1367:
1949:
of the original structure. It has the same expressive power as the original structure, in the sense that a set is definable over the enlarged structure from a set of parameters if and only if it is definable over the original structure from that same set of parameters.
1174:
1943:
1590:. Although the usual ordering relation is not directly included in the structure, there is a formula that defines the set of nonnegative reals, since these are the only reals that possess square roots:
1486:
1580:
2460:
1644:
2509:
1418:
954:
1084:
659:
1714:
1362:{\displaystyle \varphi =\exists x_{0}\cdots \exists x_{n-1}(x_{0}<x_{1}\land \cdots \land x_{n-1}<x\land \forall y(y<x\rightarrow (y\equiv x_{0}\lor \cdots \lor y\equiv x_{n-1})))}
2381:
559:
2317:
497:
445:
1986:
2695:
1816:
2723:
1675:
2145:
1488:
be the first-order structure consisting of the natural numbers and their usual arithmetic operations and order relation. The sets definable in this structure are known as the
259:
2635:
2581:
2557:
2533:
2245:
2169:
2066:
2042:
1782:
1738:
978:
896:
843:
796:
772:
741:
715:
285:
134:
110:
86:
2603:
2112:
2221:
1142:
1027:
1758:
393:
1842:
1113:
872:
1868:
2265:
2189:
2086:
1162:
998:
819:
691:
305:
218:
198:
174:
154:
2535:
is an automorphism preserving the empty set of parameters, and thus it is impossible to define any particular integer in this structure without parameters in
1873:
2559:. In fact, since any two integers are carried to each other by a translation and its inverse, the only sets of integers definable in
1435:
1523:
2386:
1596:
2470:
1379:
915:
2467:
This result can sometimes be used to classify the definable subsets of a given structure. For example, in the case of
1039:
956:
be the structure consisting of the natural numbers with the usual ordering. Then every natural number is definable in
33:
1500:
instead of first-order logic, the definable sets of natural numbers in the resulting structure are classified in the
564:
1680:
2328:
506:
37:
2270:
450:
398:
1960:
41:
2808:
2644:
1954:
1787:
2700:
1652:
2117:
231:
2616:
2562:
2538:
2514:
2226:
2150:
2047:
2023:
1763:
1719:
959:
877:
824:
777:
753:
722:
696:
266:
115:
91:
67:
1946:
2586:
2091:
2738:
2194:
1118:
1003:
846:
56:, which are elements of the domain that can be referenced in the formula defining the relation.
2734:
2725:. In particular, any automorphism (translation) preserves the "distance" between two elements.
1989:
1509:
1493:
29:
1743:
311:
2798:
1821:
1092:
1505:
1501:
851:
45:
8:
1997:
1847:
1583:
1504:. These hierarchies reveal many relationships between definability in this structure and
2250:
2174:
2071:
1497:
1147:
983:
804:
676:
290:
203:
183:
159:
139:
17:
1716:. In conjunction with a formula that defines the additive inverse of a real number in
49:
1489:
2605:
itself. In contrast, there are infinitely many definable sets of pairs (or indeed
2774:
2803:
2778:
221:
2792:
1993:
664:
2764:
2013:
2001:
1421:
2012:
An important result about definable sets is that they are preserved under
1996:
of solutions to polynomial equalities and inequalities; these are called
1587:
1427:
1938:{\displaystyle {\mathcal {R}}^{\leq }=(\mathbb {R} ,0,1,+,\cdot ,\leq )}
1420:
consisting of the integers with the usual ordering (see the section on
774:(with parameters) if its graph is definable (with those parameters) in
744:
2000:. Generalizing this property of the real line leads to the study of
907:
663:
The bracket notation here indicates the semantic evaluation of the
1373:
177:
1481:{\displaystyle {\mathcal {N}}=(\mathbb {N} ,+,\cdot ,<)}
1575:{\displaystyle {\mathcal {R}}=(\mathbb {R} ,0,1,+,\cdot )}
2455:{\displaystyle (\pi (a_{1}),\ldots ,\pi (a_{m}))\in A.}
1428:
The natural numbers with their arithmetical operations
747:(that is, with no parameters in the defining formula).
