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because that element would be definable and so the extension would not be elementary. In reality, by compactness, we can realize the type in some elementary extension. As the property "There exists a unique element x such that for all y we have x not equal to s(y)" is first order and true in true arithemetic (or PA if you want), it will be true in the elementary extension given by compactness. So, I'm guessing your confusion was just with you thinking the non-example was an example.
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haven't read
Wgunther's example yet, but I think for a first example it is probably just as well to stick to ordering only, as long as it's very clear to the reader that that's what we're doing. (In some sense I belive there is no "explicit" example of a proper elementary extension of the naturals with plus and times — what "explicit" means is usually not very well specified, which is a recurring problem in WP articles where the axiom of choice comes up.) --
2215:
just assumed that Z is disjoint from N to make it just a union. Then defining the ordering in the extended structure is a bit simpler to write down. Enderton would probably be a good enough source. Chapter 3 talks about these models of the natural number. 3.2 talks about (essentially) our specific example, and how the nonstandard models of N with < and S are the same as with just N with S, which is just N with any Z chains you want.
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The notion of an elementary extension apparently includes also the arithmetic operations. My impression is that constructing such an extension is not a simple matter and may involve the axiom of choice or some weaker form of it. The example you presented seems to simple to have a chance of being an
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Right, our model isn't so complicated. It's just the theory of omega with membership. It is finitely axiomatizable and decidable. It actually would still be even if you added the successor function for ordinals too. And yes, the disjoint union of N and Z does the trick. That is what my example is, I
2184:
Which operations are included in the notion depends on the language. If you're working in a language whose only non-logical symbol is the ordering, then you only have to worry about the ordering. If the language includes plus and times, then you have to worry about addition and multiplication. I
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As far as I can tell, there is no example where an elementary extension is constructed. Just one that says "Here's a type, if we could realize this in some elementary extension this is what we'd end up with." And then a warning that this isn't the same as just adding one element to realize the type
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Given a complete n-type p one can ask if there is a model of the theory that omits p, in other words there is no n-tuple in the model which realizes p. If p is an isolated point in the Stone space, i.e., if {p} is an open set. It is easy to see that every model realizes p (at least if the theory is
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This is really unhelpful to me and I know at least basic model theory. There needs to be an explanation of the relationship between topology and model theory - the link to the topological notion of an open point doesn't help me to understand how an element of a model's domain can be isolated. What
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1. Yes, I have corrected the example and noted it on the discussion page because I felt that a little explanation is in order. Sorry, if it confused anyone. I still think that the definitions could be cleared up a bit (preferably by an expert). 2. The lecture on logic that we have had (at
Charles
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is often used to mean complete type, and partial type is used for the general notion. Unfortunately there is not widespread agreement on this, different papers have a different convention. In my experience, the latter view of types being complete is more widespread, as in many areas one need only
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I'm quite confident that the example given is not an elementary extension. If the language only contains the ordering, then adding the integers on top of the naturals would make it an elementary extension, as no new element without immediate predecessors would be added. If the language contains
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I agree that the article as it stands lacks sufficient context, and I've tried to add a better lede, but what I'm missing is _why_ types are significant, or some kind of analogy as to what they're _like_ -- for example it was explained to me that the diagram of a structure is 'analogous' to the
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This line is hard to understand because it is complicated and imprecise. The way it is stated, it can be interpreted as "a type is just a set of satisfiable formulas". Is the "sequence of elements" fixed for a given type? Is a type a set of _all_ formulas that are true of a sequence?
1931:; that is what's really going on anyway, and there's a lot of expense for novices reading this as written I'm sure. The language I think is relatively clear ("take the natural numbers with the ordering"). But, there's really no cost in making it more explicit (something like "take
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is just another copy of the naturals. This is intentionally suggestive. The article even makes a point that you can't just add finitely many elements. The ordering though is not the standard ordering. I guess that could be clearer in the article, or we could just call it
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multiplication table for a group -- I cannot find any similar useful and correct analogy for types that would provide motivation for why we study them. The rest of the article does (I think) a good job explaining what types are and what you can do with them. Help!
