91:
2258:
926:
50:
688:
153:
2004:
1629:
2667:
487:
510:
2253:{\displaystyle \omega ^{\omega ^{\omega ^{\varepsilon _{0}+1}}}=\omega ^{{\varepsilon _{0}}^{\omega }}=\omega ^{{\varepsilon _{0}}^{1+\omega }}=\omega ^{(\varepsilon _{0}\cdot {\varepsilon _{0}}^{\omega })}={(\omega ^{\varepsilon _{0}})}^{{\varepsilon _{0}}^{\omega }}={\varepsilon _{0}}^{{\varepsilon _{0}}^{\omega }}\,,}
2833:. It might appear that this is the non-constructive equivalent of the constructive definition using iterated exponentiation; but the two definitions are equally non-constructive at steps indexed by limit ordinals, which represent transfinite recursion of a higher order than taking the supremum of an exponential series.
2442:
1440:
1434:
1998:
4239:
4083:
2506:
335:
683:{\displaystyle \varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{\omega },\varepsilon _{\omega +1},\ldots ,\varepsilon _{\varepsilon _{0}},\ldots ,\varepsilon _{\varepsilon _{1}},\ldots ,\varepsilon _{\varepsilon _{\varepsilon _{\cdot _{\cdot _{\cdot }}}}},\ldots }
1861:
3004:
2305:
1738:
1624:{\displaystyle \varepsilon _{\beta }=\sup \left\lbrace {\varepsilon _{\beta -1}+1},\omega ^{\varepsilon _{\beta -1}+1},\omega ^{\omega ^{\varepsilon _{\beta -1}+1}},\omega ^{\omega ^{\omega ^{\varepsilon _{\beta -1}+1}}},\ldots \right\rbrace \,,}
1200:
1317:
3303:
3114:
1867:
4094:
240:. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by
2662:{\displaystyle \varepsilon _{\beta }=\sup \left\{1,\varepsilon _{\beta -1},\varepsilon _{\beta -1}^{\varepsilon _{\beta -1}},\varepsilon _{\beta -1}^{\varepsilon _{\beta -1}^{\varepsilon _{\beta -1}}},\dots \right\}.}
3960:
3556:
is represented by a tree containing a root and a single leaf.) An order on the set of finite rooted trees is defined recursively: we first order the subtrees joined to the root in decreasing order, and then use
1085:
482:{\displaystyle \varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdot ^{\cdot ^{\cdot }}}}}=\sup \left\{\omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\omega ^{\omega ^{\omega ^{\omega }}},\dots \right\}\,,}
3409:
2831:
3954:
It is natural to consider any fixed point of this expanded map to be an epsilon number, whether or not it happens to be strictly an ordinal number. Some examples of non-ordinal epsilon numbers are
296:
2736:
3834:
3757:
3162:
3037:
2928:
1770:
1271:
3521:
3475:
3357:
3217:
3634:
3056:
2896:
2786:
2697:
2472:
3941:
2437:{\displaystyle \varepsilon _{1}=\sup \left\{1,\varepsilon _{0},{\varepsilon _{0}}^{\varepsilon _{0}},{\varepsilon _{0}}^{{\varepsilon _{0}}^{\varepsilon _{0}}},\ldots \right\}.}
4269:
2865:
2288:
1028:
1309:
1682:
3554:
1429:{\displaystyle \varepsilon _{0}=\sup \left\lbrace 1,\omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\omega ^{\omega ^{\omega ^{\omega }}},\ldots \right\rbrace \,,}
1138:
2498:
1993:{\displaystyle \omega ^{\omega ^{\varepsilon _{0}+1}}=\omega ^{(\varepsilon _{0}\cdot \omega )}={(\omega ^{\varepsilon _{0}})}^{\omega }=\varepsilon _{0}^{\omega }\,,}
1675:
1131:
4234:{\displaystyle \varepsilon _{1/2}=\left\{\varepsilon _{0}+1,\omega ^{\varepsilon _{0}+1},\ldots \mid \varepsilon _{1}-1,\omega ^{\varepsilon _{1}-1},\ldots \right\}.}
3907:
3429:
2916:
2759:
1758:
1649:
1220:
1105:
3222:
3689:
for progressively larger ordinals α (including, by this rarefied form of transfinite recursion, limit ordinals), with progressively larger least fixed points
2741:
Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinal numbers themselves. For any ordinal number
4279:. Conway goes on to define a broader class of "irreducible" surreal numbers that includes the epsilon numbers as a particularly interesting subclass.
841:
171:
4078:{\displaystyle \varepsilon _{-1}=\left\{0,1,\omega ,\omega ^{\omega },\ldots \mid \varepsilon _{0}-1,\omega ^{\varepsilon _{0}-1},\ldots \right\}}
17:
4386:
3477:
in turn has a similar Cantor normal form. We obtain the finite rooted tree representing α by joining the roots of the trees representing
1034:
753:
4526:
3362:
2791:
4502:
3888:
provided a number of examples of concepts that had natural extensions from the ordinals to the surreals. One such function is the
1282:
264:
2702:
3797:
3720:
3134:
3009:
1856:{\displaystyle \omega ^{\varepsilon _{0}+1}=\omega ^{\varepsilon _{0}}\cdot \omega ^{1}=\varepsilon _{0}\cdot \omega \,,}
4334:
973:
806:
207:
189:
134:
112:
77:
1243:
955:
105:
4379:
3480:
3434:
3316:
3189:
3043:
947:
63:
4427:
3760:
3606:
951:
737:
3523:
to a new root. (This has the consequence that the number 0 is represented by a single root while the number
4345:
2874:
2764:
2675:
2450:
3948:
3913:
4466:
4372:
3164:, which means that the Cantor normal form is not very useful for epsilon numbers. The ordinals less than
2999:{\displaystyle \varepsilon _{\omega }=\sup\{\varepsilon _{0},\varepsilon _{1},\varepsilon _{2},\ldots \}}
1733:{\displaystyle \varepsilon _{\beta }=\sup \lbrace \varepsilon _{\delta }\mid \delta <\beta \rbrace }
4247:
3173:, however, can be usefully described by their Cantor normal forms, which leads to a representation of
2843:
2699:
is a fixed point not only of base ω exponentiation but also of base δ exponentiation for all ordinals
2266:
996:
4559:
3183:
1288:
1278:
233:
4584:
3526:
936:
707:
ordinals also exist, along with uncountable epsilon numbers whose index is an uncountable ordinal.
507:
Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in
99:
1195:{\displaystyle \alpha ^{\beta }=\sup \lbrace \alpha ^{\delta }\mid 0<\delta <\beta \rbrace }
4395:
4293:
3702:. The least ordinal not reachable from 0 by this procedure—i. e., the least ordinal α for which
3561:
on these ordered sequences of subtrees. In this way the set of all finite rooted trees becomes a
940:
4341:
3582:
987:
741:
720:
116:
3298:{\displaystyle \alpha =\omega ^{\beta _{1}}+\omega ^{\beta _{2}}+\cdots +\omega ^{\beta _{k}}}
3109:{\displaystyle \alpha \geq 1\Rightarrow \varepsilon _{\omega _{\alpha }}=\omega _{\alpha }\,.}
2477:
1654:
1110:
4483:
3892:
3558:
3414:
757:
724:
326:
2901:
2744:
1743:
1634:
1205:
1090:
4545:
4456:
4446:
3944:
3889:
3876:
8:
69:
38:
3636:
form a normal function, whose fixed points form a normal function; this is known as the
2788:
is the least epsilon number (fixed point of the exponential map) not already in the set
167:
4531:
4288:
3885:
3129:
789:
system, consisting of all surreals that are fixed points of the base ω exponential map
778:
245:
229:
760:
of this ordering (it is in fact the least ordinal with this property, and as such, in
4436:
4330:
3566:
1238:
4276:
764:
749:
30:
This article is about a type of ordinal in mathematics. For the physical constant
4406:
3637:
3598:
3050:
3047:
1274:
771:
3881:
3769:. In a set theory where such an ordinal can be proved to exist, one has a map
3310:
786:
249:
4578:
1223:
4364:
3943:; this mapping generalises naturally to include all surreal numbers in its
3586:
782:
767:, is used as a measure of the strength of the theory of Peano arithmetic).