2783:
Mathematical Logic: The
Berkeley Undergraduate Course
2703:
2647:
2619:
2589:
2565:
2541:
2517:
2473:
2389:
2331:
2273:
2253:
2229:
2197:
2177:
2153:
2120:
2094:
2074:
2050:
2026:
1963:
1876:
1850:
1824:
1790:
1766:
1746:
1722:
1683:
1655:
1639:{\displaystyle \varphi =\exists y(y\cdot y\equiv x).}
1599:
1526:
1438:
1382:
1177:
1150:
1121:
1095:
1042:
1006:
986:
962:
918:
880:
854:
827:
807:
780:
756:
725:
699:
679:
567:
509:
453:
401:
314:
293:
269:
234:
206:
186:
162:
142:
118:
94:
70:
2717:
2689:
2629:
2597:
2575:
2551:
2527:
2504:{\displaystyle {\mathcal {Z}}=(\mathbb {Z} ,<)}
2503:
2454:
2375:
2311:
2259:
2239:
2215:
2183:
2163:
2139:
2106:
2080:
2060:
2036:
1980:
1937:
1862:
1836:
1810:
1776:
1752:
1732:
1708:
1669:
1638:
1574:
1480:
1413:{\displaystyle {\mathcal {Z}}=(\mathbb {Z} ,<)}
1412:
1361:
1156:
1136:
1107:
1078:
1021:
992:
972:
949:{\displaystyle {\mathcal {N}}=(\mathbb {N} ,<)}
948:
890:
866:
837:
813:
790:
766:
735:
709:
685:
653:
553:
491:
439:
387:
299:
279:
253:
212:
192:
168:
148:
128:
104:
80:
2007:
2790:
1079:{\displaystyle \varphi =\neg \exists y(y<x),}
908:The natural numbers with only the order relation
1029:stating that there exist no elements less than
654:{\displaystyle {\mathcal {M}}\models \varphi .}
1709:{\displaystyle {\mathcal {R}}\models \varphi }
2684:
2648:
1515:
1372:In contrast, one cannot define any specific
861:
855:
2376:{\displaystyle (a_{1},\ldots ,a_{m})\in A}
554:{\displaystyle (a_{1},\ldots ,a_{m})\in A}
2711:
2591:
2583:without parameters are the empty set and
2488:
1898:
1804:
1663:
1541:
1453:
1397:
933:
2312:{\displaystyle a_{1},\ldots ,a_{m}\in M}
492:{\displaystyle a_{1},\ldots ,a_{m}\in M}
440:{\displaystyle b_{1},\ldots ,b_{n}\in X}
1870:is nonnegative. The enlarged structure
2791:
2641:= 2) Boolean combinations of the sets
1981:{\displaystyle {\mathcal {R}}^{\leq }}
308:if and only if there exists a formula
2728:
1496:. If the structure is considered in
1376:without parameters in the structure
2769:Principles of Mathematical Analysis
2690:{\displaystyle \{(a,b)\mid a-b=m\}}
1811:{\displaystyle a,b\in \mathbb {R} }
1582:be the structure consisting of the
13:
2752:Fundamentals of Mathematical Logic
2622:
2568:
2544:
2520:
2476:
2232:
2156:
2053:
2029:
1967:
1880:
1769:
1725:
1686:
1606:
1529:
1441:
1385:
1279:
1200:
1184:
1052:
1049:
965:
921:
883:
830:
783:
759:
728:
702:
570:
272:
121:
97:
73:
14:
2820:
2718:{\displaystyle m\in \mathbb {Z} }
1670:{\displaystyle a\in \mathbb {R} }
1144:stating that there exist exactly
2140:{\displaystyle A\subseteq M^{m}}
1760:to define the usual ordering in
254:{\displaystyle A\subseteq M^{m}}
980:without parameters. The number
52:can be defined with or without
2663:
2651:
2630:{\displaystyle {\mathcal {Z}}}
2576:{\displaystyle {\mathcal {Z}}}
2552:{\displaystyle {\mathcal {Z}}}
2528:{\displaystyle {\mathcal {Z}}}
2498:
2484:
2440:
2437:
2424:
2409:
2396:
2390:
2364:
2332:
2240:{\displaystyle {\mathcal {M}}}
2207:
2164:{\displaystyle {\mathcal {M}}}
2061:{\displaystyle {\mathcal {L}}}
2037:{\displaystyle {\mathcal {M}}}
2008:Invariance under automorphisms
1992:. Thus the definable sets are
1932:
1894:
1777:{\displaystyle {\mathcal {R}}}