550:. 2. In general a (partial) type is not required to be a filter. Any (partial) type has however its set of concequences which is a filter of the set of formulae. I prefer to use (partial) types this way, it is not however totally standard. Something else: The word
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One sense of "explicit" is provably impossible by
Tenenbaum's theorem. At any rate, your point should be clearly spelled out in the article. If all one is looking for is the order, than the disjoint union of N and Z seems to do the trick, no?
2625:
Anyway, wouldn't it make more sense to talk about the basic idea of types in the first sentence in order to build up some intuition before giving a formal definition? That would mitigate such imprecisions intrinsic to natural language. --
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I appreciate a lot the content of this article, but the language looks pretty obscure and at times just awkward. I wonder if somebody could improve it. Not being a specialist in type theory, I'd rather abstain from doing it myself.
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2025:, you're suggesting that addition and multiplication are around, whereas if I now understand correctly, you just want the order. In that case it would probably be best to call it ω, which is the order type of the naturals.
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confused me, and probably will confuse other people as well. Also, there's no mention of the language of the model, and as only the ordering is mentioned, I think most people will assume that the language is just
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I'm not sure why you think the other model satisfies that sentence. If you're correct about that, then of course you're also correct that it's not an elementary extension, but I don't think you are.
1848:. In this case, just putting a copy of the integers on top of the naturals suffices to give an elementary extension, and the ultrapower construction or the integers times the rationals is not needed.
1965:, the natural numbers with it's standard well ordering"). You're certainly right that this example can be made clearer and encyclopedic. I'll try to change it around later today if no one else does.
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Please explain what you mean by "the example given". As far as I can tell, 24.131.192.199 is correct; no well-specified "example" is given. In particular, nothing in the text says what
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are meant as a type in model of rationals then we would need examples of two conditions that are also met by some rational number. Or maybe what is meant by the example is a type in the
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complete). The omitting types theorem says that conversely if p is not isolated then there is a countable model omitting p (provided that the language is countable).
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I rewrote the example. I think it's more understandable now and the structures are less nebulous. People can feel free to revert if they think it was better before.
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has in common with the naturals is cardinality; that is, you really just mean some countably infinite set. Then an "intentionally suggestive" notation of
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is "just a copy" of the naturals, you seem to be allowing yourself to re-interpret the non-logical symbols on it. In that case, the only thing
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No, I'm not confusing elementary equivalence with order isomorphism. :) The article reads: "This can be remedied by a new structure,
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And if this is meant to be some kind of "nonexample" of elementary equivalence, then this is not clear from the article.
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of arithmetics, because being consistent with a model of a theory and being consistent with a theory are differnt things?
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is supposed to be; it just says that it's the part of the elementary extension that lives above the standard naturals. --
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and the ordering on it by attaching two copies of the naturals end to end. After the construction it is claimed that "
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being a copy of the naturals. I see your point now, and everything makes sense. Denoting the added elements by
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addition, multiplication and so forth, then constructing an elementary extension would not be so easy.
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all have the property that there is only one element that does not have an immediate predecessor. --
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1534:)". But these two structures are not elementarily equivalent, as the formula I wrote above shows.
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of the natural numbers with order is not actually elementary, as the model satisfies the sentence
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There should also be a link/mention of saturation since that concept is defined in terms of types
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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for you to conclude that the structures are not elementarily equivalent. Are you assuming that
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Hmm, honestly I don't think this is a good idea. First of all, by calling the standard model
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No, the article does not in fact do that. I suspect you're assuming that
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are elementarily equivalent is not true. Now for example the formula
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Two replies to the above. 1. As regards the example, it now reads
1711:. This is not the case. One can arrange for it to be a copy of
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I am not an expert on this, so a second opinion would be helpful.
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exactly are the conditions under which a type is omitted?
