180:. Briefly, an ordinal notation is either the name zero (describing the ordinal 0), or the successor of an ordinal notation (describing the successor of the ordinal described by that notation), or a Turing machine (computable function) that produces an increasing sequence of ordinal notations (that describe the ordinal that is the limit of the sequence), and ordinal notations are (partially) ordered so as to make the successor of
6725:. Every recursively enumerable (or even hyperarithmetic) nonempty subset of this total ordering has a least element. So it resembles a well-ordering in some respects. For example, one can define the arithmetic operations on it. Yet it is not possible to effectively determine exactly where the initial well-ordered part ends and the part lacking a least element begins.
604:
165:. There are several equivalent definitions of this: the simplest is to say that a computable ordinal is the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an ordinal is recursive when we can present the set of smaller ordinals in such a way that a computer (
1699:. It can be described as the set of all ordinals that can be written as finite expressions, starting from zero, using only the Veblen hierarchy and addition. The Feferman–Schütte ordinal is important because, in a sense that is complicated to make precise, it is the smallest (infinite) ordinal that cannot be ("
3619:
axioms, one gets some extremely large recursive ordinals. (Strictly speaking it is not known that all of these really are ordinals: by construction, the ordinal strength of a theory can only be proved to be an ordinal from an even stronger theory. So for the large cardinal axioms this becomes quite
1794:
To go far beyond the
Feferman–Schütte ordinal, one needs to introduce new methods. Unfortunately there is not yet any standard way to do this: every author in the subject seems to have invented their own system of notation, and it is quite hard to translate between the different systems. The first
203:
It is tempting to forget about ordinal notations, and only speak of the recursive ordinals themselves: and some statements are made about recursive ordinals which, in fact, concern the notations for these ordinals. This leads to difficulties, however, as even the smallest infinite ordinal, ω, has
3602:
By dropping the requirement of having a concrete description, even larger recursive countable ordinals can be obtained as the ordinals measuring the strengths of various strong theories; roughly speaking, these ordinals are the smallest order types of "natural" ordinal notations that the theories
4870:
into a smaller ordinal. (This is trivially true for regular cardinals; however, we are mainly interested in countable ordinals.) Being nonprojectible is a much stronger condition than being admissible, recursively inaccessible, or even recursively Mahlo. By Jensen's method of projecta, this
1827:) is defined to be the smallest ordinal that cannot be constructed by starting with 0, 1, ω and Ω, and repeatedly applying addition, multiplication and exponentiation, and ψ to previously constructed ordinals (except that ψ can only be applied to arguments less than
461:
1815:, and Pohlers. However most systems use the same basic idea, of constructing new countable ordinals by using the existence of certain uncountable ordinals. Here is an example of such a definition, described in much greater detail in the article on
192:
of ordinal notations itself is highly non-recursive, owing to the impossibility of deciding whether a given Turing machine does indeed produce a sequence of notations); a recursive ordinal is then an ordinal described by some ordinal notation.
307:, and, in fact, merely adding to Peano's axioms the axioms that state the well-ordering of all ordinals below the Bachmann–Howard ordinal is sufficient to obtain all arithmetical consequences of Kripke–Platek set theory.
3412:
Next is the last unnamed ordinal, referred by David Madore as the proof-theoretic ordinal of
Stability. This is the proof-theoretic ordinal of Stability, an extension of Kripke-Platek set theory. Its value is equal to
1428:
1876:
To construct still larger ordinals, we can extend the definition of ψ by throwing in more ways of constructing uncountable ordinals. There are several ways to do this, described to some extent in the article on
2433:
5951:
4438:
is recursively Mahlo and more, but even if we limit ourselves to countable ordinals, ZFC proves the existence of recursively Mahlo ordinals. They are, however, beyond the reach of Kripke–Platek set theory.)
5407:
2364:
7313:
1032:
6666:
6461:
6341:
6217:
6060:
1906:. Indeed, the main importance of these large ordinals, and the reason to describe them, is their relation to certain formal systems as explained above. However, such powerful formal systems as full
6872:
3456:
2486:
4405:
2579:
1616:
1548:
832:
599:{\displaystyle \varepsilon _{0}+1,\qquad \omega ^{\varepsilon _{0}+1}=\varepsilon _{0}\cdot \omega ,\qquad \omega ^{\omega ^{\varepsilon _{0}+1}}=(\varepsilon _{0})^{\omega },\qquad {\text{etc.}}}
1719:
It is, of course, possible to describe ordinals beyond the
Feferman–Schütte ordinal. One could continue to seek fixed points in a more and more complicated manner: enumerate the fixed points of
983:
4066:
4001:
3293:
1750:
6688:
some represent ordinals and some do not. One can define a recursive total ordering that is a subset of the Kleene notations and has an initial segment which is well-ordered with order-type
747:
1666:
1967:
874:
6723:
3945:
3812:
3717:
3680:
1259:
1223:
1088:
449:
7819:
5231:
4129:
2526:
783:
3156:
3018:
2912:
2695:
2619:
660:
368:
2169:
2003:
5689:
5599:
5490:
6802:
3192:
687:
133:, but they are large among those that have constructive notations (descriptions). Larger and larger ordinals can be defined, but they become more and more difficult to describe.
6926:
4247:
2051:
5756:
These are weakened variants of stable ordinals. There are ordinals with these properties smaller than the aforementioned least nonprojectible ordinal, for example an ordinal is
3054:
2948:
2778:
2226:
1574:
1506:
1312:
1285:
1161:
1135:
921:
6612:
1480:
415:
60:). However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the
2088:
5721:
5631:
2870:
1454:
2652:
2133:
5815:
4359:
4161:
3244:
3110:
2834:
2288:
2258:
1342:
7845:
5658:
5568:
5521:
5290:
5133:
5057:
5034:
4958:
4931:
4904:
4535:
3904:
3839:
3775:
3748:
3503:
3340:
2722:
1697:
260:
can easily be arithmetically described (it is countable), the Peano axioms are not strong enough to show that it is indeed an ordinal; in fact, transfinite induction on ε
714:
5888:
3570:
3550:
3407:
3387:
5066:
work on projecta. The least ordinals that are nonprojectible relative to a given set are tied to
Harrington's construction of the smallest reflecting Spector 2-class.
4796:
4737:
4700:
4649:
4614:
6524:
6501:
6404:
6381:
6284:
6261:
6160:
6127:
6104:
5973:
5859:
5741:
5450:
5346:
5326:
5263:
5186:
5106:
4868:
4848:
4824:
4555:
4506:
4485:
4465:
4319:
4299:
4279:
4204:
4184:
4087:
4021:
2308:
1367:
1187:
1108:
1052:
388:
6481:
6361:
6237:
6080:
6003:
941:
894:
627:
6946:
6892:
4575:
4425:
3530:
3367:
6586:
6553:
5783:
3212:
4984:
5007:
5835:
5541:
5430:
5156:
4769:
4673:
3476:
3313:
3074:
2968:
2798:
2742:
7087:
3641:
be described in a recursive way. (It is not the order type of any recursive well-ordering of the integers.) That ordinal is a countable ordinal called the
5088:(a hypothesis stronger than the mere hypothesis of consistency, and implied by the existence of an inaccessible cardinal), then there exists a countable
6980:(for Veblen hierarchy and some impredicative ordinals). This is probably the most readable book on large countable ordinals (which is not saying much).