1733:{\displaystyle {\mathcal {R}}}
1703:
1697:
1677:is nonnegative if and only if
1630:
1612:
1569:
1537:
1508:, and are also of interest in
1475:
1449:
1407:
1393:
1356:
1353:
1350:
1300:
1297:
1285:
1219:
1131:
1125:
1070:
1058:
1016:
1010:
973:{\displaystyle {\mathcal {N}}}
943:
929:
891:{\displaystyle {\mathcal {M}}}
838:{\displaystyle {\mathcal {M}}}
791:{\displaystyle {\mathcal {M}}}
767:{\displaystyle {\mathcal {M}}}
736:{\displaystyle {\mathcal {M}}}
710:{\displaystyle {\mathcal {M}}}
645:
581:
542:
510:
382:
318:
280:{\displaystyle {\mathcal {M}}}
129:{\displaystyle {\mathcal {L}}}
105:{\displaystyle {\mathcal {M}}}
81:{\displaystyle {\mathcal {L}}}
1:
2771:, 3rd. ed. McGraw-Hill, 1976.
2759:Model Theory: An Introduction
2744:
2613:> 1) of elements of
59:
2737:is used to characterize the
2598:{\displaystyle \mathbb {Z} }
2107:{\displaystyle X\subseteq M}
1492:, and are classified in the
40:whose elements satisfy some
7:
2216:{\displaystyle \pi :M\to M}
1137:{\displaystyle \varphi (x)}
1022:{\displaystyle \varphi (x)}
902:
750:A function is definable in
88:be a first-order language,
10:
2825:
2511:above, any translation of
1115:is defined by the formula
1000:is defined by the formula
1516:The field of real numbers
845:(with parameters) if the
743:with parameters from the
2739:elementary substructures
2247:that is the identity on
1753:{\displaystyle \varphi }
898:(with those parameters).
388:{\displaystyle \varphi }
2068:-structure with domain
1837:{\displaystyle a\leq b}
136:-structure with domain
2741:of a given structure.
2719:
2691:
2631:
2609:-tuples for any fixed
2599:
2577:
2553:
2529:
2505:
2456:
2377:
2313:
2261:
2241:
2223:be an automorphism of
2217:
2185:
2165:
2141:
2108:
2082:
2062:
2038:
1990:quantifier elimination
1982:
1947:definitional extension
1939:
1864:
1838:
1812:
1778:
1754:
1734:
1710:
1671:
1640:
1576:
1510:descriptive set theory
1494:arithmetical hierarchy
1482:
1414:
1363:
1158:
1138:
1109:
1108:{\displaystyle n>0}
1080:
1023:
994:
974:
950:
892:
868:
839:
815:
792:
768:
737:
719:if it is definable in
711:
687:
655:
555:
493:
441:
389:
301:
281:
255:
214:
194:
170:
150:
130:
106:
82:
2720:
2692:
2632:
2600:
2578:
2554:
2530:
2506:
2457:
2378:
2314:
2262:
2242:
2218:
2186:
2171:with parameters from
2166:
2142:
2109:
2083:
2063:
2039:
1983:
1940:
1865:
1839:
1813:
1779:
1755:
1735:
1711:
1672:
1641:
1577:
1483:
1415:
1364:
1159:
1139:
1110:
1089:and a natural number
1081:
1024:
995:
975:
951:
893:
869:
867:{\displaystyle \{a\}}
840:
816:
793:
769:
738:
712:
688:
656:
556:
494:
442:
390:
302:
287:with parameters from
282:
256:
215:
195:
171:
151:
131:
107:
83:
48:of that structure. A
2701:
2645:
2617:
2587:
2563:
2539:
2515:
2471:
2387:
2329:
2271:
2251:
2227:
2195:
2175:
2151:
2118:
2092:
2072:
2048:
2024:
1994:Boolean combinations
1961:
1874:
1848:
1822:
1788:
1764:
1744:
1720:
1681:
1653:
1597:
1524:
1506:computability theory
1502:analytical hierarchy
1436:
1380:
1175:
1148:
1119:
1093:
1040:
1004:
984:
960:
916:
878:
852:
825:
805:
778:
754:
723:
697:
677:
565:
507:
451:
399:
312:
291:
267:
232:
204:
184:
160:
140:
116:
92:
68:
46:first-order language
2775:Slaman, Theodore A.