761:
241:
704:
221:
2836:
The following facts about epsilon numbers are straightforward to prove:
3562:
2922:
2868:
700:
501:
3657:). In the notation of the Veblen hierarchy, the epsilon mapping is
925:
3836:; these are all still epsilon numbers, as they lie in the image of
1080:{\displaystyle \alpha ^{\beta }=\alpha ^{\beta -1}\cdot \alpha \,,}
497:
256:
745:
3404:{\displaystyle \alpha >\beta _{1}\geq \cdots \geq \beta _{k}}
2826:{\displaystyle \{\varepsilon _{\delta }\mid \delta <\beta \}}
1311:, these fixed points are precisely the ordinal epsilon numbers.
777:
A more general class of epsilon numbers has been identified by
2672:
In particular, whether or not the index β is a limit ordinal,
2290:, is obtained by starting from 0 and exponentiating with base
2925:
set of epsilon numbers is an epsilon number; so for instance
1230:
From this definition, it follows that for any fixed ordinal
504:
in the case of the von
Neumann representation of ordinals.
2871:, being a countable union of countable ordinals; in fact,
3947:, which in turn provides a natural generalisation of the
3585:, which represents decreasing sequences of ordinals as a
2474:
indexed by any ordinal that has an immediate predecessor
3119:
291:{\displaystyle \varepsilon =\omega ^{\varepsilon },\,}
4250:
4097:
3963:
3916:
3895:
3800:
3723:
3609:
3529:
3483:
3437:
3417:
3365:
3319:
3225:
3192:
3137:
3059:
3012:
2931:
2904:
2877:
2846:
2794:
2767:
2747:
2731:{\displaystyle 1<\delta <\varepsilon _{\beta }}
2705:
2678:
2509:
2480:
2453:
2308:
2269:
2007:
1870:
1773:
1746:
1685:
1657:
1637:
1443:
1320:
1291:
1246:
1208:
1141:
1113:
1093:
1037:
999:
770:
Many larger epsilon numbers can be defined using the
703:, as is any epsilon number whose index is countable.
513:
338:
267:
3829:{\displaystyle \alpha \mapsto \varphi _{\alpha }(0)}
3752:{\displaystyle \alpha \mapsto \varphi _{\alpha }(0)}
3717:, or equivalently the first fixed point of the map
3581:This representation is related to the proof of the
3157:{\displaystyle \varepsilon =\omega ^{\varepsilon }}
3032:{\displaystyle \beta \mapsto \varepsilon _{\beta }}
162:
may be too technical for most readers to understand
4350:(2nd ed.), PWN – Polish Scientific Publishers
4263:
4233:
4077:
3935:
3901:
3828:
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3628:
3548:
3515:
3469:
3423:
3403:
3351:
3297:
3211:
3156:
3108:
3031:
2998:
2910:
2890:
2859:
2825:
2780:
2753:
2730:
2691:
2661:
2492:
2466:
2436:
2282:
2252:
1992:
1855:
1752:
1732:
1669:
1643:
1623:
1428:
1303:
1265:
1214:
1194:
1125:
1099:
1079:
1022:
682:
481:
325:), which can be viewed as the "limit" obtained by
290:
4576:
2945:
2523:
2322:
1699:
1457:
1334:
1155:
393:
4527:the theories of iterated inductive definitions
1266:{\displaystyle \beta \mapsto \alpha ^{\beta }}
906:, and his delta numbers are those of the form
756:, show that Peano arithmetic cannot prove the
4380:
3678:Continuing in this vein, one can define maps
3516:{\displaystyle \beta _{1},\ldots ,\beta _{k}}
3470:{\displaystyle \beta _{1},\ldots ,\beta _{k}}
3352:{\displaystyle \beta _{1},\ldots ,\beta _{k}}
2263:a different sequence with the same supremum,
301:in which ω is the smallest infinite ordinal.
4394:
2993:
2948:
2820:
2795:
1727:
1702:
1189:
1158:
3212:{\displaystyle \alpha <\varepsilon _{0}}
954:. Unsourced material may be challenged and
329:from a sequence of smaller limit ordinals:
78:Learn how and when to remove these messages
4387:
4373:
4354:
4340:
3603:The fixed points of the "epsilon mapping"
900:. His gamma numbers are those of the form
802:
3666:, and its fixed points are enumerated by
3629:{\displaystyle x\mapsto \varepsilon _{x}}
3102:
3006:is an epsilon number. Thus, the mapping
2246:
1986:
1849:
1617:
1422:
1073:
1016:
974:Learn how and when to remove this message
475:
287:
232:whose defining property is that they are
208:Learn how and when to remove this message
190:Learn how and when to remove this message
174:, without removing the technical details.
135:Learn how and when to remove this message
4359:. Göttingen: Vandenhoeck & Ruprecht.
98:This article includes a list of general
842:multiplicatively indecomposable ordinal
14:
4577:
1283:fixed-point lemma for normal functions
4368:
4314:Subsystems of Second-order Arithmetic
2891:{\displaystyle \varepsilon _{\beta }}
2840:Although it is quite a large number,
2781:{\displaystyle \varepsilon _{\beta }}
2692:{\displaystyle \varepsilon _{\beta }}
2467:{\displaystyle \varepsilon _{\beta }}
754:Gödel's second incompleteness theorem
172:make it understandable to non-experts
3936:{\displaystyle n\mapsto \omega ^{n}}
3869:
952:adding citations to reliable sources
919:
915:
872:, and epsilon numbers to be numbers
146:
84:
43:
3592:
24:
723:proofs, because for many purposes
104:it lacks sufficient corresponding
25:
4596:
4503:Takeuti–Feferman–Buchholz ordinal
4244:There is a natural way to define
3866:that enumerates epsilon numbers.
3773:that enumerates the fixed points
807:additively indecomposable ordinal
59:This article has multiple issues.
4264:{\displaystyle \varepsilon _{n}}
3640:(the Veblen functions with base
2860:{\displaystyle \varepsilon _{0}}
2283:{\displaystyle \varepsilon _{1}}
1023:{\displaystyle \alpha ^{0}=1\,,}
924:
151:
89:
48:
2921:The union (or supremum) of any
1304:{\displaystyle \alpha =\omega }
67:or discuss these issues on the
4306:
3920:
3823:
3817:
3804:
3746:
3740:
3727:
3613:
3069:
3016:
2500:can be constructed similarly.