4430:
But note that we are still talking about possibly countable ordinals here. (While the existence of inaccessible or Mahlo cardinals cannot be proved in
7146:
7047:
7019:
6965:
204:
many notations, some of which cannot be proved to be equivalent to the obvious notation (the simplest program that enumerates all natural numbers).
7071:
6735:
or another recursively axiomatizable theory that has an ω-model but no hyperarithmetical ω-models, and (if needed) conservatively extend S with
3962:, the countable admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal but for Turing machines with
196:
Any ordinal smaller than a recursive ordinal is itself recursive, so the set of all recursive ordinals forms a certain (countable) ordinal, the
1375:
7224:
7585:. In: Barwise, J. (eds) The Syntax and Semantics of Infinitary Languages. Lecture Notes in Mathematics, vol 72. Springer, Berlin, Heidelberg.
6808:) (for the first n formulas φ with one numeric free variable; ⌈φ⌉ is the Gödel number) has no inconsistency proof shorter than n. Then the
8005:
7091:
2376:
3719:
is the smallest non-recursive ordinal, and there is no hope of precisely "describing" any ordinals from this point on—we can only
269:
8145:
5896:
5135:
is a model of ZFC. Such ordinals are beyond the strength of ZFC in the sense that it cannot (by construction) prove their existence.
2097:
are isomorphic to
Buchholz's ordinal notation, the ordinals up to this point can be expressed using hydras from the game. For example
8121:
2175:
5351:
749:, but since the Greek alphabet does not have transfinitely many letters it is better to use a more robust notation: define ordinals
2313:
17:
7254:
4069:
is the smallest limit of admissible ordinals (mentioned later), yet the ordinal itself is not admissible. It is also the smallest
6729:
1704:
988:
7661:
6617:
6412:
6292:
6168:
6011:
7234:
6818:
3416:
2441:
4364:
2534:
1579:
1511:
788:
1922:
Beyond this, there are multiple recursive ordinals which aren't as well known as the previous ones. The first of these is
946:
129:
is used throughout, except where otherwise noted. The ordinals described here are not as large as the ones described in
3863:) was the smallest ordinal for which KP does not prove transfinite induction, the Church–Kleene ordinal is the smallest
3575:
Next is a group of ordinals which not that much are known about, but are still fairly significant (in ascending order):
4034:
3969:
3253:
1722:
7930:
7769:
7347:
7121:
7041:
7013:
6995:
6977:
4702:-reflecting ordinal is also the supremum of the closure ordinals of monotonic inductive definitions whose graphs are
1839:
is the first uncountable ordinal. It is put in because otherwise the function ψ gets "stuck" at the smallest ordinal
6672:
Stronger weakenings of stability have appeared in proof-theoretic publications, including analysis of subsystems of
3586:
A possible limit of
Taranovsky's C ordinal notation. (Conjectural, assuming well-foundedness of the notation system)
719:
7998:
1629:
1929:
837:
6691:
5081:
4431:
3913:
3780:
3685:
3648:
3612:
3590:
1911:
1228:
1192:
1057:
420:
8046:
7793:
5191:
4092:
2491:
1669:
752:
6956:
Most books describing large countable ordinals are on proof theory, and unfortunately tend to be out of print.
3119:
2981:
2875:
2657:
2584:
1716:
enumerates the ordinals that cannot be obtained from smaller ordinals using addition and the Veblen functions.
632:
340:
2138:
1972:
7056:
5666:
5576:
331:
250:
7558:
5455:
4616:-reflection schema. They can also be considered "recursive analogues" of some uncountable cardinals such as
2840:
augmented by the recursive inaccessibility of the class of ordinals (KPi), or, on the arithmetical side, of
6742:
3856:
3247:
3164:
3113:
2975:
2837:
1903:
665:
300:
7906:
327:
8085:
7991:
6897:
4986:) is not a strong enough axiom schema to imply nonprojectibility, in fact there are transitive models of
4213:
2008:
1886:
1878:
1816:
1789:
304:
7498:
3026:
2920:
2750:
2370:
2181:
1553:
1485:
1290:
1264:
1140:
1113:
899:
299:
But we can do this for systems far beyond Peano's axioms. For example, the proof-theoretic strength of
188:
and to make the limit greater than any term of the sequence (this order is computable; however, the set
67:
Since there are only countably many notations, all ordinals with notations are exhausted well below the
7026:
Math. Intelligencer 4 (1982), no. 4, 182–189; contains an informal description of the Veblen hierarchy.
6591:
5075:
1459:
393:
7975:
4249:. An ordinal that is both recursively inaccessible and a limit of recursively inaccessibles is called
2060:
8178:
7438:
7250:
5694:
5604:
5159:
3642:
2843:
1902:) with the notation above) is an important one, because it describes the proof-theoretic strength of
1433:
197:
68:
4253:. There exists a theory of large ordinals in this manner that is highly parallel to that of (small)
227:
Certain computable ordinals are so large that while they can be given by a certain ordinal notation
8203:
6809:
4586:
4434:, that of recursively inaccessible or recursively Mahlo ordinals is a theorem of ZFC: in fact, any
3947:
is the smallest admissible ordinal (beyond ω in case the axiom of infinity is not included in KP).
2624:
2366:. It is the supremum of the range of Buchholz's psi functions. It was first named by David Madore.
2100:
293:
7734:
6739:. Let T be the tree of (essentially) finite partial ω-models of S: A sequence of natural numbers
5788:
4328:
4134:
3217:
3083:
2807:
2263:
2231:
1317:
7823:
7601:
6673:
5636:
5546:
5499:
5268:
5111:
5039:
5012:
4936:
4909:
4882:
4617:
4513:
4254:
3882:
3817:
3753:
3726:
3604:
3580:
3481:
3318:
3077:
2700:
2053:, a first-order theory of arithmetic allowing quantification over the natural numbers as well as
1907:
1869:≤Ω. However the fact that we included Ω allows us to get past this point: ψ(Ω+1) is greater than
1675:
273:
64:); various more-concrete ways of defining ordinals that definitely have notations are available.
692:
161:(or recursive ordinals) are certain countable ordinals: loosely speaking those represented by a
5864:
5744:
5293:
4872:
4703:
4621:
3868:
3555:
3535:
3392:
3372:
277:
4774:
4715:
4678:
4627:
4592:
8208:
8102:
6509:
6486:
6389:
6366:
6269:
6246:
6132:
6112:
6089:
5958:
5844:
5726:
5435:
5331:
5311:
5248:
5171:
5091:
4853:
4833:
4809:
4740:
4540:
4491:
4470:
4450:
4304:
4284:
4264:
4189:
4169:
4072:
4006:
2801:
2293:
1352:
1172:
1093:
1037:
373:
236:
7559:"Ordinal proof theory of Kripke-Platek set theory augmented by strong reflection principles"
6466:
6346:
6222:
6065:
5978:
5304:; the existence of these ordinals can be proved in ZFC, and they are closely related to the
926:
879:
612:
8164:
8075:
8065:
7809:, Studies in Logic and the Foundations of Mathematics (vol. 79, 1974). Accessed 2022-12-04.
