3349:
365:
46:
1149:. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.
1152:
When cardinality is studied in ZF without the axiom of choice, it is no longer possible to prove that each infinite set has some aleph number as its cardinality; the sets whose cardinality is an aleph number are exactly the infinite sets that can be well-ordered. The method of
1418:
His new numbers deserved something unique. ... Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet ... the aleph could be taken to represent new beginnings ...
1250:
In older mathematics books, the letter aleph is often printed upside down by accident â for example, in SierpiĆski (1958) the letter aleph appears both the right way up and upside down â partly because a
730:, and moreover it is possible to assume that 2 is as least as large as any cardinal number we like. The main restriction ZFC puts on the value of 2 is that it cannot equal certain special cardinals with
1728:
534:, etc.) because in those cases we only have to close with respect to finite operations â sums, products, etc. The process involves defining, for each countable ordinal, via
518:
is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the
2403:
2486:
1627:
590:) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity
149:
503:(this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in â”
159:(â) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme
2800:
1157:
is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF. For example, one can define card(
997:
2958:
1746:
2813:
2136:
602:
The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers. CH is independent of
2398:
2818:
2808:
2545:
1751:
2296:
1742:
2954:
1411:
1356:
412:
3051:
2795:
1620:
1526:
2356:
2049:
1790:
583:
465:
3312:
3014:
2777:
2772:
2597:
2018:
1702:
390:
3307:
3090:
3007:
2720:
2651:
2528:
1770:
1579:
1348:
386:
314:
2378:
3232:
3058:
2744:
1977:
1319:
575:
2383:
2715:
2454:
1712:
1613:
1574:
1145:
is an aleph number is equivalent to the existence of a well-ordering of every set, which in turn is equivalent to the
3110:
3105:
375:
3039:
2629:
2023:
1991:
1682:
426:
1756:
394:
379:
3373:
3329:
3278:
3175:
2673:
2634:
2111:
993:
340:
299:
180:
156:
3170:
1785:
3378:
3100:
2639:
2491:
2474:
2197:
1677:
3002:
2979:
2940:
2826:
2767:
2413:
2333:
2177:
2121:
1734:
491:
is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set Ï
3292:
3019:
2997:
2964:
2857:
2703:
2688:
2661:
2612:
2496:
2431:
2256:
2222:
2217:
2091:
1922:
1899:
1537:
195:(aleph-nought, aleph-zero, or aleph-null) is the cardinality of the set of all natural numbers, and is an
168:
35:
1324:. Polska Akademia Nauk Monografie Matematyczne. Vol. 34. Warsaw, PL: PaĆstwowe Wydawnictwo Naukowe.
3222:
3075:
2867:
2585:
2321:
2227:
2086:
2071:
1952:
1927:
1345:
Mathematics into type: Copy editing and proofreading of mathematics for editorial assistants and authors
1030:
639:
535:
39:
3348:
3195:
3157:
3034:
2838:
2678:
2602:
2580:
2408:
2366:
2265:
2232:
2096:
1884:
1795:
935:
712:
is the first uncountable cardinal number that can be demonstrated within
ZermeloâFraenkel set theory
555:
271:
3324:
3215:
3200:
3180:
3137:
3024:
2974:
2900:
2845:
2782:
2575:
2570:
2518:
2286:
2275:
1947:
1847:
1775:
1766:
1762:
1697:
1692:
926:â”: On â Cd is a bijection from the ordinals to the infinite cardinals. Formally, in
785:
351:
is smaller than any other infinite cardinal, and is therefore the (unique) least infinite ordinal.