2754:, A K Peters, 2005.
1998:semi-algebraic sets
1863:{\displaystyle b-a}
1164:elements less than
2809:Mathematical logic
2735:Tarski–Vaught test
2729:Additional results
2715:
2687:
2627:
2595:
2573:
2549:
2525:
2501:
2452:
2373:
2309:
2257:
2237:
2213:
2181:
2161:
2137:
2104:
2078:
2058:
2034:
1978:
1935:
1860:
1834:
1808:
1774:
1750:
1730:
1706:
1667:
1636:
1572:
1498:second-order logic
1478:
1410:
1359:
1154:
1134:
1105:
1076:
1019:
990:
970:
946:
888:
864:
835:
811:
788:
764:
733:
717:without parameters
707:
683:
651:
551:
489:
447:such that for all
437:
385:
297:
277:
251:
210:
190:
166:
146:
126:
102:
78:
18:mathematical logic
2761:, Springer, 2002.
2260:{\displaystyle X}
2184:{\displaystyle X}
2081:{\displaystyle M}
1490:arithmetical sets
1157:{\displaystyle n}
993:{\displaystyle 0}
814:{\displaystyle a}
686:{\displaystyle A}
300:{\displaystyle X}
213:{\displaystyle m}
193:{\displaystyle M}
169:{\displaystyle X}
149:{\displaystyle M}
2816:
2724:
2722:
2721:
2716:
2714:
2696:
2694:
2693:
2688:
2636:
2634:
2633:
2628:
2626:
2625:
2604:
2602:
2601:
2596:
2594:
2582:
2580:
2579:
2574:
2572:
2571:
2558:
2556:
2555:
2550:
2548:
2547:
2534:
2532:
2531:
2526:
2524:
2523:
2510:
2508:
2507:
2502:
2491:
2480:
2479:
2461:
2459:
2458:
2453:
2436:
2435:
2408:
2407:
2382:
2380:
2379:
2374:
2363:
2362:
2344:
2343:
2318:
2316:
2315:
2310:
2302:
2301:
2283:
2282:
2266:
2264:
2263:
2258:
2246:
2244:
2243:
2238:
2236:
2235:
2222:
2220:
2219:
2214:
2190:
2188:
2187:
2182:
2170:
2168:
2167:
2162:
2160:
2159:
2146:
2144:
2143:
2138:
2136:
2135:
2113:
2111:
2110:
2105:
2087:
2085:
2084:
2079:
2067:
2065:
2064:
2059:
2057:
2056:
2043:
2041:
2040:
2035:
2033:
2032:
1987:
1985:
1984:
1979:
1977:
1976:
1971:
1970:
1944:
1942:
1941:
1936:
1901:
1890:
1889:
1884:
1883:
1869:
1867:
1866:
1861:
1843:
1841:
1840:
1835:
1817:
1815:
1814:
1809:
1807:
1783:
1781:
1780:
1775:
1773:
1772:
1759:
1757:
1756:
1751:
1739:
1737:
1736:
1731:
1729:
1728:
1715:
1713:
1712:
1707:
1690:
1689:
1676:
1674:
1673:
1668:
1666:
1645:
1643:
1642:
1637:
1581:
1579:
1578:
1573:
1544:
1533:
1532:
1487:
1485:
1484:
1479:
1456:
1445:
1444:
1419:
1417:
1416:
1411:
1400:
1389:
1388:
1368:
1366:
1365:
1360:
1349:
1348:
1318:
1317:
1269:
1268:
1244:
1243:
1231:
1230:
1218:
1217:
1196:
1195:
1163:
1161:
1160:
1155:
1143:
1141:
1140:
1135:
1114:
1112:
1111:
1106:
1085:
1083:
1082:
1077:
1028:
1026:
1025:
1020:
999:
997:
996:
991:
979:
977:
976:
971:
969:
968:
955:
953:
952:
947:
936:
925:
924:
897:
895:
894:
889:
887:
886:
874:is definable in
873:
871:
870:
865:
844:
842:
841:
836:
834:
833:
821:is definable in
820:
818:
817:
812:
797:
795:
794:
789:
787:
786:
773:
771:
770:
765:
763:
762:
742:
740:
739:
734:
732:
731:
716:
714:
713:
708:
706:
705:
693:is definable in
692:
690:
689:
684:
660:
658:
657:
652:
644:
643:
625:
624:
612:
611:
593:
592:
574:
573:
560:
558:
557:
552:
541:
540:
522:
521:
498:
496:
495:
490:
482:
481:
463:
462:
446:
444:
443:
438:
430:
429:
411:
410:
394:
392:
391:
386:
381:
380:
362:
361:
349:
348:
330:
329:
306:
304:
303:
298:
286:
284:
283:
278:
276:
275:
260:
258:
257:
252:
250:
249:
219:
217:
216:
211:
199:
197:
196:
191:
175:
173:
172:
167:
155:
153:
152:
147:
135:
133:
132:
127:
125:
124:
111:
109:
108:
103:
101:
100:
87:
85:
84:
79:
77:
76:
2824:
2823:
2819:
2818:
2817:
2815:
2814:
2813:
2789:
2788:
2779:Woodin, W. Hugh
2757:Marker, David.
2750:Hinman, Peter.
2747:
2731:
2710:
2702:
2699:
2698:
2646:
2643:
2642:
2637:: (in the case
2621:
2620:
2618:
2615:
2614:
2590:
2588:
2585:
2584:
2567:
2566:
2564:
2561:
2560:
2543:
2542:
2540:
2537:
2536:
2519:
2518:
2516:
2513:
2512:
2487:
2475:
2474:
2472:
2469:
2468:
2431:
2427:
2403:
2399:
2388:
2385:
2384:
2383:if and only if
2358:
2354:
2339:
2335:
2330:
2327:
2326:
2297:
2293:
2278:
2274:
2272:
2269:
2268:
2267:. Then for all
2252:
2249:
2248:
2231:
2230:
2228:
2225:
2224:
2196:
2193:
2192:
2176:
2173:
2172:
2155:
2154:
2152:
2149:
2148:
2131:
2127:
2119:
2116:
2115:
2093:
2090:
2089:
2073:
2070:
2069:
2052:
2051:
2049:
2046:
2045:
2028:
2027:
2025:
2022:
2021:
2010:
1972:
1966:
1965:
1964:
1962:
1959:
1958:
1897:
1885:
1879:
1878:
1877:
1875:
1872:
1871:
1849:
1846:
1845:
1844:if and only if
1823:
1820:
1819:
1803:
1789:
1786:
1785:
1768:
1767:
1765:
1762:
1761:
1745:
1742:
1741:
1724:
1723:
1721:
1718:
1717:
1685:
1684:
1682:
1679:
1678:
1662:
1654:
1651:
1650:
1598:
1595:
1594:
1540:
1528:
1527:
1525:
1522:
1521:
1518:
1452:
1440:
1439:
1437:
1434:
1433:
1430:
1396:
1384:
1383:
1381:
1378:
1377:
1338:
1334:
1313:
1309:
1258:
1254:
1239:
1235:
1226:
1222:
1207:
1203:
1191:
1187:
1176:
1173:
1172:
1149:
1146:
1145:
1120:
1117:
1116:
1094:
1091:
1090:
1041:
1038:
1037:
1005:
1002:
1001:
985:
982:
981:
964:
963:
961:
958:
957:
932:
920:
919:
917:
914:
913:
910:
905:
882:
881:
879:
876:
875:
853:
850:
849:
829:
828:
826:
823:
822:
806:
803:
802:
782:
781:
779:
776:
775:
758:
757:
755:
752:
751:
727:
726:
724:
721:
720:
701:
700:
698:
695:
694:
678:
675:
674:
667:in the formula.