7357:
7332:
Iterated
Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies
6931:
6877:
4560:
4410:
3508:
3345:
1873:. The key property of Ω that we used is that it is greater than any ordinal produced by ψ.
1777:
1773:
6562:
6529:
5759:
3197:
8:
7453:"Collapsing functions based on recursively large ordinals: A well-ordering proof for KPM"
4963:
4166:
An ordinal that is both admissible and a limit of admissibles, or equivalently such that
3161:
Next is another unnamed ordinal, referred by David Madore as the "countable" collapse of
3023:
Next is another unnamed ordinal, referred by David Madore as the "countable" collapse of
2917:
Next is another unnamed ordinal, referred by David Madore as the "countable" collapse of
173:
162:
50:
4989:
122:). Countable ordinals larger than this may still be defined, but do not have notations.
8150:
7955:
7691:
7480:
7110:
7061:
7029:
6240:
6083:
5820:
5526:
5415:
5141:
4754:
4658:
3850:
3608:
3461:
3298:
3059:
2953:
2783:
2727:
158:
142:
126:
42:
7922:
7753:
7633:
8055:
7936:
7926:
7775:
7765:
7615:
7596:
7472:
7343:
7230:
7205:
7200:
7183:
7117:
7037:
7009:
6991:
6973:
3634:
243:
7484:
7410:
2747:
Next is an unnamed ordinal, referred by David Madore as the "countable" collapse of
7918:
7757:
7749:
7610:
7526:
7464:
7335:
7327:
7195:
5085:
4748:
4435:
2531:
This next ordinal is, once again, mentioned in this same piece of code, defined as
2438:
The next ordinal is mentioned in the same piece of code as earlier, and defined as
1803:
manner), and different extensions and variations of it were described by
Buchholz,
1703:") described using smaller ordinals. It measures the strength of such systems as "
1165:
177:
153:
57:
53:
41:. The smallest ones can be usefully and non-circularly expressed in terms of their
7162:
8025:
7353:
7334:. Lecture Notes in Mathematics. Vol. 897. Springer-Verlag, Berlin-New York.
7150:
6736:
3951:
1345:
321:
265:
130:
61:
7680:, Studies in Logic and the Foundation of Mathematics vol. 94 (1978), pp.147--183
7001:
3603:
cannot prove are well ordered. By taking stronger and stronger theories such as
1812:
272:, Peano's axioms cannot formalize that reasoning. (This is at the basis of the
8034:
7859:
7806:
6556:
6407:
6287:
6163:
6006:
5891:
4322:
3963:
3616:
2971:
2094:
1796:
166:
38:
5080:
We can imagine even larger ordinals that are still countable. For example, if
1923:
8197:
7940:
7779:
7476:
7229:. Perspectives in Logic (2 ed.). Cambridge: Cambridge University Press.
7209:
6983:
6685:
5063:
3955:
3859:, but now in a different way: whereas the Bachmann–Howard ordinal (described
1804:
1700:
213:
35:
7711:
7983:
7872:
7690:
Arai, Toshiyasu (2015). "A simplified analysis of first-order reflection".
3959:
2310:-times iterated inductive definitions". In this notation, it is defined as
246:
221:
46:
4321:
contains an admissible ordinal (a recursive analog of the definition of a
1169:(there are inessential variations in the definition, such as letting, for
7105:
4023:-th ordinal that is either admissible or a limit of smaller admissibles.
1808:
1423:{\displaystyle \varphi _{\alpha }(\beta )<\varphi _{\gamma }(\delta )}
7735:"Inductive Definitions and Reflecting Properties of Admissible Ordinals"
7676:
A. Kechris, "Spector Second-order
Classes and Reflection". Appearing in
7076:
231:, a given formal system might not be sufficiently powerful to show that
7761:
7678:
Generalized
Recursion Theory II: Proceedings of the 1977 Oslo Symposium
7468:
7339:
7154:
7081:
7065:
3750:. However, as its symbol suggests, it behaves in many ways rather like
217:
31:
7713:
Inductive Definitions and Reflection Properties of Admissible Ordinals
7696:
7582:
7452:
1287:: this essentially just shifts the indices by 1, which is harmless).
7960:
7954:
Arai, Toshiyasu (1996). "Introducing the hardline in proof theory".
7860:
An introduction to the fine structure of the constructible hierarchy
7807:
An introduction to the fine structure of the constructible hierarchy
2428:{\displaystyle \psi _{0}(\Omega _{\omega +1}\cdot \varepsilon _{0})}
2005:, using the previous notation. It is the proof-theoretic ordinal of
7876:
3723:
them. But it is still far less than the first uncountable ordinal,
315:
75:
7112:
Admissible Sets and Structures: an Approach to Definability Theory
3777:. For instance, one can define ordinal collapsing functions using
2090:, the "formal theory of finitely iterated inductive definitions".
5946:{\displaystyle L_{\alpha }\prec _{\Sigma _{1}}L_{\alpha +\beta }}
3246:-indescribable cardinal. This is the proof-theoretic ordinal of
3112:-indescribable) cardinal. This is the proof-theoretic ordinal of
2836:-indescribable) cardinal. This is the proof-theoretic ordinal of
4577:. These ordinals appear in ordinal analysis of theories such as
7880:
5402:{\displaystyle L_{\alpha }\prec _{\Sigma _{1}}L_{\omega _{1}}}
3597:
2974:. This is the proof-theoretic ordinal of KPM, an extension of
2872:-comprehension + transfinite induction. Its value is equal to
7077:
Transfinite Ordinals and Their Notations: For The Uninitiated
6728:
For an example of a recursive pseudo-well-ordering, let S be
2359:{\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})}
169:, say) can manipulate them (and, essentially, compare them).
7308:{\displaystyle (\Pi _{1}^{1}{\mathsf {-CA}}){\mathsf {+BI}}}
7044:(describes recursive ordinals and the Church–Kleene ordinal)
5245:, can be defined by indescribability conditions or as those
2369:
The next ordinal is mentioned in a piece of code describing
451:, ... The next ordinal satisfying this equation is called ε
212:
There is a relation between computable ordinals and certain
7325:
7140:
6363:
is the least recursively inaccessible ordinal larger than
1917:
1768:
steps of this process, and continue diagonalizing in this
1621:
1027:{\displaystyle \varphi _{1}(\beta )=\varepsilon _{\beta }}
235:
is, indeed, an ordinal notation: the system does not show
7034:
Theory of Recursive Functions and Effective Computability
6661:{\displaystyle L_{\alpha }\prec _{\Sigma _{1}}L_{\beta }}
6456:{\displaystyle L_{\alpha }\prec _{\Sigma _{1}}L_{\beta }}
6336:{\displaystyle L_{\alpha }\prec _{\Sigma _{1}}L_{\beta }}
6212:{\displaystyle L_{\alpha }\prec _{\Sigma _{1}}L_{\beta }}
6055:{\displaystyle L_{\alpha }\prec _{\Sigma _{1}}L_{\beta }}
97:(not to be confused with the first uncountable ordinal, ω
4739:-reflecting ordinals also have a characterization using
4210:, and the least recursively inaccessible may be denoted
207:
7116:. Perspectives in Mathematical Logic. Springer-Verlag.