292:
160:
3353:
3122:
3085:
3070:
3063:
3046:
2832:
2698:
2624:
2607:
2560:
2373:
2282:
2116:
2101:
2061:
2013:
1998:
1986:
1942:
1917:
1687:
1636:
1505:
1226:
1058:
818:
227:
196:
2850:
2306:
1569:
606:: It can be neither proven nor disproven within the context of that axiom system (provided that
3383:
3288:
3095:
2905:
2895:
2787:
2668:
2503:
2479:
2260:
2244:
2149:
2126:
2003:
1972:
1937:
1832:
1667:
1588:
1407:
1352:
1119:
is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its
1048:
687:
285:
3302:
3297:
3190:
3147:
2969:
2930:
2925:
2910:
2736:
2693:
2590:
2388:
2338:
1912:
1874:
1216:
1154:
278:
176:
164:
1366:
1329:
3283:
3273:
3227:
3210:
3165:
3127:
3029:
2949:
2756:
2683:
2656:
2644:
2550:
2464:
2438:
2393:
2361:
2162:
1964:
1907:
1857:
1822:
1780:
1435:
1362:
1325:
1146:
1120:
1033:
is also a fixed point of the aleph function. This can be shown in ZFC as follows. Suppose
904:
834:
587:
523:
484:
480:
449:
344:
264:
239:
234:(one-to-one correspondence) between it and the natural numbers. Examples of such sets are
125:
1480:
1453:
507:: Every finite set of natural numbers has a maximum which is also a natural number, and
3268:
3247:
3205:
3185:
3080:
2935:
2533:
2523:
2513:
2508:
2442:
2316:
2192:
2081:
2076:
2054:
1655:
1400:
1231:
1211:
1116:
978:
437:
253:
200:
135:
93:
1541:
3367:
3242:
2920:
2427:
2212:
2202:
2172:
2157:
1827:
1066:
892:
619:
508:
257:
3142:
2989:
2890:
2882:
2762:
2710:
2619:
2555:
2538:
2469:
2187:
1889:
1672:
1142:
1124:
682:
where the smallest infinite ordinal is denoted as Ï. That is, the cardinal number â”
531:
527:
145:
82:
78:
74:
519:
3252:
3132:
2311:
2301:
2248:
1932:
1852:
1837:
1717:
1662:
1206:
717:
611:
571:
567:
561:
364:
70:
54:
2182:
2037:
2008:
1814:
1135:
1128:
1074:
826:
731:
631:
526:). This is harder than most explicit descriptions of "generation" in algebra (
321:
113:
58:
49:
Aleph-nought, aleph-zero, or aleph-null, the smallest infinite cardinal number
1591:
3334:
3237:
2290:
2207:
2167:
2131:
2067:
1879:
1869:
1842:
1596:
231:
1294:
1343:
Swanson, Ellen; O'Sean, Arlene Ann; Schleyer, Antoinette
Tingley (2000) .
634:
in 1963, when he showed conversely that the CH itself is not a theorem of
179:
to infinity" or "increases without bound"), or as an extreme point of the
3319:
3117:
2565:
2270:
1864:
1252:
1221:
324:Ïâ
2) of all positive odd integers followed by all positive even integers
172:
85:
and are named after the symbol he used to denote them, the Hebrew letter
66:
17:
483:
is used, it can be further proved that the class of cardinal numbers is
448:
is itself an ordinal number larger than all countable ones, so it is an
2915:
1707:
1275:
320:
are among the countably infinite sets. For example, the sequence (with
246:
542:
unions and complements, and taking the union of all that over all of Ï
1605:
1381:
306:
2459:
1805:
1650:
1510:
1138:
is well-orderable, but does not have an aleph as its cardinality.
792:
is the limit of a countable-length sequence of smaller cardinals.
215:
86:
45:
44:
31:
124:
and so on. Continuing in this manner, it is possible to define a
1609:
436:
is, by definition, the cardinality of the set of all countable
927:
635:
627:
623:
622:
in 1940, when he showed that its negation is not a theorem of
615:
607:
603:
579:
358:
1255:
matrix for aleph was mistakenly constructed the wrong way up.
1402:
Georg Cantor: His mathematics and philosophy of the infinite
1091:
would not be regular and thus not weakly inaccessible. Thus
934:, but a function-like class, as it is not a set (due to the
313:
These infinite ordinals: Ï, Ï + 1, Ïâ
2, Ï, Ï, and
252:
any infinite subset of the integers, such as the set of all
1504:
Chow, Timothy Y. (2007). "A beginner's guide to forcing".
1167:
of minimum possible rank. This has the property that card(
522:
generated by an arbitrary collection of subsets (see e.g.
148:, who defined the notion of cardinality and realized that
992:
holds. There are, however, some limit ordinals which are
471:
the axiom of choice) that no cardinal number is between â”
1434:. Springer Monographs in Mathematics. Berlin, New York:
1163:) to be the set of sets with the same cardinality as
1141:
Over ZF, the assumption that the cardinality of each
724: â„ 1, we can consistently assume that 2 = â”
3261:
3156:
2988:
2881:
2733:
2426:
2349:
2243:
2147:
2036:
1963:
1898:
1813:
1804:
1726:
1643:
837:holds, this is the (unique) next larger cardinal).
