7:
921:"An Invitation to Tame Optimization"
596:Examples of o-minimal theories are:
510: âȘ {±â} and a finite set
305:is the projection map on the first
1236:-adic and real subanalytic sets".
902:Garrigos, Guillaume (2015-11-02).
390:is called a set of parameters for
86:is o-minimal if and only if every
25:
373:-definable if it is contained in
350:such that every finite subset of
169: = 0,1,2,... such that
3029:
776:Pfaffian sets and o-minimality,
1:
2990:History of mathematical logic
1224:10.1090/S0894-0347-96-00216-0
1027:Model theory: An introduction
1017:10.1090/S0273-0979-00-00866-1
810:Annales de l'Institut Fourier
2915:Primitive recursive function
1275:Model Theory preprint server
925:SIAM Journal on Optimization
421:the set < (={(
63:(with parameters taken from
962:van den Dries, Lou (1998).
804:Kurdyka, Krzysztof (1998).
784:10.1007/978-1-4614-4042-0_5
614:The complete theory of the
102:. This is analogous to the
31:, and more specifically in
3087:
1979:SchröderâBernstein theorem
1706:Monadic predicate calculus
1365:Foundations of mathematics
968:Cambridge University Press
873:10.1007/s10208-018-09409-5
696:Weakly o-minimal structure
462:Model theoretic definition
3025:
3012:Philosophy of mathematics
2961:Automated theorem proving
2132:
2086:Von NeumannâBernaysâGödel
1727:
450:the definable subsets of
144:Set-theoretic definition
3066:Real algebraic geometry
2662:Self-verifying theories
2483:Tarski's axiomatization
1434:Tarski's undefinability
1429:incompleteness theorems
691:Strongly minimal theory
686:Real algebraic geometry
651:real algebraic geometry
419:
138:strongly minimal theory
3036:Mathematics portal
2647:Proof of impossibility
2295:propositional variable
1605:Propositional calculus
1025:Marker, David (2002).
993:Marker, David (2000).
669:Ćojasiewicz inequality
582:
80:quantifier elimination
2905:Kolmogorov complexity
2858:Computably enumerable
2758:Model complete theory
2550:Principia Mathematica
1610:Propositional formula
1439:BanachâTarski paradox
1239:Annals of Mathematics
919:Ioffe, A. D. (2009).
583:
51:if and only if every
47:by < is called an
2853:ChurchâTuring thesis
2840:Computability theory
2049:continuum hypothesis
1567:Square of opposition
1425:Gödel's completeness
774:Patrick Speisseger,
627:exponential function
524:
398:-definable for some
3007:Mathematical object
2898:P versus NP problem
2863:Computable function
2657:Reverse mathematics
2583:Logical consequence
2460:primitive recursive
2455:elementary function
2228:Free/bound variable
2081:TarskiâGrothendieck
1600:Logical connectives
1530:Logical equivalence
1380:Logical consequence
794:Marker (2002) p.103
49:o-minimal structure
43:,<,...) that is
3056:Mathematical logic
2805:Transfer principle
2768:Semantics of logic
2753:Categorical theory
2729:Non-standard model
2243:Logical connective
1370:Information theory
1319:Mathematical logic
765:Marker (2002) p.99
733:Marker (2002) p.81
647:semialgebraic sets
635:Pfaffian functions
620:analytic functions
609:real closed fields
578:
309:coordinates, then
94:and parameters in
29:mathematical logic
3043:
3042:
2975:Abstract category
2778:Theories of truth
2588:Rule of inference
2578:Natural deduction
2559:
2558:
2104:
2103:
1809:Cartesian product
1714:
1713:
1620:Many-valued logic
1595:Boolean functions
1478:Russell's paradox
1453:diagonal argument
1350:First-order logic
1040:978-0-387-98760-6
977:978-0-521-59838-5
937:10.1137/080722059
681:Semialgebraic set
474:,<,...) is an
16:(Redirected from
3078:
3034:
3033:
2985:History of logic
2980:Category of sets
2873:Decision problem
2652:Ordinal analysis
2593:Sequent calculus
2491:Boolean algebras
2431:
2430:
2405:
2376:logical/constant
2130:
2116:
2039:ZermeloâFraenkel
1790:Set operations:
1725:
1662:
1493:
1473:LöwenheimâSkolem
1360:Formal semantics
1312:
1305:
1298:
1289:
1263:
1228:
1226:
1217:(4): 1051â1095.