5158:
is a recursively enumerable set theory consistent with
264:
proves the consistency of Peano's axioms (a theorem by
119:
7135:. Perspectives in Mathematical Logic. Springer-Verlag.
7016:(for Veblen hierarchy and some impredicative ordinals)
6867:{\displaystyle \omega _{1}^{CK}\times (1+\eta )+\rho }
5233:
is less than the least stable ordinal, which follows.
3451:{\displaystyle \Psi _{X}^{\varepsilon _{\Upsilon +1}}}
2481:{\displaystyle \psi _{0}(\Omega _{\omega ^{\omega }})}
7976:
The countable admissible ordinal equivalence relation
7826:
7257:
6934:
6900:
6880:
6821:
6745:
6694:
6620:
6594:
6565:
6532:
6512:
6489:
6469:
6415:
6392:
6369:
6349:
6295:
6272:
6249:
6225:
6171:
6135:
6115:
6092:
6068:
6014:
5981:
5961:
5899:
5867:
5847:
5823:
5791:
5762:
5729:
5697:
5669:
5639:
5607:
5579:
5549:
5529:
5502:
5458:
5438:
5418:
5354:
5334:
5314:
5305:
5271:
5251:
5194:
5174:
5144:
5114:
5094:
5042:
5015:
4992:
4966:
4939:
4912:
4885:
4856:
4836:
4812:
4777:
4757:
4718:
4681:
4661:
4630:
4595:
4563:
4543:
4516:
4494:
4473:
4453:
4413:
4400:{\displaystyle L_{\rho }\cap {\mathcal {P}}(\omega )}
4367:
4331:
4307:
4287:
4267:
4216:
4192:
4172:
4137:
4095:
4075:
4037:
4009:
3972:
3916:
3885:
3820:
3783:
3756:
3729:
3688:
3651:
3558:
3538:
3511:
3484:
3464:
3419:
3395:
3375:
3348:
3321:
3301:
3256:
3220:
3200:
3167:
3122:
3086:
3062:
3029:
2984:
2956:
2923:
2878:
2846:
2810:
2786:
2753:
2730:
2703:
2660:
2627:
2587:
2574:{\displaystyle \psi _{0}(\Omega _{\varepsilon _{0}})}
2537:
2494:
2444:
2379:
2316:
2296:
2266:
2234:
2184:
2141:
2103:
2063:
2011:
1975:
1932:
1725:
1678:
1632:
1611:{\displaystyle \varphi _{\alpha }(\beta )<\delta }
1582:
1556:
1543:{\displaystyle \beta <\varphi _{\gamma }(\delta )}
1514:
1488:
1462:
1436:
1378:
1355:
1320:
1293:
1267:
1231:
1195:
1175:
1143:
1116:
1096:
1060:
1040:
991:
949:
929:
902:
882:
840:
827:{\displaystyle \varphi _{0}(\beta )=\omega ^{\beta }}
791:
755:
722:
695:
668:
635:
615:
464:
423:
396:
376:
343:
136:
7499:"Ordinal notations based on a weakly Mahlo cardinal"
7060:, volume 41, number 2, June 1976, pages 439 to 459,
2228:; and another subsystem of second-order arithmetic:
7911:
Studies in Logic and the Foundations of Mathematics
7742:
Studies in Logic and the Foundations of Mathematics
7184:"A new system of proof-theoretic ordinal functions"
7052:
Normal Functions and Constructive Ordinal Notations
6483:is the least recursively Mahlo ordinal larger than
978:{\displaystyle \varphi _{\gamma }(\alpha )=\alpha }
249:do not prove transfinite induction for (or beyond)
7839:
7441:(1984) (lemmata 1.3 and 1.8). Accessed 2022-05-04.
7307:
7109:
6940:
6920:
6886:
6866:
6796:
6717:
6660:
6606:
6580:
6547:
6518:
6495:
6475:
6455:
6398:
6375:
6355:
6335:
6278:
6255:
6231:
6211:
6154:
6121:
6098:
6074:
6054:
5997:
5967:
5945:
5882:
5853:
5829:
5809:
5777:
5735:
5715:
5683:
5652:
5625:
5593:
5562:
5535:
5515:
5484:
5444:
5432:has some definability-related properties. Letting
5424:
5401:
5340:
5320:
5308:from a model-theoretic perspective. For countable
5284:
5257:
5225:
5180:
5150:
5127:
5100:
5069:
5051:
5028:
5001:
4978:
4952:
4925:
4898:
4871:statement is equivalent to the statement that the
4862:
4842:
4818:
4790:
4763:
4731:
4694:
4667:
4643:
4608:
4569:
4549:
4529:
4500:
4479:
4459:
4419:
4399:
4353:
4313:
4293:
4273:
4241:
4198:
4178:
4155:
4123:
4081:
4060:
4015:
3995:
3939:
3898:
3833:
3806:
3769:
3742:
3711:
3674:
3564:
3544:
3524:
3497:
3470:
3450:
3401:
3381:
3361:
3334:
3307:
3287:
3238:
3206:
3186:
3150:
3104:
3068:
3048:
3012:
2962:
2942:
2906:
2864:
2828:
2792:
2772:
2736:
2716:
2689:
2646:
2613:
2573:
2520:
2480:
2427:
2358:
2302:
2282:
2252:
2220:
2163:
2127:
2082:
2045:
1997:
1961:
1744:
1691:
1660:
1610:
1568:
1542:
1500:
1474:
1448:
1422:
1361:
1336:
1306:
1279:
1253:
1217:
1181:
1155:
1129:
1102:
1082:
1046:
1026:
977:
935:
915:
888:
868:
826:
777:
741:
708:
681:
654:
621:
598:
443:
409:
382:
362:
7796:" (2013, unpublished). Accessed 18 November 2022.
7728:
7726:
7724:
7722:
7439:A new system of proof-theoretic ordinal functions
7161:. (Cambridge University Press, 1999) 219–279. At
4537:satisfies a certain reflection property for each
4061:{\displaystyle \omega _{\omega }^{\mathrm {CK} }}
3996:{\displaystyle \omega _{\alpha }^{\mathrm {CK} }}
3288:{\displaystyle \Psi _{X}^{\varepsilon _{\Xi +1}}}
2978:based on a Mahlo cardinal. Its value is equal to
1756:, and so on, and then look for the first ordinal
8195:
7794:Indescribable Cardinals and Admissible Analogues
7597:"Countable admissible ordinals and hyperdegrees"
1745:{\displaystyle \alpha \mapsto \Gamma _{\alpha }}
337:, which is the smallest satisfying the equation
316:Predicative definitions and the Veblen hierarchy
5743:-recursively enumerable, in the terminology of
5036:-separation of any countable admissible height
4960:-separation on its own (not in the presence of
3020:using one of Buchholz's various psi functions.