639:
635:
620:
616:
607:
603:
588:
584:
569:
568:
566:
563:
562:
561:if and only if
536:
532:
517:
513:
508:
505:
504:
477:
473:
458:
454:
452:
449:
448:
425:
421:
406:
402:
400:
397:
396:
376:
372:
357:
353:
344:
340:
325:
321:
313:
310:
309:
292:
289:
288:
271:
270:
268:
265:
264:
245:
241:
233:
230:
229:
205:
202:
201:
185:
182:
181:
161:
158:
157:
141:
138:
137:
120:
119:
117:
114:
113:
96:
95:
93:
90:
89:
72:
71:
69:
66:
65:
62:
12:
11:
5:
2822:
2812:
2811:
2806:
2801:
2787:
2786:
2785:. Spring 2006.
2772:
2762:
2755:
2746:
2743:
2730:
2727:
2713:
2709:
2706:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2624:
2593:
2570:
2546:
2522:
2500:
2497:
2494:
2490:
2486:
2483:
2478:
2465:
2464:
2463:
2462:
2451:
2448:
2445:
2442:
2439:
2434:
2430:
2426:
2423:
2420:
2417:
2414:
2411:
2406:
2402:
2398:
2395:
2392:
2372:
2369:
2366:
2361:
2357:
2353:
2350:
2347:
2342:
2338:
2334:
2321:
2320:
2308:
2305:
2300:
2296:
2292:
2289:
2286:
2281:
2277:
2256:
2234:
2212:
2209:
2206:
2203:
2200:
2180:
2158:
2134:
2130:
2126:
2123:
2103:
2100:
2097:
2077:
2055:
2031:
2009:
2006:
1975:
1969:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1900:
1896:
1893:
1888:
1882:
1859:
1856:
1853:
1833:
1830:
1827:
1806:
1802:
1799:
1796:
1793:
1771:
1749:
1740:, one can use
1727:
1705:
1702:
1699:
1696:
1693:
1688:
1665:
1661:
1658:
1647:
1646:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1543:
1539:
1536:
1531:
1517:
1514:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1455:
1451:
1448:
1443:
1429:
1426:
1409:
1406:
1403:
1399:
1395:
1392:
1387:
1370:
1369:
1358:
1355:
1352:
1347:
1344:
1341:
1337:
1333:
1330:
1327:
1324:
1321:
1316:
1312:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1267:
1264:
1261:
1257:
1253:
1250:
1247:
1242:
1238:
1234:
1229:
1225:
1221:
1216:
1213:
1210:
1206:
1202:
1199:
1194:
1190:
1186:
1183:
1180:
1153:
1133:
1130:
1127:
1124:
1104:
1101:
1098:
1087:
1086:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1018:
1015:
1012:
1009:
989:
967:
945:
942:
939:
935:
931:
928:
923:
909:
906:
904:
901:
900:
899:
885:
863:
860:
857:
832:
810:
799:
785:
761:
748:
730:
704:
682:
669:
668:
665:free variables
661:
650:
647:
642:
638:
634:
631:
628:
623:
619:
615:
610:
606:
602:
599:
596:
591:
587:
583:
580:
577:
572:
550:
547:
544:
539:
535:
531:
528:
525:
520:
516:
512:
501:
500:
488:
485:
480:
476:
472:
469:
466:
461:
457:
436:
433:
428:
424:
420:
417:
414:
409:
405:
384:
379:
375:
371:
368:
365:
360:
356:
352:
347:
343:
339:
336:
333:
328:
324:
320:
317:
296:
274:
248:
244:
240:
237:
222:natural number
209:
189:
165:
145:
123:
99:
75:
61:
58:
9:
6:
4:
3:
2:
2821:
2810:
2807:
2805:
2802:
2800:
2797:
2796:
2794:
2784:
2780:
2776:
2773:
2770:
2766:
2765:Rudin, Walter
2763:
2760:
2756:
2753:
2749:
2748:
2742:
2740:
2736:
2726:
2707:
2704:
2681:
2678:
2675:
2672:
2669:
2666:
2660:
2657:
2654:
2640:
2612:
2608:
2495:
2492:
2481:
2449:
2446:
2443:
2432:
2428:
2421:
2418:
2415:
2412:
2404:
2400:
2393:
2370:
2367:
2359:
2355:
2351:
2348:
2345:
2340:
2336:
2325:
2324:
2323:
2322:
2306:
2303:
2298:
2294:
2290:
2287:
2284:
2279:
2275:
2254:
2210:
2204:
2201:
2198:
2178:
2147:definable in
2132:
2128:
2124:
2121:
2101:
2098:
2095:
2075:
2019:
2018:
2017:
2015:
2014:automorphisms
2005:
2003:
1999:
1995:
1991:
1973:
1956:
1951:
1948:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1891:
1886:
1857:
1854:
1851:
1831:
1828:
1825:
1800:
1797:
1794:
1791:
1747:
1700:
1694:
1691:
1659:
1656:
1633:
1627:
1624:
1621:
1618:
1615:
1609:
1603:
1600:
1593:
1592:
1591:
1589:
1585:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1534:
1513:
1511:
1507:
1503:
1499:
1495:
1491:
1472:
1469:
1466:
1463:
1460:
1457:
1446:
1425:
1423:
1422:automorphisms
1404:
1401:
1390:
1375:
1345:
1342:
1339:
1335:
1331:
1328:
1325:
1322:
1319:
1314:
1310:
1306:
1303:
1294:
1291:
1288:
1282:
1276:
1273:
1270:
1265:
1262:
1259:
1255:
1251:
1248:
1245:
1240:
1236:
1232:
1227:
1223:
1214:
1211:
1208:
1204:
1197:
1192:
1188:
1181:
1178:
1171:
1170:
1169:
1167:
1151:
1128:
1122:
1102:
1099:
1096:
1073:
1067:
1064:
1061:
1055:
1046:
1043:
1036:
1035:
1034:
1032:
1013:
1007:
987:
940:
937:
926:
858:
848:
847:singleton set
808:
800:
749:
746:
718:
680:
671:
670:
666:
662:
648:
640:
636:
632:
629:
626:
621:
617:
613:
608:
604:
600:
597:
594:
589:
585:
578:
575:
548:
545:
537:
533:
529:
526:
523:
518:
514:
503:
502:
486:
483:
478:
474:
470:
467:
464:
459:
455:
434:
431:
426:
422:
418:
415:
412:
407:
403:
395:and elements
377:
373:
369:
366:
363:
358:
354:
350:
345:
341:
337:
334:
331:
326:
322:
315:
307:
294:
263:definable in
246:
242:
238:
235:
227:
226:
225:
223:
207:
187:
179:
163:
143:
57:
55:
51:
47:
43:
39:
35:
31:
27:
23:
22:definable set
19:
2799:Model theory
2782:
2768:
2758:
2751:
2732:
2638:
2610:
2606:
2466:
2011:
2002:o-minimality
1952:
1945:is called a
1648:
1588:real numbers
1519:
1431:
1371:
1165:
1088:
1030:
911:
673:
262:
63:
53:
25:
21:
15:
801:An element
2793:Categories
2745:References
60:Definition
54:parameters
2708:∈
2673:−
2667:∣
2444:∈
2422:π
2416:…
2394:π
2368:∈
2349:…
2304:∈
2288:…
2208:→
2199:π
2125:⊆
2099:⊆
1974:≤
1930:≤
1924:⋅
1887:≤
1855:−
1829:≤
1801:∈
1748:φ
1695:φ
1692:⊨
1660:∈
1649:Thus any
1625:≡
1619:⋅
1607:∃
1601:φ
1567:⋅
1467:⋅
1343:−
1332:≡
1326:∨
1323:⋯
1320:∨
1307:≡
1298:→
1280:∀
1277:∧
1263:−
1252:∧
1249:⋯
1246:∧
1212:−
1201:∃
1198:⋯
1185:∃
1179:φ
1123:φ
1053:∃
1050:¬
1044:φ
1008:φ
745:empty set
630:…
598:…
579:φ
576:⊨
546:∈
527:…
484:∈
468:…
432:∈
416:…
367:…
335:…
316:φ
239:⊆
38:structure
1424:below).