1206:
1193:
1175:
1148:
1130:
1100:
1082:
1062:
1052:
1021:
1019:
999:
989:
949:
948:
931:(4): 1894â1917.
916:
910:
909:
899:
893:
892:
866:
842:
836:
835:
825:
823:10.5802/aif.1638
801:
795:
792:
786:
772:
766:
763:
757:
740:
734:
731:
725:
722:
701:C-minimal theory
631:Wilkie's theorem
618:with restricted
587:
585:
584:
579:
574:
573:
555:
554:
542:
541:
437: <
386:); in that case
354:is contained in
118:o-minimal theory
21:
3086:
3085:
3081:
3080:
3079:
3077:
3076:
3075:
3046:
3045:
3044:
3039:
3028:
3021:
2966:Category theory
2956:Algebraic logic
2939:
2910:Lambda calculus
2848:Church encoding
2834:
2810:Truth predicate
2666:
2632:Complete theory
2555:
2424:
2420:
2416:
2411:
2403:
2123: and
2119:
2114:
2100:
2076:New Foundations
2044:axiom of choice
2027:
1989:Gödel numbering
1929: and
1921:
1825:
1710:
1660:
1641:
1590:Boolean algebra
1576:
1540:Equiconsistency
1505:Classical logic
1482:
1463:Halting problem
1451: and
1427: and
1415: and
1414:
1409:Theorems (
1404:
1321:
1316:
1270:
1252:10.2307/1971463
1231:
1204:
1196:
1173:10.2307/2000920
1151:
1128:10.2307/2000053
1103:
1080:10.2307/2000052
1060:
1055:
1041:
1031:Springer-Verlag
1024:
997:
992:
978:
961:
958:
953:
952:
918:
917:
913:
901:
900:
896:
844:
843:
839:
803:
802:
798:
793:
789:
773:
769:
764:
760:
741:
737:
732:
728:
723:
719:
714:
677:
594:
565:
546:
533:
522:
521:
516:
505:
496:
464:
447:
381:
360:
325:
292:
276:
267:
258:
247:
238:
229:
203:
183:boolean algebra
180:
164:
146:
134:complete theory
45:totally ordered
23:
22:
15:
12:
11:
5:
3084:
3082:
3074:
3073:
3068:
3063:
3058:
3048:
3047:
3041:
3040:
3026:
3023:
3022:
3020:
3019:
3014:
3009:
3004:
2999:
2998:
2997:
2987:
2982:
2977:
2968:
2963:
2958:
2953:
2951:Abstract logic
2947:
2945:
2941:
2940:
2938:
2937:
2932:
2930:Turing machine
2927:
2922:
2917:
2912:
2907:
2902:
2901:
2900:
2895:
2890:
2885:
2880:
2870:
2868:Computable set
2865:
2860:
2855:
2850:
2844:
2842:
2836:
2835:
2833:
2832:
2827:
2822:
2817:
2812:
2807:
2802:
2797:
2796:
2795:
2790:
2785:
2775:
2770:
2765:
2763:Satisfiability
2760:
2755:
2750:
2749:
2748:
2738:
2737:
2736:
2726:
2725:
2724:
2719:
2714:
2709:
2704:
2694:
2693:
2692:
2687:
2680:Interpretation
2676:
2674:
2668:
2667:
2665:
2664:
2659:
2654:
2649:
2644:
2634:
2629:
2628:
2627:
2626:
2625:
2615:
2610:
2600:
2595:
2590:
2585:
2580:
2575:
2569:
2567:
2561:
2560:
2557:
2556:
2554:
2553:
2545:
2544:
2543:
2542:
2537:
2536:
2535:
2530:
2525:
2505:
2504:
2503:
2501:minimal axioms
2498:
2487:
2486:
2485:
2474:
2473:
2472:
2467:
2462:
2457:
2452:
2447:
2434:
2432:
2413:
2412:
2410:
2409:
2408:
2407:
2395:
2390:
2389:
2388:
2383:
2378:
2373:
2363:
2358:
2353:
2348:
2347:
2346:
2341:
2331:
2330:
2329:
2324:
2319:
2314:
2304:
2299:
2298:
2297:
2292:
2287:
2277:
2276:
2275:
2270:
2265:
2260:
2255:
2250:
2240:
2235:
2230:
2225:
2224:
2223:
2218:
2213:
2208:
2198:
2193:
2191:Formation rule
2188:
2183:
2182:
2181:
2176:
2166:
2165:
2164:
2154:
2149:
2144:
2139:
2133:
2127:
2110:Formal systems
2106:
2105:
2102:
2101:
2099:
2098:
2093:
2088:
2083:
2078:
2073:
2068:
2063:
2058:
2053:
2052:
2051:
2046:
2035:
2033:
2029:
2028:
2026:
2025:
2024:
2023:
2013:
2008:
2007:
2006:
1999:Large cardinal
1996:
1991:
1986:
1981:
1976:
1962:
1961:
1960:
1955:
1950:
1935:
1933:
1923:
1922:
1920:
1919:
1918:
1917:
1912:
1907:
1897:
1892:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1842:
1836:
1834:
1827:
1826:
1824:
1823:
1822:
1821:
1816:
1811:
1806:
1801:
1796:
1788:
1787:
1786:
1781:
1771:
1766:
1764:Extensionality
1761:
1759:Ordinal number
1756:
1746:
1741:
1740:
1739:
1728:
1722:
1716:
1715:
1712:
1711:
1709:
1708:
1703:
1698:
1693:
1688:
1683:
1678:
1677:
1676:
1666:
1665:
1664:
1651:
1649:
1643:
1642:
1640:
1639:
1638:
1637:
1632:
1627:
1617:
1612:
1607:
1602:
1597:
1592:
1586:
1584:
1578:
1577:
1575:
1574:
1569:
1564:
1559:
1554:
1549:
1544:
1543:
1542:
1532:
1527:
1522:
1517:
1512:
1507:
1501:
1499:
1490:
1484:
1483:
1481:
1480:
1475:
1470:
1465:
1460:
1455:
1443:Cantor's
1441:
1436:
1431:
1421:
1419:
1406:
1405:
1403:
1402:
1397:
1392:
1387:
1382:
1377:
1372:
1367:
1362:
1357:
1352:
1347:
1342:
1341:
1340:
1329:
1327:
1323:
1322:
1317:
1315:
1314:
1307:
1300:
1292:
1286:
1285:
1278:
1269:
1268:External links
1266:
1265:
1264:
1229:
1194:
1166:(2): 469â476.
1149:
1121:(2): 593â605.
1101:
1073:(2): 565â592.
1053:
1039:
1022:
1010:(3): 351â357.
990:
976:
957:
954:
951:
950:
911:
894:
857:(1): 119â154.
837:
816:(3): 769â783.