8146:the theories of iterated inductive definitions
7719:
7672:
7670:
7098:
6243:larger than an admissible ordinal larger than
5751:
3855:The Church–Kleene ordinal is again related to
3615:, or Zermelo–Fraenkel set theory with various
1772:manner. This leads to the definition of the "
742:{\displaystyle \varepsilon _{\alpha }=\alpha }
310:
102:
7999:
7907:"Short Course on Admissible Recursion Theory"
7883:page), Państwowe Wydawn. Accessed 2022-12-01.
4026:
2621:. In general, the proof-theoretic ordinal of
2260:- comprehension + transfinite induction, and
1661:{\displaystyle \varphi _{\alpha }(0)=\alpha }
220:, that is, at least a reasonable fragment of
45:. Beyond that, many ordinals of relevance to
34:, there are many ways of describing specific
8013:
7732:
4743:on ordinal functions, lending them the name
4325:). The 1-section of Harrington's functional
3628:
3623:
2371:large countable ordinals and numbers in Agda
1962:{\displaystyle \psi _{0}(\Omega _{\omega })}
1163:. This family of functions is known as the
869:{\displaystyle \varphi _{\gamma +1}(\beta )}
7733:Richter, Wayne; Aczel, Peter (1974-01-01).
7667:
7664:" (1976), p.387. Accessed 13 February 2023.
7330:; Pohlers, Wolfram; Sieg, Wilfried (1981).
6815:Any such construction must have order type
6718:{\displaystyle \omega _{1}^{\mathrm {CK} }}
5241:Even larger countable ordinals, called the
3940:{\displaystyle \omega _{1}^{\mathrm {CK} }}
3860:
3807:{\displaystyle \omega _{1}^{\mathrm {CK} }}
3712:{\displaystyle \omega _{1}^{\mathrm {CK} }}
3675:{\displaystyle \omega _{1}^{\mathrm {CK} }}
1254:{\displaystyle \varphi _{\gamma }(\alpha )}
1218:{\displaystyle \varphi _{\delta }(\alpha )}
1083:{\displaystyle \varphi _{\delta }(\alpha )}
444:{\displaystyle \omega ^{\omega ^{\omega }}}
370:, so it is the limit of the sequence 0, 1,
8006:
7992:
7978:(2017), p.1233. Accessed 28 December 2022.
5226:{\displaystyle (L_{\alpha },\in )\vDash T}
4124:{\displaystyle L_{\alpha }\cap P(\omega )}
2521:{\displaystyle ID_{<\omega ^{\omega }}}
1807:(ordinal diagrams), Feferman (θ systems),
778:{\displaystyle \varphi _{\gamma }(\beta )}
7959:
7695:
7614:
7199:
6905:
6679:
5677:
5587:
3151:{\displaystyle \Psi (\varepsilon _{K+1})}
3013:{\displaystyle \psi (\varepsilon _{M+1})}
2907:{\displaystyle \psi (\varepsilon _{I+1})}
2690:{\displaystyle \psi _{0}(\Omega _{\nu })}
2614:{\displaystyle ID_{<\varepsilon _{0}}}
1783:
785:by transfinite induction as follows: let
655:{\displaystyle \omega ^{\alpha }=\alpha }
363:{\displaystyle \omega ^{\alpha }=\alpha }
7181:
7141:Both recursive and nonrecursive ordinals
6959:
6812:of T is a recursive pseudowellordering.
4427:is the least recursively Mahlo ordinal.
2164:{\displaystyle \psi (\Omega _{\omega })}
1998:{\displaystyle \psi (\Omega _{\omega })}
125:Due to the focus on countable ordinals,
7904:
7634:"Subsystems of Second-Order Arithmetic"
7556:
7450:
7222:
7104:
5684:{\displaystyle x\subseteq \mathbb {N} }
5594:{\displaystyle x\subseteq \mathbb {N} }
2488:. It is the proof-theoretic ordinal of
1918:Beyond even the Bachmann-Howard ordinal
1622:The Feferman–Schütte ordinal and beyond
14:
8196:
7405:
7403:
7401:
7399:
7397:
7395:
7393:
7391:
7389:
7387:
7300:
7297:
7294:
7284:
7281:
7278:
7149:, "The realm of ordinal analysis." in
7130:
5485:{\displaystyle L_{\sigma }\prec _{1}L}
4850:-recursive injective function mapping
4771:, a similar property corresponding to
4301:-recursive closed unbounded subset of
3844:
2697:— note that in this certain instance,
27:Ordinals in mathematics and set theory
7987:
7900:
7898:
7628:
7626:
7594:
7552:
7550:
7385:
7383:
7381:
7379:
7377:
7375:
7373:
7371:
7369:
7367:
7226:Subsystems of Second Order Arithmetic
6797:{\displaystyle x_{1},x_{2},...,x_{n}}
3187:{\displaystyle \varepsilon _{\Xi +1}}
1831:, to ensure that it is well defined).
1752:, then enumerate the fixed points of
682:{\displaystyle \varepsilon _{\iota }}
270:Gödel's second incompleteness theorem
208:Relationship to systems of arithmetic
7953:
7689:
7169:
7024:The varieties of arboreal experience
5062:Nonprojectible ordinals are tied to
4801:
3458:using Stegert's Psi function, where
3295:using Stegert's Psi function, where
2581:, is the proof-theoretic ordinal of
1914:, seem beyond reach for the moment.
147:
6921:{\displaystyle (\mathbb {Q} ,<)}
4242:{\displaystyle \omega _{1}^{E_{1}}}
2373:, and defined by "AndrasKovacs" as
2046:{\displaystyle \Pi _{1}^{1}-CA_{0}}
24:
7905:Simpson, Stephen G. (1978-01-01).
7895:
7862:(1974). Accessed 21 February 2023.
7828:
7623:
7547:
7364:
7262:
7080:, expository article (8 pages, in
6709:
6706:
6637:
6432:
6312:
6188:
6031:
5916:
5793:
5699:
5609:
5504:
5371:
5236:
5017:
4941:
4914:
4779:
4720:
4683:
4632:
4597:
4544:
4495:
4454:
4383:
4346:
4206:-th admissible ordinal, is called
4139:
4052:
4049:
3987:
3984:
3931:
3928:
3867:such that the construction of the
3798:
3795:
3703:
3700:
3666:
3663:
3435:
3421:
3272:
3258:
3222:
3201:
3173:
3123:
3116:+ Π3 - Ref. Its value is equal to
3088:
3049:{\displaystyle \varepsilon _{K+1}}
2943:{\displaystyle \varepsilon _{M+1}}
2848:
2812:
2773:{\displaystyle \varepsilon _{I+1}}
2705:
2675:
2552:
2459:
2394:
2336:
2236:
2221:{\displaystyle \Pi _{1}^{1}-CA+BI}
2186:
2149:
2013:
1983:
1947:
1733:
1705:arithmetical transfinite recursion
1680:
1569:{\displaystyle \alpha >\gamma }
1501:{\displaystyle \alpha <\gamma }
1307:{\displaystyle \varphi _{\gamma }}
1280:{\displaystyle \gamma <\delta }
1156:{\displaystyle \gamma <\delta }
1130:{\displaystyle \varphi _{\gamma }}
916:{\displaystyle \varphi _{\gamma }}
716:as the smallest ordinal such that
455:: it is the limit of the sequence
284:prove that any ordinal less than ε
137:Generalities on recursive ordinals
30:In the mathematical discipline of
25:
8220:
8122:Takeuti–Feferman–Buchholz ordinal
7581:Friedman, H., Jensen, R. (1968).