2447:Generally, the epsilon number
2182:
2162:
2152:
2117:
1958:
1938:
1928:
1909:
1277:, so it has arbitrarily large
1250:
13:
1:
4534: < ω
4357:Grundbegriffe der Mengenlehre
4299:
3549:{\displaystyle 1=\omega ^{0}}
1651:has an immediate predecessor
1107:has an immediate predecessor
18:Epsilon numbers (mathematics)
4525:Proof-theoretic ordinals of
4355:Hessenberg, Gerhard (1906).
4347:Cardinal and ordinal numbers
3880:, the classic exposition on
2898:is countable if and only if
748:to prove the consistency of
710:The smallest epsilon number
7:
4282:
3186:, as follows. Any ordinal
986:The standard definition of
805:defined gamma numbers (see
738:Gentzen's consistency proof
27:Type of transfinite numbers
10:
4601:
4548: ≥ ω
3596:
3182:as the ordered set of all
304:The least such ordinal is
29:
4560:First uncountable ordinal
4402:
4271:for every surreal number
3411:, uniquely determined by
3128:Any epsilon number ε has
840:, and delta numbers (see
500:, which is equivalent to
4428:Feferman–Schütte ordinal
4396:Large countable ordinals
3761:Feferman–Schütte ordinal
2493:{\displaystyle \beta -1}
1670:{\displaystyle \beta -1}
1126:{\displaystyle \beta -1}
4467:Bachmann–Howard ordinal
4294:Large countable ordinal
3902:{\displaystyle \omega }
3857:, including of the map
3431:. Each of the ordinals
3424:{\displaystyle \alpha }
3219:has Cantor normal form
727:is only required up to
321:(chiefly American), or
119:more precise citations.
4407:First infinite ordinal
4329:(1976) Academic Press
4275:, and the map remains
4265:
4235:
4079:
3937:
3903:
3830:
3753:
3630:
3550:
3517:
3471:
3425:
3405:
3353:
3299:
3213:
3158:
3110:
3053:is an epsilon number.
3033:
3000:
2912:
2911:{\displaystyle \beta }
2892:
2861:
2827:
2782:
2755:
2754:{\displaystyle \beta }
2732:
2693:
2663:
2494:
2468:
2438:
2284:
2254:
1994:
1857:
1754:
1753:{\displaystyle \beta }
1734:
1671:
1645:
1644:{\displaystyle \beta }
1625:
1430:
1305:
1267:
1216:
1215:{\displaystyle \beta }
1196:
1127:
1101:
1100:{\displaystyle \beta }
1081:
1024:
988:ordinal exponentiation
684:
483:
292:
4266:
4236:
4080:
3951:for surreal numbers.
3938:
3904:
3831:
3754:
3631:
3551:
3518:
3472:
3426:
3406:
3354:
3300:
3214:
3159:
3111:
3039:is a normal function.
3034:
3001:
2913:
2893:
2862:
2828:
2783:
2756:
2733:
2694:
2664:
2495:
2469:
2439:
2285:
2255:
1995:
1858:
1755:
1735:
1672:
1646:
1626:
1431:
1306:
1268:
1217:
1197:
1128:
1102:
1082:
1025:
725:transfinite induction
685:
484:
327:transfinite recursion
293:
4546:Nonrecursive ordinal
4457:large Veblen ordinal
4447:small Veblen ordinal
4327:On Numbers and Games
4312:Stephen G. Simpson,
4248:
4095:
3961:
3914:
3893:
3877:On Numbers and Games
3798:
3721:
3607:
3527:
3481:
3435:
3415:
3363:
3317:
3223:
3190:
3135:
3057:
3010:
2929:
2902:
2875:
2844:
2792:
2765:
2745:
2703:
2676:
2507:
2478:
2451:
2306:
2267:
2005:
1868:
1771:
1744:
1683:
1655:
1635:
1441:
1318:
1289:
1244:
1206:
1139:
1111:
1091:
1035:
997:
948:improve this section
511:
336:
265:
228:are a collection of
4532:Computable ordinals
3559:lexicographic order
3184:finite rooted trees
3120:Representation of ε
2644:
2642:
2589:
1985:
1760:is a limit ordinal.
742:Goodstein's theorem
317:(chiefly British),
230:transfinite numbers
39:Vacuum permittivity
4484:Buchholz's ordinal
4342:Sierpiński, Wacław
4339:Section XIV.20 of
4289:Ordinal arithmetic
4261:
4231:
4075:
3949:Cantor normal form
3933:
3899:
3886:John Horton Conway
3826:
3749:
3626:
3546:
3513:
3467:
3421:
3401:
3359:are ordinals with
3349:
3295:
3209:
3154:
3130:Cantor normal form
3106:
3029:
2996:
2908:
2888:
2857:
2823:
2778:
2751:
2728:
2689:
2659:
2609:
2593:
2556:
2490:
2464:
2434:
2280:
2250:
1990:
1971:
1853:
1750:
1730:
1667:
1641:
1621:
1426:
1301:
1263:
1212:
1192:
1123:
1097:
1077:
1020:
779:John Horton Conway
680:
479:
288:
246:ordinal arithmetic
244:in the context of
4572:
4571:
4437:Ackermann ordinal
3870:Surreal ε numbers
984:
983:
976:
916:Ordinal ε numbers
803:Hessenberg (1906)
740:and the proof of
255:that satisfy the
218:
217:
210:
200:
199:
192:
145:
144:
137:
82:
16:(Redirected from
4592:
4556:
4555:
4542:
4541:
4389:
4382:
4375:
4366:
4365:
4360:
4351:
4317:
4310:
4277:order-preserving
4270:
4268:
4267:
4262:
4260:
4259:
4240:
4238:
4237:
4232:
4227:
4223:
4216:
4215:
4208:
4207:
4184:
4183:
4165:
4164:
4157:
4156:
4133:
4132:
4115:
4114:
4110:
4084:
4082:
4081:
4076:
4074:
4070:
4063:
4062:
4055:
4054:
4031:
4030:
4012:
4011:
3976:
3975:
3942:
3940:
3939:
3934:
3932:
3931:
3908:
3906:
3905:
3900:
3865:
3856:
3846:
3835:
3833:
3832:
3827:
3816:
3815:
3793:
3786:
3779:
3772:
3768:
3758:
3756:
3755:
3750:
3739:
3738:
3716:
3701:
3688:
3674:
3665:
3656:
3638:Veblen hierarchy
3635:
3633:
3632:
3627:
3625:
3624:
3593:Veblen hierarchy
3577:
3567:order isomorphic
3563:well-ordered set
3555:
3553:
3552:
3547:
3545:
3544:
3522:
3520:
3519:
3514:
3512:
3511:
3493:
3492:
3476:
3474:
3473:
3468:
3466:
3465:
3447:
3446:
3430:
3428:
3427:
3422:
3410:
3408:
3407:
3402:
3400:
3399:
3381:
3380:
3358:
3356:
3355:
3350:
3348:
3347:
3329:
3328:
3304:
3302:
3301:
3296:
3294:
3293:
3292:
3291:
3268:
3267:
3266:
3265:
3248:
3247:
3246:
3245:
3218:
3216:
3215:
3210:
3208:
3207:
3181:
3172:
3163:
3161:
3160:
3155:
3153:
3152:
3115:
3113:
3112:
3107:
3101:
3100:
3088:
3087:
3086:
3085:
3038:
3036:
3035:
3030:
3028:
3027:
3005:
3003:
3002:
2997:
2986:
2985:
2973:
2972:
2960:
2959:
2941:
2940:
2917:
2915:
2914:
2909:
2897:
2895:
2894:
2889:
2887:
2886:
2866:
2864:
2863:
2858:
2856:
2855:
2832:
2830:
2829:
2824:
2807:
2806:
2787:
2785:
2784:
2779:
2777:
2776:
2760:
2758:
2757:
2752:
2737:
2735:
2734:
2729:
2727:
2726:
2698:
2696:
2695:
2690:
2688:
2687:
2668:
2666:
2665:
2660:
2655:
2651:
2643:
2641:
2640:
2639:
2623:
2607:
2588:
2587:
2586:
2570:
2552:
2551:
2519:
2518:
2499:
2497:
2496:
2491:
2473:
2471:
2470:
2465:
2463:
2462:
2443:
2441:
2440:
2435:
2430:
2426:
2419:
2418:
2417:
2416:
2415:
2414:
2404:
2403:
2402:
2390:
2389:
2388:
2374:
2373:
2372:
2371:
2361:
2360:
2359:
2345:
2344:
2318:
2317:
2298:
2289:
2287:
2286:
2281:
2279:
2278:
2259:
2257:
2256:
2251:
2245:
2244:
2243:
2242:
2237:
2236:
2235:
2223:
2222:
2221:
2207:
2206:
2205:
2204:
2199:
2198:
2197:
2185:
2181:
2180:
2179:
2178:
2156:
2155:
2151:
2150:
2145:
2144:
2143:
2129:
2128:
2108:
2107:
2106:
2105:
2094:
2093:
2092:
2073:
2072:
2071:
2070:
2065:
2064:
2063:
2044:
2043:
2042:
2041:
2040:
2039:
2032:
2031:
1999:
1997:
1996:
1991:
1984:
1979:
1967:
1966:
1961:
1957:
1956:
1955:
1954:
1932:
1931:
1921:
1920:
1900:
1899:
1898:
1897:
1890:
1889:
1862:
1860:
1859:
1854:
1842:
1841:
1829:
1828:
1816:
1815:
1814:
1813:
1796:
1795:
1788:
1787:
1759:
1757:
1756:
1751:
1739:
1737:
1736:
1731:
1714:
1713:
1695:
1694:
1676:
1674:
1673:
1668:
1650:
1648:
1647:
1642:
1630:
1628:
1627:
1622:
1616:
1612:
1605:
1604:
1603:
1602:
1601:
1600:
1593:
1592:
1559:
1558:
1557:
1556:
1549:
1548:
1520:
1519:
1512:
1511:
1488:
1481:
1480:
1453:
1452:
1435:
1433:
1432:
1427:
1421:
1417:
1410:
1409:
1408:
1407:
1406:
1405:
1383:
1382:
1381:
1380:
1363:
1362:
1330:
1329:
1310:
1308:
1307:
1302:
1272:
1270:
1269:
1264:
1262:
1261:
1236:
1221:
1219:
1218:
1213:
1201:
1199:
1198:
1193:
1170:
1169:
1151:
1150:
1132:
1130:
1129:
1124:
1106:
1104:
1103:
1098:
1086:
1084:
1083:
1078:
1066:
1065:
1047:
1046:
1029:
1027:
1026:
1021:
1009:
1008:
990:with base α is:
979:
972:
968:
965:
959:
928:
920:
911:
905:
899:
888:
878:
871:
860:
850:
844:) to be numbers
839:
829:
815:
809:) to be numbers
798:
765:ordinal analysis
758:well-foundedness
750:Peano arithmetic
735:
719:appears in many
718:
698:
689:
687:
686:
681:
673:
672:
671:
670:
669:
668:
667:
666:
665:
664:
626:
625:
624:
623:
600:
599:
598:
597:
574:
573:
555:
554:
536:
535:
523:
522:
495:
488:
486:
485:
480:
474:
470:
463:
462:
461:
460:
459:
458:
436:
435:
434:
433:
416:
415:
389:
388:
387:
386:
385:
384:
383:
382:
381:
380:
348:
347:
297:
295:
294:
289:
283:
282:
213:
206:
195:
188:
184:
181:
175:
155:
154:
147:
140:
133:
129:
126:
120:
115:this article by
106:inline citations
93:
92:
85:
74:
52:
51:
44:
21:
4600:
4599:
4595:
4594:
4593:
4591:
4590:
4589:
4585:Ordinal numbers
4575:
4574:
4573:
4568:
4554:
4551:
4550:
4549:
4540:
4537:
4536:
4535:
4521:
4519:
4498:
4492:
4479:
4433:
4424:
4416:Epsilon numbers
4398:
4393:
4363:
4321:
4320:
4311:
4307:
4302:
4285:
4255:
4251:
4249:
4246:
4245:
4203:
4199:
4198:
4194:
4179:
4175:
4152:
4148:
4147:
4143:
4128:
4124:
4123:
4119:
4106:
4102:
4098:
4096:
4093:
4092:
4050:
4046:
4045:
4041:
4026:
4022:
4007:
4003:
3984:
3980:
3968:
3964:
3962:
3959:
3958:
3927:
3923:
3915:
3912:
3911:
3894:
3891:
3890:
3882:surreal numbers
3872:
3864:
3858:
3855:
3848:
3845:
3837:
3811:
3807:
3799:
3796:
3795:
3792:
3788:
3785:
3781:
3778:
3774:
3770:
3767:
3763:
3734:
3730:
3722:
3719:
3718:
3711:
3703:
3699:
3690:
3687:
3679:
3673:
3667:
3664:
3658:
3647:
3641:
3620:
3616:
3608:
3605:
3604:
3601:
3599:Veblen function
3595:
3587:graph-theoretic
3576:
3570:
3540:
3536:
3528:
3525:
3524:
3507:
3503:
3488:
3484:
3482:
3479:
3478:
3461:
3457:
3442:
3438:
3436:
3433:
3432:
3416:
3413:
3412:
3395:
3391:
3376:
3372:
3364:
3361:
3360:
3343:
3339:
3324:
3320:
3318:
3315:
3314:
3287:
3283:
3282:
3278:
3261:
3257:
3256:
3252:
3241:
3237:
3236:
3232:
3224:
3221:
3220:
3203:
3199:
3191:
3188:
3187:
3180:
3174:
3171:
3165:
3148:
3144:
3136:
3133:
3132:
3126:
3124:by rooted trees
3123:
3096:
3092:
3081:
3077:
3076:
3072:
3058:
3055:
3054:
3044:initial ordinal
3023:
3019:
3011:
3008:
3007:
2981:
2977:
2968:
2964:
2955:
2951:
2936:
2932:
2930:
2927:
2926:
2903:
2900:
2899:
2882:
2878:
2876:
2873:
2872:
2851:
2847:
2845:
2842:
2841:
2802:
2798:
2793:
2790:
2789:
2772:
2768:
2766:
2763:
2762:
2746:
2743:
2742:
2722:
2718:
2704:
2701:
2700:
2683:
2679:
2677:
2674:
2673:
2629:
2625:
2624:
2613:
2608:
2597:
2576:
2572:
2571:
2560:
2541:
2537:
2530:
2526:
2514:
2510:
2508:
2505:
2504:
2479:
2476:
2475:
2458:
2454:
2452:
2449:
2448:
2410:
2406:
2405:
2398:
2394:
2393:
2392:
2391:
2384:
2380:
2379:
2378:
2367:
2363:
2362:
2355:
2351:
2350:
2349:
2340:
2336:
2329:
2325:
2313:
2309:
2307:
2304:
2303:
2297:
2291:
2274:
2270:
2268:
2265:
2264:
2238:
2231:
2227:
2226:
2225:
2224:
2217:
2213:
2212:
2211:
2200:
2193:
2189:
2188:
2187:
2186:
2174:
2170:
2169:
2165:
2161:
2160:
2146:
2139:
2135:
2134:
2133:
2124:
2120:
2116:
2112:
2095:
2088:
2084:
2083:
2082:
2081:
2077:
2066:
2059:
2055:
2054:
2053:
2052:
2048:
2027:
2023:
2022:
2018:
2017:
2013:
2012:
2008:
2006:
2003:
2002:
1980:
1975:
1962:
1950:
1946:
1945:
1941:
1937:
1936:
1916:
1912:
1908:
1904:
1885:
1881:
1880:
1876:
1875:
1871:
1869:
1866:
1865:
1837:
1833:
1824:
1820:
1809:
1805:
1804:
1800:
1783:
1779:
1778:
1774:
1772:
1769:
1768:
1745:
1742:
1741:
1709:
1705:
1690:
1686:
1684:
1681:
1680:
1656:
1653:
1652:
1636:
1633:
1632:
1582:
1578:
1577:
1573:
1572:
1568:
1567:
1563:
1538:
1534:
1533:
1529:
1528:
1524:
1501:
1497:
1496:
1492:
1470:
1466:
1465:
1464:
1460:
1448:
1444:
1442:
1439:
1438:
1401:
1397:
1396:
1392:
1391:
1387:
1376:
1372:
1371:
1367:
1358:
1354:
1341:
1337:
1325:
1321:
1319:
1316:
1315:
1290:
1287:
1286:
1275:normal function
1257:
1253:
1245:
1242:
1241:
1231:
1207:
1204:
1203:
1165:
1161:
1146:
1142:
1140:
1137:
1136:
1112:
1109:
1108:
1092:
1089:
1088:
1055:
1051:
1042:
1038:
1036:
1033:
1032:
1004:
1000:
998:
995:
994:
980:
969:
963:
960:
945:
929:
918:
907:
901:
890:
880:
873:
862:
852:
845:
831:
817:
810:
790:
772:Veblen function
762:proof-theoretic
744:). Its use by
734:
728:
717:
711:
697:
691:
690:. The ordinal
660:
656:
655:
651:
650:
646:
645:
641:
640:
636:
619:
615:
614:
610:
593:
589:
588:
584:
563:
559:
550:
546:
531:
527:
518:
514:
512:
509:
508:
493:
454:
450:
449:
445:
444:
440:
429:
425:
424:
420:
411:
407:
400:
396:
376:
372:
371:
367:
366:
362:
361:
357:
356:
352:
343:
339:
337:
334:
333:
311:
278:
274:
266:
263:
262:
250:ordinal numbers
248:; they are the
238:exponential map
226:epsilon numbers
214:
203:
202:
201:
196:
185:
179:
176:
168:help improve it
165:
156:
152:
141:
130:
124:
121:
111:Please help to
110:
94:
90:
53:
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
4598:
4588:
4587:
4570:
4569:
4567:
4566:
4557:
4552:
4543:
4538:
4529:
4523:
4515:
4513:
4500:
4494:
4490:
4481:
4477:
4464:
4454:
4444:
4434:
4431:
4425:
4422:
4413:
4403:
4400:
4399:
4392:
4391:
4384:
4377:
4369:
4362:
4361:
4352:
4337:
4322:
4319:
4318:
4304:
4303:
4301:
4298:
4297:
4296:
4291:
4284:
4281:
4258:
4254:
4242:
4241:
4230:
4226:
4222:
4219:
4214:
4211:
4206:
4202:
4197:
4193:
4190:
4187:
4182:
4178:
4174:
4171:
4168:
4163:
4160:
4155:
4151:
4146:
4142:
4139:
4136:
4131:
4127:
4122:
4118:
4113:
4109:
4105:
4101:
4086:
4085:
4073:
4069:
4066:
4061:
4058:
4053:
4049:
4044:
4040:
4037:
4034:
4029:
4025:
4021:
4018:
4015:
4010:
4006:
4002:
3999:
3996:
3993:
3990:
3987:
3983:
3979:
3974:
3971:
3967:
3930:
3926:
3922:
3919:
3898:
3871:
3868:
3862:
3853:
3841:
3825:
3822:
3819:
3814:
3810:
3806:
3803:
3790:
3783:
3776:
3765:
3748:
3745:
3742:
3737:
3733:
3729:
3726:
3707:
3694:
3683:
3671:
3662:
3645:
3623:
3619:
3615:
3612:
3597:Main article:
3594:
3591:
3574:
3543:
3539:
3535:
3532:
3510:
3506:
3502:
3499:
3496:
3491:
3487:
3464:
3460:
3456:
3453:
3450:
3445:
3441:
3420:
3398:
3394:
3390:
3387:
3384:
3379:
3375:
3371:
3368:
3346:
3342:
3338:
3335:
3332:
3327:
3323:
3311:natural number
3290:
3286:
3281:
3277:
3274:
3271:
3264:
3260:
3255:
3251:
3244:
3240:
3235:
3231:
3228:
3206:
3202:
3198:
3195:
3178:
3169:
3151:
3147:
3143:
3140:
3125:
3121:
3118:
3117:
3116:
3105:
3099:
3095:
3091:
3084:
3080:
3075:
3071:
3068:
3065:
3062:
3040:
3026:
3022:
3018:
3015:
2995:
2992:
2989:
2984:
2980:
2976:
2971:
2967:
2963:
2958:
2954:
2950:
2947:
2944:
2939:
2935:
2919:
2907:
2885:
2881:
2854:
2850:
2822:
2819:
2816:
2813:
2810:
2805:
2801:
2797:
2775:
2771:
2750:
2725:
2721:
2717:
2714:
2711:
2708:
2686:
2682:
2670:
2669:
2658:
2654:
2650:
2647:
2638:
2635:
2632:
2628:
2622:
2619:
2616:
2612:
2606:
2603:
2600:
2596:
2592:
2585:
2582:
2579:
2575:
2569:
2566:
2563:
2559:
2555:
2550:
2547:
2544:
2540:
2536:
2533:
2529:
2525:
2522:
2517:
2513:
2489:
2486:
2483:
2461:
2457:
2445:
2444:
2433:
2429:
2425:
2422:
2413:
2409:
2401:
2397:
2387:
2383:
2377:
2370:
2366:
2358:
2354:
2348:
2343:
2339:
2335:
2332:
2328:
2324:
2321:
2316:
2312:
2295:
2277:
2273:
2261:
2260:
2249:
2241:
2234:
2230:
2220:
2216:
2210:
2203:
2196:
2192:
2184:
2177:
2173:
2168:
2164:
2159:
2154:
2149:
2142:
2138:
2132:
2127:
2123:
2119:
2115:
2111:
2104:
2101:
2098:
2091:
2087:
2080:
2076:
2069:
2062:
2058:
2051:
2047:
2038:
2035:
2030:
2026:
2021:
2016:
2011:
2000:
1989:
1983:
1978:
1974:
1970:
1965:
1960:
1953:
1949:
1944:
1940:
1935:
1930:
1927:
1924:
1919:
1915:
1911:
1907:
1903:
1896:
1893:
1888:
1884:
1879:
1874:
1863:
1852:
1848:
1845:
1840:
1836:
1832:
1827:
1823:
1819:
1812:
1808:
1803:
1799:
1794:
1791:
1786:
1782:
1777:
1762:
1761:
1749:
1729:
1726:
1723:
1720:
1717:
1712:
1708:
1704:
1701:
1698:
1693:
1689:
1678:
1666:
1663:
1660:
1640:
1620:
1615:
1611:
1608:
1599:
1596:
1591:
1588:
1585:
1581:
1576:
1571:
1566:
1562:
1555:
1552:
1547:
1544:
1541:
1537:
1532:
1527:
1523:
1518:
1515:
1510:
1507:
1504:
1500:
1495:
1491:
1487:
1484:
1479:
1476:
1473:
1469:
1463:
1459:
1456:
1451:
1447:
1436:
1425:
1420:
1416:
1413:
1404:
1400:
1395:
1390:
1386:
1379:
1375:
1370:
1366:
1361:
1357:
1353:
1350:
1347:
1344:
1340:
1336:
1333:
1328:
1324:
1300:
1297:
1294:
1260:
1256:
1252:
1249:
1228:
1227:
1211:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1168:
1164:
1160:
1157:
1154:
1149:
1145:
1134:
1122:
1119:
1116:
1096:
1076:
1072:
1069:
1064:
1061:
1058:
1054:
1050:
1045:
1041:
1030:
1019:
1015:
1012:
1007:
1003:
982:
981:
932:
930:
923:
917:
914:
787:surreal number
732:
715:
695:
679:
676:
663:
659:
654:
649:
644:
639:
635:
632:
629:
622:
618:
613:
609:
606:
603:
596:
592:
587:
583:
580:
577:
572:
569:
566:
562:
558:
553:
549:
545:
542:
539:
534:
530:
526:
521:
517:
490:
489:
478:
473:
469:
466:
457:
453:
448:
443:
439:
432:
428:
423:
419:
414:
410:
406:
403:
399:
395:
392:
379:
375:
370:
365:
360:
355:
351:
346:
342:
319:epsilon naught
315:epsilon nought
309:
299:
298:
286:
281:
277:
273:
270:
216:
215:
198:
197:
159:
157:
150:
143:
142:
97:
95:
88:
83:
57:
56:
54:
47:
33:
26:
9:
6:
4:
3:
2:
4597:
4586:
4583:
4582:
4580:
4565:
4561:
4558:
4547:
4544:
4533:
4530:
4528:
4524:
4518:
4512:
4508:
4504:
4501:
4497:
4489:
4485:
4482:
4476:
4472:
4468:
4465:
4462:
4458:
4455:
4452:
4448:
4445:
4442:
4438:
4435:
4429:
4426:
4421:
4417:
4414:
4412:
4408:
4405:
4404:
4401:
4397:
4390:
4385:
4383:
4378:
4376:
4371:
4370:
4367:
4358:
4353:
4349:
4348:
4343:
4338:
4336:
4335:0-12-186350-6
4332:
4328:
4325:J.H. Conway,
4324:
4323:
4316:(2009, p.387)
4315:
4309:
4305:
4295:
4292:
4290:
4287:
4286:
4280:
4278:
4274:
4256:
4252:
4228:
4224:
4220:
4217:
4212:
4209:
4204:
4200:
4195:
4191:
4188:
4185:
4180:
4176:
4172:
4169:
4166:
4161:
4158:
4153:
4149:
4144:
4140:
4137:
4134:
4129:
4125:
4120:
4116:
4111:
4107:
4103:
4099:
4091:
4090:
4089:
4071:
4067:
4064:
4059:
4056:
4051:
4047:
4042:
4038:
4035:
4032:
4027:
4023:
4019:
4016:
4013:
4008:
4004:
4000:
3997:
3994:
3991:
3988:
3985:
3981:
3977:
3972:
3969:
3965:
3957:
3956:
3955:
3952:
3950:
3946:
3928:
3924:
3917:
3910:
3896:
3887:
3883:
3879:
3878:
3867:
3861:
3851:
3844:
3840:
3820:
3812:
3808:
3801:
3762:
3743:
3735:
3731:
3724:
3715:
3710:
3706:
3697:
3693:
3686:
3682:
3676:
3670:
3661:
3655:
3651:
3644:
3639:
3621:
3617:
3610:
3600:
3590:
3588:
3584:
3583:hydra theorem
3579:
3573:
3568:
3564:
3560:
3541:
3537:
3533:
3530:
3508:
3504:
3500:
3497:
3494:
3489:
3485:
3462:
3458:
3454:
3451:
3448:
3443:
3439:
3418:
3396:
3392:
3388:
3385:
3382:
3377:
3373:
3369:
3366:
3344:
3340:
3336:
3333:
3330:
3325:
3321:
3312:
3308:
3288:
3284:
3279:
3275:
3272:
3269:
3262:
3258:
3253:
3249:
3242:
3238:
3233:
3229:
3226:
3204:
3200:
3196:
3193:
3185:
3177:
3168:
3149:
3145:
3141:
3138:
3131:
3103:
3097:
3093:
3089:
3082:
3078:
3073:
3066:
3063:
3060:
3052:
3049:
3045:
3041:
3024:
3020:
3013:
2990:
2987:
2982:
2978:
2974:
2969:
2965:
2961:
2956:
2952:
2942:
2937:
2933:
2924:
2920:
2918:is countable.