1399:
1382:"Earliest uses of symbols of set theory and logic"
746:means that there is a (countable-length) sequence
840:We can then define the aleph numbers as follows:
716:to be equal to the cardinality of the set of all
425:"Aleph One" redirects here. For other uses, see
1000:. The first such is the limit of the sequence
242:, irrespective of including or excluding zero,
150:infinite sets can have different cardinalities
1621:
332:is an ordering of the set (with cardinality â”
8:
977:. For example, it is true for any successor
780:whose limit (i.e. its least upper bound) is
81:. They were introduced by the mathematician
1447:
1445:
1123:. Any set whose cardinality is an aleph is
393:. Unsourced material may be challenged and
2447:
2042:
1810:
1628:
1614:
1606:
1509:
1187:have the same cardinality. (The set card(
413:Learn how and when to remove this message
328:{1, 3, 5, 7, 9, ...; 2, 4, 6, 8, 10, ...}
1193:) does not have the same cardinality of
704: â {0, 1, 2, ...} }.
30:"â”" redirects here. For the letter, see
1267:
1243:
821:, which assigns to any cardinal number
678: â {0, 1, 2, ...} }
214:(where Ï is the lowercase Greek letter
1197:in general, but all its elements do.)
1061:and hence not weakly inaccessible. If
1043:is a weakly inaccessible cardinal. If
998:fixed-point lemma for normal functions
996:of the omega function, because of the
538:, a set by "throwing in" all possible
578:) is 2. It cannot be determined from
7:
1525:Harris, Kenneth A. (April 6, 2009).
1347:(updated ed.). Providence, RI:
391:adding citations to reliable sources
309:of any given countably infinite set.
144:The concept and notation are due to
112:), the next larger cardinality of a
738:. An uncountably infinite cardinal
668: â Ï } = sup{ â”
155:The aleph numbers differ from the
25:
788:). As per the definition above, â”
69:of numbers used to represent the
3347:
1452:Szudzik, Mattew (31 July 2018).
1115:The cardinality of any infinite
638:â by the (then-novel) method of
363:
1540:. Math 582. Archived from
985: + 1 < Ï
1406:. Princeton University Press.
1398:Dauben, Joseph Warren (1990).
1107:which makes it a fixed point.
913:. Its cardinality is written â”
767: †... of cardinals
614:). That CH is consistent with
1:
3308:History of mathematical logic
1532:. Department of Mathematics.
1349:American Mathematical Society
1077:(and thus the cofinality of â”
813:for arbitrary ordinal number
3233:Primitive recursive function
1321:Cardinal and Ordinal Numbers
1127:with an ordinal and is thus
1031:weakly inaccessible cardinal
1019:which is sometimes denoted Ï
819:successor cardinal operation
626:. That it is independent of
576:cardinality of the continuum
1575:Encyclopedia of Mathematics
1318:SierpiĆski, WacĆaw (1958).
1280:Encyclopedia of Mathematics
584:ZermeloâFraenkel set theory
511:of finite sets are finite.
495:: Any countable subset of Ï
466:ZermeloâFraenkel set theory
444:(or sometimes Ω). The set Ï
3400:
2297:SchröderâBernstein theorem
2024:Monadic predicate calculus
1683:Foundations of mathematics
720:2: For any natural number
559:
553:
440:. This set is denoted by Ï
427:Aleph One (disambiguation)
424:
29:
3343:
3330:Philosophy of mathematics
3279:Automated theorem proving
2450:
2404:Von NeumannâBernaysâGödel
2045:
973:is strictly greater than
343:(a weaker version of the
341:axiom of countable choice
274:(in the geometric sense),
222:. A set has cardinality â”
181:extended real number line
336:) of positive integers.
199:. The set of all finite
27:Infinite cardinal number
2980:Self-verifying theories
2801:Tarski's axiomatization
1752:Tarski's undefinability
1747:incompleteness theorems
1536:. Intro to Set Theory.