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has a least element? It doesn't assert that anywhere. --
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1035:By the example given I mean the model
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2489:{\displaystyle b}
2345:Given a language
1767:Ah, I did assume
1567:comment added by
1540:comment added by
1308:say, with domain
1189:comment added by
976:comment added by
682:comment added by
660:
643:comment added by
547:0\land y^{2}: -->
540:0\land y^{2}: -->
497:2\implies }": -->
496:0\land y^{2}: -->
464:
242:
215:
185:
169:Example of a type
166:
165:
162:
161:
158:
157:
2692:
2617:
2615:
2614:
2609:
2607:
2606:
2588:
2587:
2575:
2574:
2558:
2556:
2555:
2550:
2548:
2547:
2529:
2528:
2516:
2515:
2495:
2493:
2492:
2487:
2471:
2469:
2468:
2463:
2461:
2460:
2447:
2445:
2444:
2439:
2434:
2433:
2415:
2414:
2402:
2401:
2376:
2374:
2373:
2368:
2366:
2365:
2337:Given paragraph
2328:
2326:
2325:
2320:
2318:
2317:
2108:
2106:
2105:
2100:
2098:
2097:
2081:
2079:
2078:
2073:
2071:
2070:
2054:
2052:
2051:
2046:
2044:
2043:
2024:
2022:
2021:
2016:
2014:
2002:
2000:
1999:
1994:
1992:
1991:
1964:
1962:
1961:
1956:
1945:
1930:
1928:
1927:
1922:
1920:
1907:
1905:
1904:
1899:
1897:
1893:
1847:
1845:
1844:
1839:
1820:
1818:
1817:
1812:
1810:
1806:
1793:
1791:
1790:
1785:
1783:
1779:
1732:
1730:
1729:
1724:
1722:
1710:
1708:
1707:
1702:
1700:
1688:
1686:
1685:
1680:
1678:
1674:
1635:
1633:
1632:
1627:
1625:
1621:
1608:
1606:
1605:
1600:
1598:
1594:
1579:
1552:
1533:
1531:
1530:
1525:
1520:
1502:
1500:
1499:
1494:
1492:
1491:
1478:
1476:
1475:
1470:
1468:
1467:
1462:
1461:
1447:
1445:
1444:
1439:
1437:
1433:
1424:
1414:
1400:
1399:
1383:
1381:
1380:
1375:
1367:
1363:
1354:
1342:
1340:
1339:
1334:
1332:
1328:
1319:
1307:
1305:
1304:
1299:
1297:
1296:
1291:
1290:
1256:
1254:
1253:
1248:
1246:
1245:
1232:
1230:
1229:
1224:
1222:
1221:
1220:
1201:
1182:
1180:
1179:
1174:
1169:
1151:
1149:
1148:
1143:
1141:
1140:
1127:
1125:
1124:
1119:
1117:
1116:
1111:
1110:
1096:
1094:
1093:
1088:
1086:
1085:
1080:
1079:
1065:
1063:
1062:
1057:
1055:
1054:
1049:
1048:
1020:
1018:
1017:
1012:
1010:
1009:
1008:
988:
969:
967:
966:
961:
959:
958:
945:
943:
942:
937:
935:
931:
930:
916:
914:
913:
908:
894:
877:
796:
794:
793:
788:
786:
785:
784:
767:
765:
764:
759:
757:
756:
736:
694:
659:
637:
549:
545:
544:
537:
522:
521:
475:
473:
472:
467:
465:
460:
451:
449:
448:
443:
425:
423:
422:
417:
369:
367:
366:
361:
340:
338:
337:
332:
308:
306:2\implies y: -->
304:
303:
299:2\implies y: -->
296:
272:
271:
258:2\implies y: -->
253:
251:
250:
245:
243:
238:
226:
224:
223:
218:
216:
211:
196:
194:
193:
188:
186:
181:
134:
133:
130:
127:
124:
103:
98:
97:
87:
80:
79:
74:
66:
59:
42:
33:
32:
25:
24:
16:
2700:
2699:
2695:
2694:
2693:
2691:
2690:
2689:
2640:
2639:
2598:
2579:
2566:
2561:
2560:
2539:
2520:
2507:
2502:
2501:
2478:
2477:
2450:
2449:
2425:
2406:
2393:
2379:
2378:
2377:and a sequence
2355:
2354:
2307:
2306:
2300:
2291:
2284:
2255:
2090:
2084:
2083:
2063:
2057:
2056:
2036:
2030:
2029:
2005:
2004:
1981:
1980:
1933:
1932:
1911:
1910:
1888:
1883:
1882:
1824:
1823:
1801:
1796:
1795:
1774:
1769:
1768:
1713:
1712:
1691:
1690:
1669:
1664:
1663:
1616:
1611:
1610:
1589:
1584:
1583:
1562:
1535:
1505:
1504:
1481:
1480:
1455:
1450:
1449:
1428:
1407:
1391:
1386:
1385:
1358:
1345:
1344:
1323:
1310:
1309:
1284:
1279:
1278:
1235:
1234:
1213:
1206:
1205:
1184:
1154:
1153:
1130:
1129:
1104:
1099:
1098:
1073:
1068:
1067:
1042:
1037:
1036:
1001:
994:
993:
971:
948:
947:
924:
919:
918:
887:
870:
799:
798:
777:
770:
769:
746:
745:
726:
677:
666:
638:
627:
608:
588:
586:lede needs help
513:
494:
493:
454:
453:
428:
427:
372:
371:
343:
342:
314:
313:
263:
256:
255:
229:
228:
199:
198:
175:
174:
171:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
2698:
2696:
2688:
2687:
2682:
2677:
2672:
2667:
2662:
2657:
2652:
2642:
2641:
2628:188.192.83.224
2620:
2619:
2605:
2601:
2597:
2594:
2591:
2586:
2582:
2578:
2573:
2569:
2546:
2542:
2538:
2535:
2532:
2527:
2523:
2519:
2514:
2510:
2485:
2459:
2437:
2432:
2428:
2424:
2421:
2418:
2413:
2409:
2405:
2400:
2396:
2392:
2389:
2386:
2364:
2331:
2330:
2316:
2296:
2289:
2282:
2254:
2251:
2250:
2249:
2248:
2247:
2246:
2245:
2244:
2243:
2242:
2241:
2240:
2239:
2238:
2237:
2236:
2235:
2234:
2233:
2232:
2231:
2230:
2229:
2228:
2227:
2150:
2149:
2148:
2147:
2146:
2145:
2144:
2143:
2142:
2141:
2140:
2139:
2138:
2137:
2136:
2135:
2096:
2093:
2069:
2066:
2042:
2039:
2026:
2013:
1990:
1954:
1951:
1948:
1944:
1940:
1919:
1896:
1892:
1869:
1868:
1867:
1866:
1865:
1864:
1863:
1862:
1861:
1860:
1837:
1834:
1831:
1809:
1805:
1782:
1778:
1756:
1755:
1754:
1753:
1752:
1751:
1750:
1749:
1721:
1699:
1677:
1673:
1653:
1652:
1651:
1650:
1649:
1648:
1624:
1620:
1597:
1593:
1556:
1555:
1554:
1553:
1523:
1519:
1515:
1512:
1490:
1466:
1460:
1436:
1432:
1427:
1423:
1417:
1413:
1410:
1406:
1403:
1398:
1394:
1373:
1370:
1366:
1362:
1357:
1353:
1331:
1327:
1322:
1318:
1295:
1289:
1272:
1271:
1270:
1269:
1244:
1219:
1216:
1172:
1168:
1164:
1161:
1139:
1115:
1109:
1084:
1078:
1053:
1047:
1033:
1007:
1004:
978:128.214.20.122
957:
934:
929:
906:
903:
900:
897:
893:
890:
886:
883:
880:
876:
873:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
783:
780:
755:
738:
737:
731:comment added
719:24.131.192.199
714:
665:
662:
626:
625:Omitting Types
623:
607:
604:
587:
584:
578:Vlad Patryshev
574:
533:
528:
525:
520:
516:
512:
509:
506:
503:
463:
441:
438:
435:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
359:
356:
353:
350:
330:
327:
324:
321:
294:
291:
288:
283:
278:
275:
270:
266:
241:
236:
214:
209:
206:
184:
170:
167:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
2697:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2647:
2645:
2638:
2637:
2633:
2629:
2623:
2603:
2599:
2595:
2592:
2589:
2584:
2580:
2576:
2571:
2567:
2544:
2540:
2536:
2533:
2530:
2525:
2521:
2517:
2512:
2508:
2499:
2483:
2475:
2430:
2426:
2422:
2419:
2416:
2411:
2407:
2403:
2398:
2394:
2387:
2384:
2352:
2348:
2344:
2343:
2342:
2340:
2335:
2304:
2299:
2295:
2288:
2281:
2277:
2273:
2269:
2265:
2261:
2257:
2256:
2252:
2226:
2222:
2218:
2213:
2212:
2211:
2207:
2203:
2198:
2197:
2196:
2192:
2188:
2183:
2182:
2181:
2177:
2173:
2168:
2167:
2166:
2165:
2164:
2163:
2162:
2161:
2160:
2159:
2158:
2157:
2156:
2155:
2154:
2153:
2152:
2151:
2134:
2130:
2126:
2122:
2121:
2120:
2116:
2112:
2094:
2067:
2040:
2027:
1978:
1977:
1976:
1972:
1968:
1949:
1946:
1894:
1881:
1880:
1879:
1878:
1877:
1876:
1875:
1874:
1873:
1872:
1871:
1870:
1859:
1855:
1851:
1832:
1807:
1780:
1766:
1765:
1764:
1763:
1762:
1761:
1760:
1759:
1758:
1757:
1748:
1744:
1740:
1736:
1675:
1661:
1660:
1659:
1658:
1657:
1656:
1655:
1654:
1647:
1643:
1639:
1622:
1595:
1581:
1580:
1578:
1574:
1570:
1566:
1560:
1559:
1558:
1557:
1551:
1547:
1543:
1539:
1510:
1464:
1434:
1425:
1415:
1411:
1408:
1404:
1401:
1396:
1392:
1368:
1364:
1355:
1329:
1320:
1293:
1276:
1275:
1274:
1273:
1268:
1264:
1260:
1217:
1203:
1202:
1200:
1196:
1192:
1188:
1159:
1113:
1082:
1051:
1034:
1032:
1028:
1024:
1005:
991:
990:
989:
987:
983:
979:
975:
932:
898:
895:
891:
888:
884:
881:
874:
871:
864:
861:
858:
852:
846:
843:
840:
834:
828:
825:
822:
819:
813:
807:
781:
742:
734:
730:
724:
720:
715:
713:
709:
705:
701:
697:
696:
695:
693:
689:
685:
681:
675:
671:
663:
661:
658:
654:
650:
646:
642:
632:
624:
622:
621:
617:
613:
605:
603:
602:
598:
594:
585:
583:
582:
579:
572:
571:
568:
562:
561:
558:
553:
526:
523:
518:
514:
510:
507:
504:
501:
490:
489:
486:
481:
479:
461:
439:
436:
433:
407:
401:
398:
392:
386:
380:
354:
348:
325:
319:
310:
292:
289:
286:
276:
273:
268:
264:
239:
234:
212:
207:
204:
182:
168:
153:
149:
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
2624:
2621:
2473:
2350:
2346:
2336:
2332:
2302:
2297:
2293:
2286:
2279:
2275:
2270:is a set of
2267:
2260:model theory
1569:84.248.64.88
1563:— Preceding
1542:84.248.64.88
1536:— Preceding
1191:84.248.64.88
1185:— Preceding
972:— Preceding
743:
739:
684:84.248.64.88
678:— Preceding
673:
669:
667:
634:
629:
609:
589:
573:
567:82.208.2.227
563:
551:
491:
485:82.208.2.227
482:
477:
311:
172:
148:Mid-priority
147:
107:
73:Mid‑priority
51:WikiProjects
34:
2498:first-order
2353:-structure
2305:-structure
2272:first-order
2264:mathematics
946:but not in
917:is true in
727:—Preceding
639:—Preceding
541:2\implies }
123:Mathematics
114:mathematics
70:Mathematics
2644:Categories
645:Dr satsuma
612:Zero sharp
593:Zero sharp
557:Thehalfone
307:x}" /: -->
2187:Trovatore
2111:Trovatore
1638:Trovatore
1259:Trovatore
1023:Trovatore
704:Trovatore
39:is rated
2217:Wgunther
2125:Wgunther
1967:Wgunther
1565:unsigned
1538:unsigned
1503:(w.r.t.