2905:
2883:
2879:
2870:
2852:
2848:
2839:
2838:
2837:
2834:
2817:
2814:
2811:
2808:
2803:
2799:
2773:
2769:
2748:
2739:
2723:
2719:
2715:
2712:
2709:
2706:
2684:
2680:
2656:
2652:
2648:
2645:
2636:
2633:
2630:
2626:
2620:
2617:
2614:
2610:
2604:
2601:
2598:
2594:
2590:
2583:
2580:
2577:
2573:
2567:
2564:
2561:
2557:
2553:
2548:
2545:
2542:
2538:
2534:
2531:
2527:
2520:
2515:
2511:
2503:
2502:
2501:
2487:
2484:
2481:
2459:
2455:
2431:
2427:
2423:
2420:
2411:
2407:
2399:
2395:
2385:
2381:
2375:
2368:
2364:
2356:
2352:
2346:
2341:
2337:
2333:
2330:
2326:
2319:
2314:
2310:
2302:
2301:
2300:
2294:
2275:
2271:
2247:
2239:
2232:
2228:
2218:
2214:
2208:
2201:
2194:
2190:
2175:
2171:
2166:
2157:
2147:
2140:
2136:
2130:
2125:
2121:
2113:
2109:
2102:
2099:
2096:
2089:
2085:
2078:
2074:
2067:
2060:
2056:
2049:
2045:
2036:
2033:
2028:
2024:
2019:
2014:
2009:
2001:
1987:
1981:
1976:
1972:
1968:
1963:
1951:
1947:
1942:
1933:
1925:
1922:
1917:
1913:
1905:
1901:
1894:
1891:
1886:
1882:
1877:
1872:
1864:
1850:
1846:
1843:
1838:
1834:
1830:
1825:
1821:
1817:
1810:
1806:
1801:
1797:
1792:
1789:
1784:
1780:
1775:
1767:
1766:
1765:
1747:
1724:
1721:
1718:
1715:
1710:
1706:
1696:
1691:
1687:
1679:
1664:
1661:
1658:
1638:
1618:
1613:
1609:
1606:
1597:
1594:
1589:
1586:
1583:
1579:
1574:
1569:
1564:
1560:
1553:
1550:
1545:
1542:
1539:
1535:
1530:
1525:
1521:
1516:
1513:
1508:
1505:
1502:
1498:
1493:
1489:
1485:
1482:
1477:
1474:
1471:
1467:
1461:
1454:
1449:
1445:
1437:
1423:
1418:
1414:
1411:
1402:
1398:
1393:
1388:
1384:
1377:
1373:
1368:
1364:
1359:
1355:
1351:
1348:
1345:
1342:
1338:
1331:
1326:
1322:
1314:
1313:
1312:
1298:
1295:
1292:
1284:
1280:
1276:
1258:
1254:
1247:
1240:
1234:
1225:
1224:limit ordinal
1209:
1186:
1183:
1180:
1177:
1174:
1171:
1166:
1162:
1152:
1147:
1143:
1135:
1120:
1117:
1114:
1094:
1074:
1070:
1067:
1062:
1059:
1056:
1052:
1048:
1043:
1039:
1031:
1017:
1013:
1010:
1005:
1001:
993:
992:
991:
989:
978:
975:
967:
964:February 2023
957:
953:
949:
943:
942:
938:
933:This section
931:
927:
922:
921:
913:
910:
904:
898:
894:
887:
883:
876:
870:
866:
859:
855:
848:
843:
838:
834:
828:
824:
820:
813:
808:
804:
800:
797:
793:
788:
784:
780:
775:
773:
768:
766:
763:
759:
755:
752:, along with
751:
747:
743:
739:
731:
726:
722:
714:
708:
706:
702:
694:
677:
674:
661:
657:
652:
647:
642:
637:
633:
630:
627:
620:
616:
611:
607:
604:
601:
594:
590:
585:
581:
578:
575:
570:
567:
564:
560:
556:
551:
547:
543:
540:
537:
532:
528:
524:
519:
515:
505:
503:
499:
476:
471:
467:
464:
455:
451:
446:
441:
437:
430:
426:
421:
417:
412:
408:
404:
401:
397:
390:
377:
373:
368:
363:
358:
353:
349:
344:
340:
332:
331:
330:
328:
324:
320:
316:
312:
308:
302:
284:
279:
275:
271:
268:
261:
260:
259:
258:
254:
251:
247:
243:
239:
235:
231:
227:
223:
212:
209:
194:
191:
183:
173:
169:
163:
160:This article
158:
149:
148:
139:
136:
128:
118:
114:
108:
107:
101:
96:
87:
86:
81:
79:
72:
71:
66:
65:
60:
55:
46:
45:
40:
36:
19:
4563:
4516:
4510:
4506:
4495:
4487:
4474:
4470:
4460:
4450:
4440:
4419:
4415:
4410:
4356:
4346:
4326:
4313:
4308:
4272:
4243:
4087:
3953:
3875:
3873:
3859:
3849:
3842:
3838:
3713:
3708:
3704:
3695:
3691:
3684:
3680:
3677:
3668:
3659:
3653:
3649:
3642:
3602:
3580:
3571:
3306:
3175:
3166:
3127:
2835:
2740:
2671:
2446:
2292:
2262:
1763:
1279:fixed points
1232:
1229:
985:
970:
961:
946:Please help
934:
908:
902:
896:
892:
885:
881:
874:
868:
864:
857:
853:
846:
836:
832:
826:
822:
818:
811:
801:
795:
791:
783:Donald Knuth
776:
769:
729:
712:
709:
692:
506:
491:
323:epsilon zero
322:
318:
314:
313:(pronounced
306:
305:
303:
300:
252:
242:Georg Cantor
237:
234:fixed points
225:
219:
204:
186:
180:January 2023
177:
161:
131:
122:
103:
75:
68:
62:
61:Please help
58:
31:
3048:uncountable
1740:, whenever
1202:, whenever
705:Uncountable
222:mathematics
117:introducing
4300:References
3847:for every
879:such that
851:such that
816:such that
100:references
64:improve it
4253:ε
4221:…
4210:−
4201:ε
4196:ω
4186:−
4177:ε
4173:∣
4170:…
4150:ε
4145:ω
4126:ε
4100:ε
4068:…
4057:−
4048:ε
4043:ω
4033:−
4024:ε
4020:∣
4017:…
4009:ω
4005:ω
3998:ω
3970:−
3966:ε
3925:ω
3921:↦
3897:ω
3813:α
3809:φ
3805:↦
3802:α
3794:, ... of
3736:α
3732:φ
3728:↦
3725:α
3618:ε
3614:↦
3565:which is
3538:ω
3505:β
3498:…
3486:β
3459:β
3452:…
3440:β
3419:α
3393:β
3389:≥
3386:⋯
3383:≥
3374:β
3367:α
3341:β
3334:…
3322:β
3285:β
3280:ω
3273:⋯
3259:β
3254:ω
3239:β
3234:ω
3227:α
3201:ε
3194:α
3150:ε
3146:ω
3139:ε
3098:α
3094:ω
3083:α
3079:ω
3074:ε
3070:⇒
3064:≥
3061:α
3025:β
3021:ε
3017:↦
3014:β
2991:…
2979:ε
2966:ε
2953:ε
2938:ω
2934:ε
2923:non-empty
2906:β
2884:β
2880:ε
2869:countable
2867:is still
2849:ε
2818:β
2812:δ
2809:∣
2804:δ
2800:ε
2774:β
2770:ε
2749:β
2724:β
2720:ε
2713:δ
2685:β
2681:ε
2649:…
2634:−
2631:β
2627:ε
2618:−
2615:β
2611:ε
2602:−
2599:β
2595:ε
2581:−
2578:β
2574:ε
2565:−
2562:β
2558:ε
2546:−
2543:β
2539:ε
2516:β
2512:ε
2485:−
2482:β
2460:β
2456:ε
2424:…
2408:ε
2396:ε
2382:ε
2365:ε
2353:ε
2338:ε
2311:ε
2299:instead:
2272:ε
2240:ω
2229:ε
2215:ε
2202:ω
2191:ε
2172:ε
2167:ω
2148:ω
2137:ε
2131:⋅
2122:ε
2114:ω
2103:ω
2086:ε
2079:ω
2068:ω
2057:ε
2050:ω
2025:ε
2020:ω
2015:ω
2010:ω
1982:ω
1973:ε
1964:ω
1948:ε
1943:ω
1926:ω
1923:⋅
1914:ε
1906:ω
1883:ε
1878:ω
1873:ω
1847:ω
1844:⋅
1835:ε
1822:ω
1818:⋅
1807:ε
1802:ω
1781:ε
1776:ω
1748:β
1725:β
1719:δ
1716:∣
1711:δ
1707:ε
1692:β
1688:ε
1662:−
1659:β
1639:β
1610:…
1587:−
1584:β
1580:ε
1575:ω
1570:ω
1565:ω
1543:−
1540:β
1536:ε
1531:ω
1526:ω
1506:−
1503:β
1499:ε
1494:ω
1475:−
1472:β
1468:ε
1450:β
1446:ε
1415:…
1403:ω
1399:ω
1394:ω
1389:ω
1378:ω
1374:ω
1369:ω
1360:ω
1356:ω
1349:ω
1323:ε
1299:ω
1293:α
1259:β
1255:α
1251:↦
1248:β
1210:β
1187:β
1181:δ
1172:∣
1167:δ
1163:α
1148:β
1144:α
1118:−
1115:β
1095:β
1071:α
1068:⋅
1060:−
1057:β
1053:α
1044:β
1040:α
1002:α
935:does not
889:whenever
861:whenever
830:whenever
721:induction
701:countable
699:is still
678:…
662:⋅
658:⋅
653:⋅
648:ε
643:ε
638:ε
631:…
617:ε
612:ε
605:…
591:ε
586:ε
579:…
565:ω
561:ε
552:ω
548:ε
541:…
529:ε
516:ε
502:set union
468:…
456:ω
452:ω
447:ω
442:ω
431:ω
427:ω
422:ω
413:ω
409:ω
402:ω
378:⋅
374:⋅
369:⋅
364:ω
359:ω
354:ω
341:ε
280:ε
276:ω
269:ε
70:talk page
4579:Category
4344:(1965),
4283:See also
3759:—is the
3051:cardinal
1764:Because
1285:. When
498:supremum
257:equation
125:May 2021
4430: Γ
3046:of any
1281:by the
1239:mapping
956:removed
941:sources
891:1 <
863:0 <
785:in the
746:Gentzen
736:(as in
496:is the
236:of an
166:Please
113:improve
4562:
4505:
4486:
4469:
4459:
4449:
4439:
4418:
4409:
4333:
3945:domain
3712:(0) =
3589:game.