903:Examples
224:. Then:
176:a fixed
30:relation
1374:integer
44:in the
42:formula
32:on the
2191:. Let
2114:, and
2044:be an
1955:theory
1818:, set
1784:: for
672:A set
228:A set
200:, and
178:subset
34:domain
24:is an
2804:Logic
1584:field
36:of a
28:-ary
2777:and
2733:The
2697:for
2496:<
2020:Let
1988:has
1953:The
1520:Let
1473:<
1432:Let
1405:<
1292:<
1271:<
1233:<
1100:>
1065:<
941:<
912:Let
64:Let
20:, a
1957:of
1586:of
261:is
180:of
112:an
50:set
16:In
2795::
2781:.
2767:.
2088:,
2016:.
2004:.
1512:.
1168::
1033::
220:a
156:,
2712:Z
2705:m
2685:}
2682:m
2679:=
2676:b
2670:a
2664:)
2661:b
2658:,
2655:a
2652:(
2649:{
2639:n
2623:Z
2611:n
2607:n
2592:Z
2569:Z
2545:Z
2521:Z
2499:)
2493:,
2489:Z
2485:(
2482:=
2477:Z
2450:.
2447:A
2441:)
2438:)
2433:m
2429:a
2425:(
2419:,
2413:,
2410:)
2405:1
2401:a
2397:(
2391:(
2371:A
2365:)
2360:m
2356:a
2352:,
2346:,
2341:1
2337:a
2333:(
2319:,
2307:M
2299:m
2295:a
2291:,
2285:,
2280:1
2276:a
2255:X
2233:M
2211:M
2205:M
2202::
2179:X
2157:M
2133:m
2129:M
2122:A
2102:M
2096:X
2076:M
2054:L
2030:M
1968:R
1933:)
1927:,
1921:,
1918:+
1915:,
1912:1
1909:,
1906:0
1903:,
1899:R
1895:(
1892:=
1881:R
1858:a
1852:b
1832:b
1826:a
1805:R
1798:b
1795:,
1792:a
1770:R
1726:R
1704:]
1701:a
1698:[
1687:R
1664:R
1657:a
1634:.
1631:)
1628:x
1622:y
1616:y
1613:(
1610:y
1604:=
1570:)
1564:,
1561:+
1558:,
1555:1
1552:,
1549:0
1546:,
1542:R
1538:(
1535:=
1530:R
1476:)
1470:,
1464:,
1461:+
1458:,
1454:N
1450:(
1447:=
1442:N
1408:)
1402:,
1398:Z
1394:(
1391:=
1386:Z
1357:)
1354:)
1351:)
1346:1
1340:n
1336:x
1329:y
1315:0
1311:x
1304:y
1301:(
1295:x
1289:y
1286:(
1283:y
1274:x
1266:1
1260:n
1256:x
1241:1
1237:x
1228:0
1224:x
1220:(
1215:1
1209:n
1205:x
1193:0
1189:x
1182:=
1166:x
1152:n
1132:)
1129:x
1126:(
1103:0
1097:n
1074:,
1071:)
1068:x
1062:y
1059:(
1056:y
1047:=
1031:x
1017:)
1014:x
1011:(
988:0
966:N
944:)
938:,
934:N
930:(
927:=
922:N
884:M
862:}
859:a
856:{
831:M
809:a
798:.
784:M
760:M
729:M
703:M
681:A
649:.
646:]
641:n
637:b
633:,
627:,
622:1
618:b
614:,
609:m
605:a
601:,
595:,
590:1
586:a
582:[
571:M
549:A
543:)
538:m
534:a
530:,
524:,
519:1
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511:(
499:,
487:M
479:m
475:a
471:,
465:,
460:1
456:a
435:X
427:n
423:b
419:,
413:,
408:1
404:b
383:]
378:n
374:y
370:,
364:,
359:1
355:y
351:,
346:m
342:x
338:,
332:,
327:1
323:x
319:[
295:X
273:M
247:m
243:M
236:A
208:m
188:M
164:X
144:M
122:L
98:M
74:L
26:n
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