796:
787:
767:
758:
735:
726:
716:
715:
713:
710:
709:
708:
703:
698:
693:
688:
683:
676:
673:
662:stratification
643:
642:
638:
623:
612:
601:
593:
590:
589:
588:
577:
572:
568:
564:
561:
558:
553:
549:
545:
540:
536:
532:
529:
514:
501:
494:
463:
460:
456:
455:
448:
445:
429:) â
377:
358:
328:
327:
321:
317:) â
287:
277:
272:
263:
256:
248:) â
243:
236:
230:
224:
199:
189:
185:of subsets of
176:
160:
156: = (
145:
142:
82:. A structure
67:) is a finite
35:, an infinite
24:
14:
13:
10:
9:
6:
4:
3:
2:
3083:
3072:
3069:
3067:
3064:
3062:
3059:
3057:
3054:
3053:
3051:
3038:
3037:
3032:
3024:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2996:
2993:
2992:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2948:
2946:
2942:
2936:
2933:
2931:
2928:
2926:
2925:Recursive set
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2875:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2845:
2843:
2841:
2837:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2794:
2791:
2789:
2786:
2784:
2781:
2780:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2747:
2744:
2743:
2742:
2739:
2735:
2734:of arithmetic
2732:
2731:
2730:
2727:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2699:
2698:
2695:
2691:
2688:
2686:
2683:
2682:
2681:
2678:
2677:
2675:
2673:
2669:
2663:
2660:
2658:
2655:
2653:
2650:
2648:
2645:
2642:
2641:from ZFC
2638:
2635:
2633:
2630:
2624:
2621:
2620:
2619:
2616:
2614:
2611:
2609:
2606:
2605:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2574:
2571:
2570:
2568:
2566:
2562:
2552:
2551:
2547:
2546:
2541:
2540:non-Euclidean
2538:
2534:
2531:
2529:
2526:
2524:
2523:
2519:
2518:
2516:
2513:
2512:
2510:
2506:
2502:
2499:
2497:
2494:
2493:
2492:
2488:
2484:
2481:
2480:
2479:
2475:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2442:
2440:
2436:
2435:
2433:
2428:
2422:
2417:Example
2414:
2406:
2401:
2400:
2399:
2396:
2394:
2391:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2369:
2368:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2345:
2342:
2340:
2337:
2336:
2335:
2332:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2309:
2308:
2305:
2303:
2300:
2296:
2293:
2291:
2288:
2286:
2283:
2282:
2281:
2278:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2245:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2222:
2219:
2217:
2214:
2212:
2209:
2207:
2204:
2203:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2180:
2177:
2175:
2174:by definition
2172:
2171:
2170:
2167:
2163:
2160:
2159:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2134:
2131:
2128:
2126:
2122:
2117:
2111:
2107:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2062:
2061:KripkeâPlatek
2059:
2057:
2054:
2050:
2047:
2045:
2042:
2041:
2040:
2037:
2036:
2034:
2030:
2022:
2019:
2018:
2017:
2014:
2012:
2009:
2005:
2002:
2001:
2000:
1997:
1995:
1992:
1990:
1987:
1985:
1982:
1980:
1977:
1974:
1970:
1966:
1963:
1959:
1956:
1954:
1951:
1949:
1946:
1945:
1944:
1940:
1937:
1936:
1934:
1932:
1928:
1924:
1916:
1913:
1911:
1908:
1906:
1905:constructible
1903:
1902:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1837:
1835:
1833:
1828:
1820:
1817:
1815:
1812:
1810:
1807:
1805:
1802:
1800:
1797:
1795:
1792:
1791:
1789:
1785:
1782:
1780:
1777:
1776:
1775:
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1751:
1747:
1745:
1742:
1738:
1735:
1734:
1733:
1730:
1729:
1726:
1723:
1721:
1717:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1675:
1672:
1671:
1670:
1667:
1663:
1658:
1657:
1656:
1653:
1652:
1650:
1648:
1644:
1636:
1633:
1631:
1628:
1626:
1623:
1622:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1587:
1585:
1583:
1582:Propositional
1579:
1573:
1570:
1568:
1565:
1563:
1560:
1558:
1555:
1553:
1550:
1548:
1545:
1541:
1538:
1537:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1510:Logical truth
1508:
1506:
1503:
1502:
1500:
1498:
1494:
1491:
1489:
1485:
1479:
1476:
1474:
1471:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1450:
1446:
1442:
1440:
1437:
1435:
1432:
1430:
1426:
1423:
1422:
1420:
1418:
1412:
1407:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1346:
1343:
1339:
1336:
1335:
1334:
1331:
1330:
1328:
1324:
1320:
1313:
1308:
1306:
1301:
1299:
1294:
1293:
1290:
1284:
1283:
1279:
1277:
1276:
1272:
1271:
1267:
1261:
1257:
1253:
1249:
1246:(1): 79â138.