6607:{\displaystyle \beta >\alpha }
3598:"Unrecursable" recursive ordinals
2178:, the proof-theoretic ordinal of
2176:Takeuti-Feferman-Buchholz ordinal
1475:{\displaystyle \beta <\delta }
410:{\displaystyle \omega ^{\omega }}
7849:" (2014). Accessed 2022 July 23.
7188:Annals of Pure and Applied Logic
6804:is in T iff S plus ∃m φ(m) ⇒ φ(x
4879:, up to stage α, yields a model
4624:. For example, an ordinal which
3906:of KP. Such ordinals are called
3250:+ Πω-Ref. Its value is equal to
2083:{\displaystyle ID_{<\omega }}
326:We have already mentioned (see
7968:
7947:
7886:
7865:
7852:
7812:
7799:
7786:
7704:
7683:
7654:
7588:
7575:
7451:Rathjen, Michael (1994-01-01).
7133:Recursion-theoretic hierarchies
5716:{\displaystyle \Sigma _{2}^{1}}
5626:{\displaystyle \Delta _{2}^{1}}
3589:The proof-theoretic ordinal of
3579:The proof-theoretic ordinal of
3158:using Rathjen's Psi function.
2865:{\displaystyle \Delta _{2}^{1}}
1626:The smallest ordinal such that
1449:{\displaystyle \alpha =\gamma }
590:
530:
484:
7660:F. G. Abramson, G. E. Sacks, "
7519:
7491:
7457:Archive for Mathematical Logic
7444:
7431:
7319:
7289:
7258:
7243:
7216:
7175:
6915:
6901:
6855:
6843:
6575:
6566:
6542:
6533:
6286:is called inaccessibly-stable
6149:
6137:
5992:
5983:
5877:
5868:
5772:
5763:
5214:
5195:
4394:
4388:
4257:. For example, we can define
4118:
4112:
3145:
3126:
3007:
2988:
2901:
2882:
2684:
2671:
2568:
2548:
2475:
2455:
2422:
2390:
2353:
2327:
2158:
2145:
2122:
2119:
2113:
2107:
1992:
1979:
1956:
1943:
1795:such system was introduced by
1729:
1649:
1643:
1599:
1593:
1537:
1531:
1417:
1411:
1395:
1389:
1248:
1242:
1212:
1206:
1110:-th common fixed point of the
1077:
1071:
1008:
1002:
966:
960:
863:
857:
808:
802:
772:
766:
578:
564:
288:is well ordered, we say that ε
13:
1:
8153: < ω
7923:10.1016/S0049-237X(08)70941-8
7754:10.1016/S0049-237X(08)70592-5
7057:The Journal of Symbolic Logic
6951:
6686:scheme of notations of Kleene
4442:
4251:recursively hyperinaccessible
3637:is the smallest ordinal that
2744:, the first nonzero ordinal.
2647:{\displaystyle ID_{<\nu }}
2128:{\displaystyle +(0(\omega ))}
1861:for any ordinal α satisfying
8144:Proof-theoretic ordinals of
7892:Barwise (1976), theorem 7.2.
7820:Locally countable models of
7616:10.1016/0001-8708(76)90187-0
7527:"Proof Theory of Reflection"
7223:Simpson, Stephen G. (2009).
7201:10.1016/0168-0072(86)90052-7
5817:-reflecting for all natural
5810:{\displaystyle \Pi _{n}^{0}}
4354:{\displaystyle {}^{2}S^{\#}}
4156:{\displaystyle \Pi _{1}^{1}}
3239:{\displaystyle \Pi _{0}^{2}}
3105:{\displaystyle \Pi _{1}^{1}}
2914:using an unknown function.
2829:{\displaystyle \Pi _{0}^{1}}
2283:{\displaystyle ID_{\omega }}
2253:{\displaystyle \Pi _{1}^{1}}
1337:{\displaystyle \gamma ^{th}}
172:A different definition uses
7:
7840:{\displaystyle \Sigma _{1}}
7583:Note on admissible ordinals
7251:An independence result for
7182:Buchholz, W. (1986-01-01).
5752:Variants of stable ordinals
5653:{\displaystyle L_{\sigma }}
5563:{\displaystyle L_{\sigma }}
5516:{\displaystyle \Sigma _{1}}
5285:{\displaystyle L_{\alpha }}
5128:{\displaystyle L_{\alpha }}
5052:{\displaystyle >\omega }
5029:{\displaystyle \Sigma _{1}}
4953:{\displaystyle \Sigma _{1}}
4926:{\displaystyle \Sigma _{1}}
4899:{\displaystyle L_{\alpha }}
4747:. An unpublished paper by
4530:{\displaystyle L_{\alpha }}
4432:Zermelo–Fraenkel set theory
3899:{\displaystyle L_{\alpha }}
3834:{\displaystyle \omega _{1}}
3770:{\displaystyle \omega _{1}}
3743:{\displaystyle \omega _{1}}
3633:The supremum of the set of
3613:Zermelo–Fraenkel set theory
3498:{\displaystyle \omega ^{+}}
3335:{\displaystyle \omega ^{+}}
2717:{\displaystyle \Omega _{0}}
1912:Zermelo–Fraenkel set theory
1890:(sometimes just called the
1879:ordinal collapsing function
1817:ordinal collapsing function
1790:Ordinal collapsing function
1692:{\displaystyle \Gamma _{0}}
1054:is a limit ordinal, define
311:Specific recursive ordinals
280:.) Since Peano arithmetic
69:first uncountable ordinal ω
10:
8225:
8167: ≥ ω
7662:Uncountable Gandy Ordinals
7557:Stegert, Jan-Carl (2010).
5412:The least stable level of
5076:Minimal model (set theory)
5073:
4751:supplies, for each finite
4653:recursively weakly compact
4259:recursively Mahlo ordinals
4027:Beyond admissible ordinals
3848:
1787:
709:{\displaystyle \zeta _{0}}
319:
151:
140:
8179:First uncountable ordinal
8021:
7595:Sacks, Gerald E. (1976).
7131:Hinman, Peter G. (1978).
7099:Beyond recursive ordinals
7094:, manuscript in progress.
6948:is a recursive ordinal.
5883:{\displaystyle (+\beta )}
3629:The Church–Kleene ordinal
3624:Beyond recursive ordinals
3565:{\displaystyle \epsilon }
3545:{\displaystyle \epsilon }
3402:{\displaystyle \epsilon }
3382:{\displaystyle \epsilon }
239:for such large ordinals.