3305:where
1237:, the
1235:> 1
877:> 2
849:> 1
814:> 0
492:where
224:, the
102:, but
37:, see
3309:is a
1631:when
1273:is a
1222:is a
1087:when
895:<
867:<
835:<
4331:ISBN
4088:and
3909:-map
3652:) =
3370:>
3313:and
3197:<
3042:The
2815:<
2716:<
2710:<
1722:<
1184:<
1178:<
939:any
937:cite
781:and
4478:Ω+1
4463:(Ω)
4453:(Ω)
4443:(Ω)
3874:In
3852:≤ Γ
3700:(0)
3569:to
2946:sup
2524:sup
2323:sup
1700:sup
1458:sup
1335:sup
1156:sup
950:by
494:sup
394:sup
220:In
170:to
4581::
4520:+1
4493:(Ω
3884:,
3787:,
3780:,
3698:+1
3675:.
3578:.
2761:,
2738:.
912:.
884:=
856:=
854:αδ
825:=
821:+
799:.
794:→
774:.
73:.
4564:Ω
4553:1
4539:1
4522:)
4517:ω
4514:Ω
4511:ε
4509:(
4507:ψ
4499:)
4496:ω
4491:0
4488:ψ
4480:)
4475:ε
4473:(
4471:ψ
4461:θ
4451:θ
4441:θ
4432:0
4423:0
4420:ε
4411:ω
4388:e
4381:t
4374:v
4273:n
4257:n
4229:.
4225:}
4218:,
4213:1
4205:1
4192:,
4189:1
4181:1
4167:,
4162:1
4159:+
4154:0
4141:,
4138:1
4135:+
4130:0
4121:{
4117:=
4112:2
4108:/
4104:1
4072:}
4065:,
4060:1
4052:0
4039:,
4036:1
4028:0
4014:,
4001:,
3995:,
3992:1
3989:,
3986:0
3982:{
3978:=
3973:1
3929:n
3918:n
3863:1
3860:φ
3854:0
3850:β
3843:β
3839:φ
3824:)
3821:0
3818:(
3791:2
3789:Γ
3784:1
3782:Γ
3777:0
3775:Γ
3771:Γ
3766:0
3764:Γ
3747:)
3744:0
3741:(
3714:α
3709:α
3705:φ
3696:α
3692:φ
3685:α
3681:φ
3672:2
3669:φ
3663:1
3660:φ
3654:ω
3650:α
3648:(
3646:0
3643:φ
3622:x
3611:x
3575:0
3572:ε
3542:0
3534:=
3531:1
3509:k
3501:,
3495:,
3490:1
3463:k
3455:,
3449:,
3444:1
3397:k
3378:1
3345:k
3337:,
3331:,
3326:1
3307:k
3289:k
3276:+
3270:+
3263:2
3250:+
3243:1
3230:=
3205:0
3179:0
3176:ε
3170:0
3167:ε
3142:=
3122:0
3104:.
3090:=
3067:1
2994:}
2988:,
2983:2
2975:,
2970:1
2962:,
2957:0
2949:{
2943:=
2853:0
2821:}
2796:{
2707:1
2657:.
2653:}
2646:,
2637:1
2621:1
2605:1
2591:,
2584:1
2568:1
2554:,
2549:1
2535:,
2532:1
2528:{
2521:=
2488:1
2432:.
2428:}
2421:,
2412:0
2400:0
2386:0
2376:,
2369:0
2357:0
2347:,
2342:0
2334:,
2331:1
2327:{
2320:=
2315:1
2296:0
2293:ε
2276:1
2248:,
2233:0
2219:0
2209:=
2195:0
2183:)
2176:0
2163:(
2158:=
2153:)
2141:0
2126:0
2118:(
2110:=
2100:+
2097:1
2090:0
2075:=
2061:0
2046:=
2037:1
2034:+
2029:0
1988:,
1977:0
1969:=
1959:)
1952:0
1939:(
1934:=
1929:)
1918:0
1910:(
1902:=
1895:1
1892:+
1887:0
1851:,
1839:0
1831:=
1826:1
1811:0
1798:=
1793:1
1790:+
1785:0
1728:}
1703:{
1697:=
1677:.
1665:1
1619:,
1614:}
1607:,
1598:1
1595:+
1590:1
1561:,
1554:1
1551:+
1546:1
1522:,
1517:1
1514:+
1509:1
1490:,
1486:1
1483:+
1478:1
1462:{
1455:=
1424:,
1419:}
1412:,
1385:,
1365:,
1352:,
1346:,
1343:1
1339:{
1332:=
1327:0
1296:=
1233:α
1226:.
1190:}
1175:0
1159:{
1153:=
1133:.
1121:1
1075:,
1063:1
1049:=
1018:,
1014:1
1011:=
1006:0
977:)
971:(
966:)
962:(
958:.
944:.
909:ω
903:ω
897:ε
893:α
886:ε
882:α
875:ε
869:δ
865:α
858:δ
847:δ
837:γ
833:α
827:γ
823:γ
819:α
812:γ
796:ω
792:x
733:0
730:ε
716:0
713:ε
696:0
693:ε
675:,
634:,
628:,
621:1
608:,
602:,
595:0
582:,
576:,
571:1
568:+
557:,
544:,
538:,
533:2
525:,
520:1
477:,
472:}
465:,
438:,
418:,
405:,
398:{
391:=
350:=
345:0
310:0
307:ε
285:,
272:=
253:ε
211:)
205:(
193:)
187:(
182:)
178:(
164:.
138:)
132:(
127:)
123:(
109:.
80:)
76:(
41:.
34:0
32:ε
20:)
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