1245:
1241:
1240:
1235:
1230:
1225:
1220:
1216:
1212:
1211:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1174:
1169:
1165:
1161:
1160:
1155:
1150:
1146:
1142:
1138:
1134:
1129:
1124:
1120:
1116:
1115:
1110:
1106:
1105:Knight, Julia
1102:
1098:
1094:
1090:
1086:
1081:
1076:
1072:
1068:
1067:
1059:
1054:
1050:
1046:
1042:
1036:
1032:
1028:
1023:
1018:
1013:
1009:
1005:
1004:
996:
991:
987:
983:
979:
973:
969:
965:
960:
959:
955:
946:
942:
938:
934:
930:
926:
922:
915:
912:
907:
906:
898:
895:
890:
886:
882:
878:
874:
870:
865:
860:
856:
852:
848:
841:
838:
833:
829:
824:
819:
815:
811:
807:
800:
797:
791:
788:
785:
781:
777:
771:
768:
762:
759:
755:
752:
748:
745:
739:
736:
730:
727:
721:
718:
711:
707:
706:Tame topology
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
678:
674:
672:
670:
665:
663:
660:
656:
652:
648:
639:
636:
632:
628:
624:
621:
617:
613:
610:
606:
602:
599:
598:
597:
591:
575:
570:
566:
562:
559:
556:
551:
547:
543:
538:
534:
530:
527:
520:
519:
518:
513:
509:
504:
500:
493:
489:
486: â
485:
481:
477:
473:
469:
461:
459:
453:
449:
444:
440:
436:
433: :
432:
428:
424:
420:
418:
416:
412:
408:
403:
401:
397:
393:
389:
385:
380:
376:
372:
368:
364:
357:
353:
349:
346:) containing
345:
341:
337:
333:
330:For a subset
324:
320:
316:
312:
308:
304:
301: â
300:
297: :
296:
290:
286:
283: â
282:
278:
275:
271:
266:
262:
259: =
255:
252: :
251:
246:
242:
235:
231:
227:
223:
219:
215:
211:
208: Ă
207:
202:
198:
195: â
194:
190:
188:
184:
179:
175:
172:
171:
170:
168:
163:
159:
155:
151:
143:
141:
139:
135:
131:
127:
123:
119:
115:
112:
107:
105:
101:
97:
93:
92:free variable
89:
85:
81:
76:
74:
70:
66:
62:
59: â
58:
54:
50:
46:
42:
38:
34:
30:
19:
3061:Model theory
3027:
2825:Ultraproduct
2672:Model theory
2637:Independence
2573:Formal proof
2565:Proof theory
2548:
2521:
2478:real numbers
2450:second-order
2361:Substitution
2238:Metalanguage
2179:conservative
2152:Axiom schema
2096:Constructive
2066:MorseâKelley
2032:Set theories
2011:Aleph number
2004:inaccessible
1910:Grothendieck
1794:intersection
1681:Higher-order
1669:Second-order
1615:Truth tables
1572:Venn diagram
1355:Formal proof
1281:
1274:
1243:
1237:
1233:
1214:
1208:
1198:Wilkie, A.J.