1187:unsigned
1152:(w.r.t.
974:unsigned
680:unsigned
653:contribs
641:unsigned
259:x}": -->
1850:Kreipas
729:undated
150:on the
41:B-class
2472:, the
2202:Tkuvho
2172:Tkuvho
1739:Tkuvho
1343:where
478:theory
47:scale.
2349:, an
524:: -->
505:: -->
290:: -->
274:: -->
28:This
2632:talk
2474:type
2292:,…,
2268:type
2266:, a
2221:talk
2206:talk
2191:talk
2176:talk
2129:talk
2115:talk
1971:talk
1854:talk
1743:talk
1642:talk
1573:talk
1546:talk
1402::=≤
1384:and
1263:talk
1195:talk
1027:talk
982:talk
896:<
885:<
862:<
768:and
723:talk
708:talk
688:talk
649:talk
616:talk
597:talk
552:type
341:and
2476:of
2258:In
2003:or
1737:.
1448:.
970:.
725:)
142:Mid
2646::
2634:)
2593:…
2534:…
2420:…
2285:,
2223:)
2208:)
2193:)
2178:)
2131:)
2117:)
1973:)
1953:⟩
1950:≤
1939:⟨
1856:)
1833:≤
1745:)
1644:)
1575:)
1548:)
1465:∗
1426:×
1416:∪
1409:≤
1405:∪
1397:∗
1393:≤
1372:∅
1356:∩
1321:∪
1294:∗
1265:)
1197:)
1114:∗
1083:∗
1052:∗
1029:)
984:)
868:∃
856:∀
853:∧
844:≤
832:∀
829:∧
823:≠
811:∃
805:∃
710:)
690:)
655:)
651:•
618:)
599:)
576:--
532:⟹
511:∧
440:ψ
437:∧
434:ϕ
402:ψ
399:∧
387:ϕ
378:∃
349:ψ
320:ϕ
300:x}
282:⟹
235:−
2630:(
2618:.
2604:n
2600:b
2596:,
2590:,
2585:2
2581:b
2577:,
2572:1
2568:b
2545:n
2541:x
2537:,
2531:,
2526:2
2522:x
2518:,
2513:1
2509:x
2484:b
2458:M
2436:)
2431:n
2427:b
2423:,
2417:,
2412:2
2408:b
2404:,
2399:1
2395:b
2391:(
2388:=
2385:b
2363:M
2351:L
2347:L
2329:.
2315:M
2303:L
2298:n
2294:x
2290:2
2287:x
2283:1
2280:x
2276:L
2219:(
2204:(
2189:(
2174:(
2127:(
2113:(
2095:′
2092:N
2068:′
2065:N
2041:′
2038:N
2012:N
1989:N
1969:(
1947:,
1943:N
1918:Z
1895:′
1891:N
1852:(
1836:}
1830:{
1808:′
1804:N
1781:′
1777:N
1741:(
1720:N
1698:N
1676:′
1672:N
1640:(
1623:′
1619:N
1596:′
1592:N
1571:(
1544:(
1522:)
1518:N
1514:(
1511:L
1489:N
1459:N
1435:′
1431:N
1422:N
1412:′
1369:=
1365:′
1361:N
1352:N
1330:′
1326:N
1317:N
1288:N
1261:(
1243:N
1218:′
1215:N
1193:(
1171:)
1167:N
1163:(
1160:L
1138:N
1108:N
1077:N
1046:N
1025:(
1006:′
1003:N
980:(
956:N
933:′
928:N
905:)
902:)
899:y
892:′
889:w
882:w
879:(
875:′
872:w
865:y
859:w
850:)
847:z
841:x
838:(
835:z
826:y
820:x
817:(
814:y
808:x
782:′
779:N
754:N
735:.
721:(
706:(
686:(
647:(
614:(
595:(
527:2
519:2
515:y
508:0
502:y
462:2
414:)
411:)
408:x
405:(
396:)
393:x
390:(
384:(
381:x
358:)
355:x
352:(
329:)
326:x
323:(
293:x
287:y
277:2
269:2
265:y
240:2
213:2
208:=
205:x
183:2
154:.
53::
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