1460:. Wolfram Web Resources
1111:Role of axiom of choice
499:has an upper bound in Ï
92:The cardinality of the
3354:Mathematics portal
2965:Proof of impossibility
2613:propositional variable
1923:Propositional calculus
1538:University of Michigan
1481:"Continuum Hypothesis"
1454:"Continuum Hypothesis"
305:the set of all finite
298:the set of all binary
230:, that is, there is a
141:, as described below.
50:
36:Aleph (disambiguation)
34:. For other uses, see
3223:Kolmogorov complexity
3176:Computably enumerable
3076:Model complete theory
2868:Principia Mathematica
1928:Propositional formula
1757:BanachâTarski paradox
1485:mathworld.wolfram.com
1430:Jech, Thomas (2003).
1299:mathworld.wolfram.com
1083:) would be less than
942:Fixed points of omega
536:transfinite induction
460:. The definition of â”
302:of finite length, and
272:constructible numbers
226:if and only if it is
48:
40:Alef (disambiguation)
3171:ChurchâTuring thesis
3158:Computability theory
2367:continuum hypothesis
1885:Square of opposition
1743:Gödel's completeness
936:Burali-Forti paradox
630:was demonstrated by
618:was demonstrated by
556:Continuum hypothesis
550:Continuum hypothesis
387:improve this section
293:computable functions
218:), has cardinality â”
3325:Mathematical object
3216:P versus NP problem
3181:Computable function
2975:Reverse mathematics
2901:Logical consequence
2778:primitive recursive
2773:elementary function
2546:Free/bound variable
2399:TarskiâGrothendieck
1918:Logical connectives
1848:Logical equivalence
1698:Logical consequence
1479:Weisstein, Eric W.
1293:Weisstein, Eric W.
817:we must define the
742:having cofinality â”
586:augmented with the
3123:Transfer principle
3086:Semantics of logic
3071:Categorical theory
3047:Non-standard model
2561:Logical connective
1688:Information theory
1637:Mathematical logic
1589:Weisstein, Eric W.
1386:jeff560.tripod.com
1227:Transfinite number
1059:successor cardinal
456:is distinct from â”
286:computable numbers
256:or the set of all
228:countably infinite
116:set is aleph-one â”
57:, particularly in
51:
3361:
3360:
3293:Abstract category
3096:Theories of truth
2906:Rule of inference
2896:Natural deduction
2877:
2876:
2422:
2421:
2127:Cartesian product
2032:
2031:
1938:Many-valued logic
1913:Boolean functions
1796:Russell's paradox
1771:diagonal argument
1668:First-order logic
1527:"Lecture 31"
1458:Wolfram Mathworld
1179:) if and only if
1099:and consequently
1049:successor ordinal
688:least upper bound
423:
422:
415:
279:algebraic numbers
197:infinite cardinal
16:(Redirected from
3391:
3374:Cardinal numbers
3352:
3351:
3303:History of logic
3298:Category of sets
3191:Decision problem
2970:Ordinal analysis
2911:Sequent calculus
2809:Boolean algebras
2749:
2748:
2723:
2694:logical/constant
2448:
2434:
2357:ZermeloâFraenkel
2108:Set operations:
2043:
1980:
1811:
1791:LöwenheimâSkolem
1678:Formal semantics
1630:
1623:
1616:
1607:
1602:
1601:
1583:
1557:
1556:
1554:
1552:
1547:on March 4, 2016
1546:
1531:
1522:
1516:
1515:
1513:
1501:
1495:
1494:
1492:
1491:
1476:
1470:
1469:
1467:
1465:
1449:
1440:
1439:
1427:
1421:
1420:
1405:
1395:
1393:
1392:
1377:
1371:
1370:
1340:
1334:
1333:
1315:
1309:
1308:
1306:
1305:
1290:
1284:
1283:
1272:
1256:
1248:
1217:Regular cardinal
1192:
1178:
1172:
1162:
946:For any ordinal
922:Informally, the
883: <
825:the next larger
786:Easton's theorem