1163:
1157:
1118:
1112:
1070:
1064:
1026:
1007:
1001:
963:
928:
924:
914:
904:
897:
854:
850:
840:
813:
809:
799:
790:
775:
770:
761:
738:
729:
720:
666:
644:
595:
511:
507:
502:
498:
491:
487:
483:
479:
475:
471:
467:
465:
457:
451:
442:
438:
434:
430:
426:
422:
414:
410:
406:
404:
399:
395:
391:
387:
383:
378:
374:
370:
366:
362:
355:
351:
347:
343:
339:
335:
331:
329:
322:
318:
314:
310:
306:
302:
298:
294:
288:
284:
280:
273:
269:
264:
260:
253:
249:
244:
240:
233:
225:
221:
217:
213:
209:
205:
200:
196:
192:
186:
177:
173:
166:
161:
157:
153:
149:
147:
129:
125:
117:
113:
108:
99:
95:
83:
77:
75:and points.
64:
60:
56:
48:
40:
33:model theory
26:
2935:Type theory
2883:undecidable
2815:Truth value
2702:equivalence
2381:non-logical
1994:Enumeration
1984:Isomorphism
1931:cardinality
1915:Von Neumann
1880:Ultrafilter
1845:Uncountable
1779:equivalence
1696:Quantifiers
1686:Fixed-point
1655:First-order
1535:Consistency
1520:Proposition
1497:Traditional
1468:Lindström's
1458:Compactness
1400:Type theory
1345:Cardinality
361:. A subset
3050:Categories
2746:elementary
2439:arithmetic
2307:Quantifier
2285:functional
2157:Expression
1875:Transitive
1819:identities
1804:complement
1737:hereditary
1720:Set theory
1190:0707.03024
1145:0662.03024
1097:0662.03023
1049:1003.03034
986:0953.03045
956:References
864:1804.07795
616:real field
517:such that
369:is called
232:the set {(
3017:Supertask
2920:Recursion
2878:decidable
2712:saturated
2690:of models
2613:deductive
2608:axiomatic
2528:Hilbert's
2515:Euclidean
2496:canonical
2419:axiomatic
2351:Signature
2280:Predicate
2169:Extension
2091:Ackermann
2016:Operation
1895:Universal
1885:Recursive
1860:Singleton
1855:Inhabited
1840:Countable
1830:Types of
1814:power set
1784:partition
1701:Predicate
1647:Predicate
1562:Syllogism
1552:Soundness
1525:Inference
1515:Tautology
1417:paradoxes
945:1052-6234
881:1615-3375
832:0373-0956
603:RCF, the
563:∪
560:…
557:∪
544:∪
441:}) is in
120:if every
90:with one
73:intervals
53:definable
37:structure
18:O-minimal
3071:Topology
3002:Logicism
2995:timeline
2971:Concrete
2830:Validity
2800:T-schema
2793:Kripke's
2788:Tarski's
2783:semantic
2773:Strength
2722:submodel
2717:spectrum
2685:function
2533:Tarski's
2522:Elements
2509:geometry
2465:Robinson
2386:variable
2371:function
2344:spectrum
2334:Sentence
2290:variable
2233:Language
2186:Relation
2147:Automata
2137:Alphabet
2121:language
1975:-jection
1953:codomain
1939:Function
1900:Universe
1870:Infinite
1774:Relation
1557:Validity
1547:Argument
1445:theorem,
1200:(1996).
675:See also
592:Examples
268:} is in
2944:Related
2741:Diagram
2639: (
2618:Hilbert
2603:Systems
2598:Theorem
2476:of the
2421:systems
2201:Formula
2196:Grammar
2112: (
2056:General
1769:Forcing
1754:Element
1674:Monadic
1449:paradox
1390:Theorem
1326:General
1260:1971463
1182:2000920
1137:2000053
1089:2000052
889:5025719
754:0943306
747:0899083
659:Verdier
655:Whitney
220:are in
216: Ă
104:minimal
88:formula
55:subset
2707:finite
2470:Skolem
2423:
2398:Theory
2366:Symbol
2356:String
2339:atomic
2216:ground
2211:closed
2206:atomic
2162:ground
2125:syntax
2021:binary
1948:domain
1865:Finite
1630:finite
1488:Logics
1447:
1395:Theory
1258:
1188:
1180:
1143:
1135:
1095:
1087:
1047:
1037:
984:
974:
943:
887:
879:
830:
637:added.
605:theory
497:,...,
116:is an
111:theory
2697:Model
2445:Peano
2302:Proof
2142:Arity
2071:Naive
1958:image
1890:Fuzzy
1850:Empty
1799:union
1744:Class
1385:Model
1375:Lemma
1333:Axiom
1256:JSTOR
1205:(PDF)
1178:JSTOR
1133:JSTOR
1085:JSTOR
1061:(PDF)
998:(PDF)
885:S2CID
859:arXiv
712:Notes
239:,...,
204:then
181:is a
122:model
69:union
2820:Type
2623:list
2427:list
2404:list
2393:Term
2327:rank
2221:open
2115:list
1927:Maps
1832:sets
1691:Free
1661:list
1411:list
1338:list
1035:ISBN
972:ISBN
941:ISSN
877:ISSN
828:ISSN
749:and
657:and
293:and
212:and
2507:of
2489:of
2437:of
1969:Sur
1943:Map
1750:Ur-
1732:Set
1248:doi
1244:128
1219:doi
1186:Zbl
1168:doi
1164:309
1141:Zbl
1123:doi
1119:295
1093:Zbl
1075:doi
1071:295
1045:Zbl
1012:doi
982:Zbl
933:doi
869:doi
818:doi
780:doi
629:by
607:of
506:in
413:on
405:If
365:of
334:of
279:if
191:if
165:),
124:of
71:of
27:In
3052::
2893:NP
2517::
2511::
2441::
2118:),
1973:Bi
1965:In
1254:.
1242:.
1213:.
1207:.
1184:.
1176:.
1162:.
1156:.
1139:.
1131:.
1117:.
1111:.
1091:.
1083:.
1069:.
1063:.
1043:.
1033:.
1008:37
1006:.
1000:.
980:.
970:.
939:.
929:19
927:.
923:.
883:.
875:.
867:.
855:20
853:.
849:.
826:.
814:48
812:.
808:.
751:MR
744:MR
402:.
291:+1
228:+1
109:A
2973:/
2888:P
2643:)
2429:)
2425:(
2322:â
2317:!
2312:â
2273:=
2268:â
2263:â
2258:â§
2253:âš
2248:ÂŹ
1971:/
1967:/
1941:/
1752:)
1748:(
1635:â
1625:3
1413:)
1311:e
1304:t
1297:v
1262:.
1250::
1234:p
1227:.
1221::
1215:9
1192:.
1170::
1147:.
1125::
1099:.
1077::
1051:.
1020:.
1014::
988:.
947:.
935::
891:.
871::
861::
834:.
820::
782::
756:.
611:.
576:.
571:r
567:I
552:1
548:I
539:0
535:X
531:=
528:X
515:0
512:X
508:M
503:r
499:I
495:1
492:I
488:M
484:X
480:M
476:L
472:M
468:L
452:M
446:2
443:S
439:y
435:x
431:M
427:y
425:,
423:x
415:M
411:S
407:M
400:A
396:A
392:D
388:A
384:A
382:(
379:n
375:S
371:A
367:M
363:D
359:1
356:S
352:A
348:S
344:A
342:(
340:S
336:M
332:A
326:.
323:n
319:S
315:D
313:(
311:Ï
307:n
303:M
299:M
295:Ï
289:n
285:S
281:D
274:n
270:S
265:n
261:x
257:1
254:x
250:M
245:n
241:x
237:1
234:x
226:n
222:S
218:M
214:D
210:D
206:M
201:n
197:S
193:D
187:M
178:n
174:S
167:n
162:n
158:S
154:S
150:M
130:T
126:T
114:T
100:M
96:M
84:M
65:M
61:M
57:X
41:M
39:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.