776: <
464:implies (in ZF,
418:
411:
407:
404:
398:
367:
359:
265:rational numbers
165:real number line
21:
3399:
3398:
3394:
3393:
3392:
3390:
3389:
3388:
3379:Hebrew alphabet
3364:
3363:
3362:
3357:
3346:
3339:
3284:Category theory
3274:Algebraic logic
3257:
3228:Lambda calculus
3166:Church encoding
3152:
3128:Truth predicate
2984:
2950:Complete theory
2873:
2742:
2738:
2734:
2729:
2721:
2441: and
2437:
2432:
2418:
2394:New Foundations
2362:axiom of choice
2345:
2307:Gödel numbering
2247: and
2239:
2143:
2028:
1978:
1959:
1908:Boolean algebra
1894:
1858:Equiconsistency
1823:Classical logic
1800:
1781:Halting problem
1769: and
1745: and
1733: and
1732:
1727:Theorems (
1722:
1639:
1634:
1587:
1586:
1568:
1565:
1560:
1550:
1548:
1544:
1529:
1524:
1523:
1519:
1503:
1502:
1498:
1489:
1487:
1478:
1477:
1473:
1463:
1461:
1451:
1450:
1443:
1436:Springer-Verlag
1429:
1428:
1424:
1414:
1397:
1390:
1388:
1379:
1378:
1374:
1359:
1342:
1341:
1337:
1317:
1316:
1312:
1303:
1301:
1292:
1291:
1287:
1274:
1273:
1269:
1265:
1260:
1259:
1249:
1245:
1240:
1203:
1188:
1174:
1168:
1158:
1147:axiom of choice
1121:initial ordinal
1113:
1082:
1056:
1042:
1025:
1024:
1014:
1013:
1007:
991:
972:
967:In many cases Ï
962:
944:
918:
912:
905:initial ordinal
878:
872:
863:
857:
847:
835:axiom of choice
812:
805:
791:
775:
766:
759:
752:
745:
737:
729:
711:
699:
685:
673:
663:
657:
650:Aleph-omega is
648:
597:
588:axiom of choice
564:
558:
552:
545:
524:Borel hierarchy
517:
506:
502:
498:
494:
490:
485:totally ordered
481:axiom of choice
478:
474:
463:
459:
455:
452:. Therefore, â”
450:uncountable set
447:
443:
438:ordinal numbers
435:
430:
419:
408:
402:
399:
384:
368:
357:
350:
347:) holds, then â”
345:axiom of choice
335:
318:
291:the set of all
284:the set of all
277:the set of all
270:the set of all
263:the set of all
245:the set of all
240:natural numbers
225:
221:
212:
194:
189:
133:
126:cardinal number
123:
119:
99:
94:natural numbers
43:
28:
23:
22:
15:
12:
11:
5:
3397:
3395:
3387:
3386:
3381:
3376:
3366:
3365:
3359:
3358:
3344:
3341:
3340:
3338:
3337:
3332:
3327:
3322:
3317:
3316:
3315:
3305:
3300:
3295:
3286:
3281:
3276:
3271:
3269:Abstract logic
3265:
3263:
3259:
3258:
3256:
3255:
3250:
3248:Turing machine
3245:
3240:
3235:
3230:
3225:
3220:
3219:
3218:
3213:
3208:
3203:
3198:
3188:
3186:Computable set
3183:
3178:
3173:
3168:
3162:
3160:
3154:
3153:
3151:
3150:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3114:
3113:
3108:
3103:
3093:
3088:
3083:
3081:Satisfiability
3078:
3073:
3068:
3067:
3066:
3056:
3055:
3054:
3044:
3043:
3042:
3037:
3032:
3027:
3022:
3012:
3011:
3010:
3005:
2998:Interpretation
2994:
2992:
2986:
2985:
2983:
2982:
2977:
2972:
2967:
2962:
2952:
2947:
2946:
2945:
2944:
2943:
2933:
2928:
2918:
2913:
2908:
2903:
2898:
2893:
2887:
2885:
2879:
2878:
2875:
2874:
2872:
2871:
2863:
2862:
2861:
2860:
2855:
2854:
2853:
2848:
2843:
2823:
2822:
2821:
2819:minimal axioms
2816:
2805:
2804:
2803:
2792:
2791:
2790:
2785:
2780:
2775:
2770:
2765:
2752:
2750:
2731:
2730:
2728:
2727:
2726:
2725:
2713:
2708:
2707:
2706:
2701:
2696:
2691:
2681:
2676:
2671:
2666:
2665:
2664:
2659:
2649:
2648:
2647:
2642:
2637:
2632:
2622:
2617:
2616:
2615:
2610:
2605:
2595:
2594:
2593:
2588:
2583:
2578:
2573:
2568:
2558:
2553:
2548:
2543:
2542:
2541:
2536:
2531:
2526:
2516:
2511:
2509:Formation rule
2506:
2501:
2500:
2499:
2494:
2484:
2483:
2482:
2472:
2467:
2462:
2457:
2451:
2445:
2428:Formal systems
2424:
2423:
2420:
2419:
2417:
2416:
2411:
2406:
2401:
2396:
2391:
2386:
2381:
2376:
2371:
2370:
2369:
2364:
2353:
2351:
2347:
2346:
2344:
2343:
2342:
2341:
2331:
2326:
2325:
2324:
2317:Large cardinal
2314:
2309:
2304:
2299:
2294:
2280:
2279:
2278:
2273:
2268:
2253:
2251:
2241:
2240:
2238:
2237:
2236:
2235:
2230:
2225:
2215:
2210:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2154:
2152:
2145:
2144:
2142:
2141:
2140:
2139:
2134:
2129:
2124:
2119:
2114:
2106:
2105:
2104:
2099:
2089:
2084:
2082:Extensionality
2079:
2077:Ordinal number
2074:
2064:
2059:
2058:
2057:
2046:
2040:
2034:
2033:
2030:
2029:
2027:
2026:
2021:
2016:
2011:
2006:
2001:
1996:
1995:
1994:
1984:
1983:
1982:
1969:
1967:
1961:
1960:
1958:
1957:
1956:
1955:
1950:
1945:
1935:
1930:
1925:
1920:
1915:
1910:
1904:
1902:
1896:
1895:
1893:
1892:
1887:
1882:
1877:
1872:
1867:
1862:
1861:
1860:
1850:
1845:
1840:
1835:
1830:
1825:
1819:
1817:
1808:
1802:
1801:
1799:
1798:
1793:
1788:
1783:
1778:
1773:
1761:Cantor's
1759:
1754:
1749:
1739:
1737:
1724:
1723:
1721:
1720:
1715:
1710:
1705:
1700:
1695:
1690:
1685:
1680:
1675:
1670:
1665:
1660:
1659:
1658:
1647:
1645:
1641:
1640:
1635:
1633:
1632:
1625:
1618:
1610:
1604:
1603:
1584:
1564:
1563:External links
1561:
1559:
1558:
1517:
1496:
1471:
1441:
1422:
1412:
1380:Miller, Jeff.
1372:
1357:
1351:. p. 16.
1335:
1310:
1285:
1266:
1264:
1261:
1258:
1257:
1242:
1241:
1239:
1236:
1235:
1234:
1232:Ordinal number
1229:
1224:
1219:
1214:
1212:Gimel function
1209:
1202:
1199:
1129:well-orderable
1117:ordinal number
1112:
1109:
1078:
1052:
1038:
1022:
1020:
1017:
1016:
1011:
1009:
1005:
986:
968:
965:
964:
958:
943:
940:
932:not a function
924:aleph function
914:
908:
897:
896:
874:
868:
865:
859:
852:
849:
845:
808:
804:
794:
789:
771:
764:
757:
750:
743:
735:
725:
709:
706:
705:
695:
683:
680:
679:
669:
659:
655:
647:
644:
600:
599:
595:
570:of the set of
554:Main article:
551:
548:
543:
515:
504:
500:
496:
492:
488:
476:
472:
461:
457:
453:
445:
441:
433:
421:
420:
371:
369:
362:
356:
353:
348:
333:
330:
329:
316:
311:
310:
303:
296:
289:
282:
275:
268:
261:
254:square numbers
250:
243:
223:
219:
210:
192:
188:
185:
167:(applied to a
136:ordinal number
129:
121:
117:
97:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3396:
3385:
3382:
3380:
3377:
3375:
3372:
3371:
3369:
3356:
3355:
3350:
3342:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3314:
3311:
3310:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3266:
3264:
3260:
3254:
3251:
3249:
3246:
3244:
3243:Recursive set
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3217:
3214:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3193:
3192:
3189:
3187:
3184:
3182:
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3163:
3161:
3159:
3155:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3112:
3109:
3107:
3104:
3102:
3099:
3098:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3072:
3069:
3065:
3062:
3061:
3060:
3057:
3053:
3052:of arithmetic
3050:
3049:
3048:
3045:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3017:
3016:
3013:
3009:
3006:
3004:
3001:
3000:
2999:
2996:
2995:
2993:
2991:
2987:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2960:
2959:from ZFC
2956:
2953:
2951:
2948:
2942:
2939:
2938:
2937:
2934:
2932:
2929:
2927:
2924:
2923:
2922:
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2902:
2899:
2897:
2894:
2892:
2889:
2888:
2886:
2884:
2880:
2870:
2869:
2865:
2864:
2859:
2858:non-Euclidean
2856:
2852:
2849:
2847:
2844:
2842:
2841:
2837:
2836:
2834:
2831:
2830:
2828:
2824:
2820:
2817:
2815:
2812:
2811:
2810:
2806:
2802:
2799:
2798:
2797:
2793:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2760:
2758:
2754:
2753:
2751:
2746:
2740:
2735:Example
2732:
2724:
2719:
2718:
2717:
2714:
2712:
2709:
2705:
2702:
2700:
2697:
2695:
2692:
2690:
2687:
2686:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2663:
2660:
2658:
2655:
2654:
2653:
2650:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2627:
2626:
2623:
2621:
2618:
2614:
2611:
2609:
2606:
2604:
2601:
2600:
2599:
2596:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2563:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2540:
2537:
2535:
2532:
2530:
2527:
2525:
2522:
2521:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2498:
2495:
2493:
2492:by definition
2490:
2489:
2488:
2485:
2481:
2478:
2477:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2452:
2449:
2446:
2444:
2440:
2435:
2429:
2425:
2415:
2412:
2410:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2379:KripkeâPlatek
2377:
2375:
2372:
2368:
2365:
2363:
2360:
2359:
2358:
2355:
2354:
2352:
2348:
2340:
2337:
2336:
2335:
2332:
2330:
2327:
2323:
2320:
2319:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2292:
2288:
2284:
2281:
2277:
2274:
2272:
2269:
2267:
2264:
2263:
2262:
2258:
2255:
2254:
2252:
2250:
2246:
2242:
2234:
2231:
2229:
2226:
2224:
2223:constructible
2221:
2220:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2155:
2153:
2151:
2146:
2138:
2135:
2133:
2130:
2128:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2109:
2107:
2103:
2100:
2098:
2095:
2094:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2069:
2065:
2063:
2060:
2056:
2053:
2052:
2051:
2048:
2047:
2044:
2041:
2039:
2035:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2005:
2002:
2000:
1997:
1993:
1990:
1989:
1988:
1985:
1981:
1976:
1975:
1974:
1971:
1970:
1968:
1966:
1962:
1954:
1951:
1949:
1946:
1944:
1941:
1940:
1939:
1936:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1905:
1903:
1901:
1900:Propositional
1897:
1891:
1888:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1859:
1856:
1855:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1828:Logical truth
1826:
1824:
1821:
1820:
1818:
1816:
1812:
1809:
1807:
1803:
1797:
1794:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1768:
1764:
1760:
1758:
1755:
1753:
1750:
1748:
1744:
1741:
1740:
1738:
1736:
1730:
1725:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1657:
1654:
1653:
1652:
1649:
1648:
1646:
1642:
1638:
1631:
1626:
1624:
1619:
1617:
1612:
1611:
1608:
1599:
1598:
1593:
1590:
1585:
1581:
1577:
1576:
1571:
1567:
1566:
1562:
1543:
1539:
1535:
1528:
1521:
1518:
1512:
1507:
1500:
1497:
1486:
1482:
1475:
1472:
1459:
1455:
1448:
1446:
1442:
1437:
1433:
1426:
1423:
1419:
1415:
1413:9780691024479
1409:
1404:
1403:
1387:
1383:
1376:
1373:
1368:
1364:
1360:
1358:0-8218-0053-1
1354:
1350:
1346:
1339:
1336:
1331:
1327:
1323:
1322:
1314:
1311:
1300:
1296:
1289:
1286:
1281:
1277:
1271:
1268:
1262:
1254:
1247:
1244:
1237:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1204:
1200:
1198:
1196:
1191:
1186:
1182:
1177:
1171:
1166:
1161:
1156:
1155:Scott's trick
1150:
1148:
1144:
1139:
1137:
1132:
1130:
1126:
1122:
1118:
1110:
1108:
1106:
1103: =
1102:
1098:
1095: â„
1094:
1090:
1086:
1081:
1076:
1072:
1068:
1067:limit ordinal
1064:
1060:
1055:
1050:
1046:
1041:
1036:
1032:
1027:
1003:
1002:
1001:
999:
995:
989:
984:
980:
976:
971:
961:
956:
953:
952:
951:
949:
941:
939:
937:
933:
929:
925:
920:
917:
911:
906:
903:-th infinite
902:
894:
893:limit ordinal
890:
886:
882:
877:
871:
866:
862:
855:
850:
843:
842:
841:
838:
836:
832:
828:
824:
820:
816:
811:
803:
799:
795:
793:
787:
783:
779:
774:
770:
763:
756:
749:
741:
733:
728:
723:
719:
715:
703:
698:
693:
692:
691:
689:
677:
672:
667:
662:
658:= sup{ â”
653:
652:
651:
645:
643:
641:
637:
633:
629:
625:
621:
617:
613:
609:
605:
593:
592:
591:
589:
585:
581:
577:
573:
569:
563:
557:
549:
547:
541:
537:
533:
529:
528:vector spaces
525:
521:
514:The ordinal Ï
512:
510:
509:finite unions
486:
482:
470:
467:
451:
439:
428:
417:
414:
406:
396:
392:
388:
382:
381:
377:
372:This section
370:
366:
361:
360:
354:
352:
346:
342:
337:
327:
326:
325:
323:
319:
308:
304:
301:
297:
294:
290:
287:
283:
280:
276:
273:
269:
266:
262:
259:
258:prime numbers
255:
251:
248:
244:
241:
237:
236:
235:
233:
229:
217:
213:
206:
202:
198:
186:
184:
182:
178:
174:
170:
166:
162:
158:
153:
151:
147:
142:
140:
137:
132:
127:
115:
111:
107:
103:
95:
90:
88:
84:
80:
76:
75:infinite sets
73:(or size) of
72:
68:
64:
63:aleph numbers
60:
56:
47:
41:
37:
33:
19:
3345:
3143:Ultraproduct
2990:Model theory
2955:Independence
2891:Formal proof
2883:Proof theory
2866:
2839:
2796:real numbers
2768:second-order
2679:Substitution
2556:Metalanguage
2497:conservative
2470:Axiom schema
2414:Constructive
2384:MorseâKelley
2350:Set theories
2329:Aleph number
2328:
2322:inaccessible
2228:Grothendieck
2112:intersection
1999:Higher-order
1987:Second-order
1933:Truth tables
1890:Venn diagram
1673:Formal proof
1595:
1573:
1570:"Aleph-zero"
1551:September 1,
1549:. Retrieved
1542:the original
1534:kaharris.org
1533:
1520:
1499:
1488:. Retrieved
1484:
1474:
1462:. Retrieved
1457:
1431:
1425:
1417:
1401:
1389:. Retrieved
1385:
1375:
1344:
1338:
1320:
1313:
1302:. Retrieved
1298:
1288:
1279:
1270:
1246:
1194:
1189:
1184:
1180:
1175:
1169:
1164:
1159:
1151:
1143:infinite set
1140:
1133:
1125:equinumerous
1114:
1104:
1100:
1096:
1092:
1088:
1084:
1079:
1070:
1062:
1053:
1044:
1039:
1034:
1028:
1018:
994:fixed points
987:
982:
974:
969:
966:
959:
954:
947:
945:
931:
923:
921:
915:
909:
907:is written Ï
900:
898:
891:an infinite
888:
887: } for
884:
880:
875:
869:
860:
853:
839:
830:
827:well-ordered
822:
814:
809:
806:
801:
800:for general
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3133:Truth value
3020:equivalence
2699:non-logical
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2249:cardinality
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2163:Uncountable
2097:equivalence
2014:Quantifiers
2004:Fixed-point
1973:First-order
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2625:Quantifier
2603:functional
2475:Expression
2193:Transitive
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2055:hereditary
2038:Set theory
1490:2020-08-12
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1735:paradoxes
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1580:EMS Press
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