105:. Ordinal numbers below ω
8047:Feferman–Schütte ordinal
8015:Large countable ordinals
7871:W. Marek, K. Rasmussen,
4791:{\displaystyle \Pi _{n}}
4732:{\displaystyle \Pi _{3}}
4695:{\displaystyle \Pi _{n}}
4644:{\displaystyle \Pi _{3}}
4618:weakly compact cardinals
4609:{\displaystyle \Pi _{3}}
4587:Kripke-Platek set theory
4208:recursively inaccessible
3857:Kripke–Platek set theory
3248:Kripke-Platek set theory
3114:Kripke-Platek set theory
2976:Kripke-Platek set theory
2838:Kripke-Platek set theory
2290:, the "formal theory of
2057:of natural numbers, and
1904:Kripke–Platek set theory
1670:Feferman–Schütte ordinal
301:Kripke–Platek set theory
294:proof-theoretic strength
18:Large countable ordinals
8086:Bachmann–Howard ordinal
7602:Advances in Mathematics
6674:second-order arithmetic
6519:{\displaystyle \alpha }
6496:{\displaystyle \alpha }
6406:is called Mahlo-stable
6399:{\displaystyle \alpha }
6376:{\displaystyle \alpha }
6279:{\displaystyle \alpha }
6256:{\displaystyle \alpha }
6155:{\displaystyle (^{++})}
6122:{\displaystyle \alpha }
6099:{\displaystyle \alpha }
5968:{\displaystyle \alpha }
5854:{\displaystyle \alpha }
5736:{\displaystyle \sigma }
5445:{\displaystyle \sigma }
5341:{\displaystyle \alpha }
5321:{\displaystyle \alpha }
5306:nonprojectible ordinals
5258:{\displaystyle \alpha }
5181:{\displaystyle \alpha }
5101:{\displaystyle \alpha }
4863:{\displaystyle \alpha }
4843:{\displaystyle \alpha }
4819:{\displaystyle \alpha }
4741:higher-type functionals
4622:indescribable cardinals
4550:{\displaystyle \Gamma }
4501:{\displaystyle \Gamma }
4480:{\displaystyle \alpha }
4460:{\displaystyle \Gamma }
4314:{\displaystyle \alpha }
4294:{\displaystyle \alpha }
4274:{\displaystyle \alpha }
4199:{\displaystyle \alpha }
4179:{\displaystyle \alpha }
4082:{\displaystyle \alpha }
4016:{\displaystyle \alpha }
3966:. One sometimes writes
3605:second-order arithmetic
3581:second-order arithmetic
2303:{\displaystyle \omega }
1908:second-order arithmetic
1887:Bachmann–Howard ordinal
1430:if and only if either (
1362:{\displaystyle \omega }
1182:{\displaystyle \delta }
1103:{\displaystyle \alpha }
1047:{\displaystyle \delta }
383:{\displaystyle \omega }
305:Bachmann–Howard ordinal
242:For example, the usual
8026:First infinite ordinal
7841:
7710:W. Richter, P. Aczel,
7641:Penn State Institution
7309:
6998:(for ordinal diagrams)
6942:
6922:
6888:
6868:
6798:
6719:
6680:A pseudo-well-ordering
6662:
6608:
6582:
6549:
6520:
6497:
6477:
6476:{\displaystyle \beta }
6457:
6400:
6377:
6357:
6356:{\displaystyle \beta }
6337:
6280:
6257:
6233:
6232:{\displaystyle \beta }
6213:
6156:
6123:
6100:
6076:
6075:{\displaystyle \beta }
6056:
5999:
5998:{\displaystyle (^{+})}
5969:
5947:
5884:
5855:
5831:
5811:
5779:
5745:alpha recursion theory
5737:
5717:
5685:
5654:
5633:iff it is a member of
5627:
5595:
5564:
5543:iff it is a member of
5537:
5517:
5486:
5446:
5426:
5403:
5342:
5322:
5286:
5259:
5227:
5182:
5152:
5129:
5102:
5053:
5030:
5003:
4980:
4954:
4933:-separation. However,
4927:
4900:
4864:
4844:
4820:
4806:An admissible ordinal
4792:
4765:
4733:
4696:
4669:
4651:-reflecting is called
4645:
4610:
4585:, a theory augmenting
4571:
4551:
4531:
4502:
4481:
4461:
4447:For a set of formulae
4421:
4401:
4355:
4315:
4295:
4275:
4243:
4200:
4180:
4157:
4125:
4083:
4062:
4017:
3997:
3941:
3900:
3835:
3808:
3771:
3744:
3713:
3676:
3566:
3546:
3526:
3499:
3472:
3452:
3403:
3383:
3363:
3336:
3309:
3289:
3240:
3208:
3188:
3152:
3106:
3070:
3050:
3014:
2964:
2944:
2908:
2866:
2830:
2794:
2774:
2738:
2718:
2691:
2648:
2615:
2575:
2522:
2482:
2429:
2360:
2304:
2284:
2254:
2222:
2165:
2129:
2093:Since the hydras from
2084:
2047:
1999:
1969:, abbreviated as just
1963:
1784:Impredicative ordinals
1746:
1693:
1672:and generally written
1662:
1612:
1570:
1544:
1502:
1476:
1450:
1424:
1363:
1338:
1308:
1281:
1255:
1219:
1183:
1157:
1131:
1104:
1084:
1048:
1028:
979:
943:-th ordinal such that
937:
936:{\displaystyle \beta }
917:
890:
889:{\displaystyle \beta }
870:
828:
779:
743:
710:
683:
656:
629:-th ordinal such that
623:
622:{\displaystyle \iota }
600:
445:
411:
384:
364:
7842:
7563:miami.uni-muenster.de
7310:
7092:Truth and provability
6960:On recursive ordinals
6943:
6941:{\displaystyle \rho }
6923:
6894:is the order type of
6889:
6887:{\displaystyle \eta }
6869:
6799:
6720:
6663:
6609:
6583:
6550:
6521:
6498:
6478:
6458:
6401:
6378:
6358:
6338:
6281:
6258:
6234:
6214:
6157:
6124:
6101:
6077:
6057:
6000:
5970:
5948:
5885:
5856:
5832:
5812:
5780:
5738:
5718:
5686:
5655:
5628:
5596:
5565:
5538:
5518:
5487:
5447:
5427:
5404:
5343:
5323:
5287:
5260:
5228:
5183:
5153:
5130:
5103:
5070:"Unprovable" ordinals
5054:
5031:
5004:
4981:
4955:
4928:
4901:
4865:
4845:
4830:if there is no total
4821:
4793:
4766:
4745:2-admissible ordinals
4734:
4697:
4670:
4646:
4611:
4572:
4570:{\displaystyle \phi }
4552:
4532:
4503:
4482:
4462:
4422:
4420:{\displaystyle \rho }
4402:
4356:
4316:
4296:
4276:
4244:
4201:
4181:
4158:
4126:
4084:
4063:
4018:
3998:
3942:
3901:
3836:
3809:
3772:
3745:
3714:
3677:
3643:Church–Kleene ordinal
3567:
3547:
3527:
3525:{\displaystyle P_{0}}
3500:
3473:
3453:
3404:
3384:
3364:
3362:{\displaystyle P_{0}}
3337:
3310:
3290:
3241:
3209:
3189:
3153:
3107:
3071:
3051:
3015:
2965:
2945:
2909:
2867:
2831:
2795:
2775:
2739:
2719:
2692:
2649:
2616:
2576:
2523:
2483:
2430:
2361:
2305:
2285:
2255:
2223:
2166:
2130:
2095:Buchholz's hydra game
2085:
2048:
2000:
1964:
1747:
1694:
1663:
1613:
1571:
1545:
1503:
1477:
1451:
1425:
1364:
1339:
1309:
1282:
1256:
1220:
1184:
1158:
1132:
1105:
1085:
1049:
1029:
980:
938:
918:
891:
871:
829:
780:
744:
711:
684:
657:
624:
601:
446:
412:
385:
365:
256:: while the ordinal ε
237:transfinite induction
198:Church–Kleene ordinal
8165:Nonrecursive ordinal
8076:large Veblen ordinal
8066:small Veblen ordinal
7824:
7326:Buchholz, Wilfried;
7255:
6932:
6898:
6878:
6819:
6810:Kleene–Brouwer order
6743:
6692:
6618:
6592:
6581:{\displaystyle (+1)}
6563:
6548:{\displaystyle (+1)}
6530:
6510:
6506:A countable ordinal
6487:
6467:
6413:
6390:
6386:A countable ordinal
6367:
6347:
6293:
6270:
6266:A countable ordinal
6247:
6223:
6169:
6133:
6113:
6109:A countable ordinal
6090:
6066:
6012:
5979:
5959:
5955:A countable ordinal
5897:
5865:
5845:
5841:A countable ordinal
5821:
5789:
5778:{\displaystyle (+1)}
5760:
5727:
5695:
5667:
5637:
5605:
5577:
5547:
5527:
5500:
5456:
5436:
5416:
5352:
5332:
5312:
5298:-elementary submodel
5269:
5249:
5192:
5172:
5142:
5112:
5092:
5040:
5013:
4990:
4964:
4937:
4910:
4883:
4854:
4834:
4810:
4775:
4755:
4716:
4679:
4659:
4628:
4593:
4561:
4541:
4514:
4492:
4471:
4451:
4411:
4365:
4329:
4305:
4285:
4265:
4214:
4190:
4170:
4135:
4093:
4073:
4035:
4007:
3970:
3914:
3883:
3818:
3781:
3754:
3727:
3686:
3649:
3556:
3536:
3509:
3482:
3462:
3417:
3393:
3373:
3346:
3319:
3299:
3254:
3218:
3207:{\displaystyle \Xi }
3198:
3165:
3120:
3084:
3060:
3027:
2982:
2954:
2921:
2876:
2844:
2808:
2784:
2751:
2728:
2701:
2658:
2625:
2585:
2535:
2492:
2442:
2377:
2314:
2294:
2264:
2232:
2182:
2139:
2101:
2061:
2009:
1973:
1930:
1723:
1676:
1630:
1580:
1554:
1512:
1486:
1460:
1434:
1376:
1353:
1318:
1291:
1265:
1229:
1225:be the limit of the
1193:
1173:
1141:
1114:
1094:
1058:
1038:
989:
947:
927:
900:
880:
838:
789:
753:
720:
693:
666:
633:
613:
609:More generally, the
462:
421:
394:
374:
341:
8151:Computable ordinals
7818:"Fred G. Abramson,
7534:University of Leeds
7506:University of Leeds
7411:"A Zoo of Ordinals"
7275:
7088:Herman Ruge Jervell
7036:McGraw-Hill (1967)
6990:, 2nd edition 1987
6839:
6714:
5806:
5712:
5622:
5452:be least such that
4979:{\displaystyle V=L}
4238:
4152:
4057:
3992:
3936:
3845:Admissible ordinals
3803:
3708:
3671:
3447:
3284:
3235:
3101:
2861:
2825:
2249:
2199:
2026:
1780:" Veblen ordinals.
896:-th fixed point of
689:. We could define
296:of Peano's axioms.
278:Goodstein sequences
274:Kirby–Paris theorem
163:computable function
159:Computable ordinals
43:Cantor normal forms
8103:Buchholz's ordinal
7837:
7469:10.1007/BF01275469
7340:10.1007/bfb0091894
7305:
7261:
7030:Hartley Rogers Jr.
6938:
6918:
6884:
6864:
6822:
6794:
6715:
6695:
6658:
6604:
6578:
6545:
6516:
6493:
6473:
6453:
6396:
6373:
6353:
6333:
6276:
6253:
6241:admissible ordinal
6229:
6209:
6152:
6119:
6096:
6084:admissible ordinal
6072:
6052:
5995:
5965:
5943:
5880:
5851:
5827:
5807:
5792:
5785:-stable iff it is
5775:
5733:
5713:
5698:
5681:
5650:
5623:
5608:
5591:
5560:
5533:
5513:
5482:
5442:
5422:
5399:
5338:
5318:
5282:
5255:
5223:
5178:
5148:
5125:
5098:
5049:
5026:
5002:{\displaystyle KP}
4999:
4976:
4950:
4923:
4896:
4860:
4840:
4816:
4788:
4761:
4729:
4692:
4665:
4641:
4606:
4567:
4547:
4527:
4498:
4477:
4467:, a limit ordinal
4457:
4417:
4397:
4351:
4311:
4291:
4271:
4239:
4217:
4196:
4176:
4153:
4138:
4121:
4079:
4058:
4038:
4013:
3993:
3973:
3937:
3917:
3896:
3851:Admissible ordinal
3831:
3804:
3784:
3767:
3740:
3709:
3689:
3672:
3652:
3635:recursive ordinals
3609:Zermelo set theory
3562:
3542:
3522:
3495:
3468:
3448:
3420:
3399:
3379:
3359:
3332:
3305:
3285:
3257:
3236:
3221:
3204:
3184:
3148:
3102:
3087:
3066:
3046:
3010:
2960:
2940:
2904:
2862:
2847:
2826:
2811:
2790:
2770:
2734:
2714:
2687:
2644:
2611:
2571:
2518:
2478:
2425:
2356:
2300:
2280:
2250:
2235:
2218:
2185:
2161:
2125:
2080:
2043:
2012:
1995:
1959:
1924:Buchholz's ordinal
1853:: in particular ψ(
1742:
1689:
1658:
1608:
1566:
1540:
1498:
1472:
1446:
1420:
1359:
1334:
1304:
1277:
1251:
1215:
1179:
1153:
1127:
1100:
1080:
1044:
1024:
985:; so for example,
975:
933:
913:
886:
866:
824:
775:
739:
706:
679:
652:
619:
596:
441:
407:
380:
360:
328:Cantor normal form
143:Computable ordinal
127:ordinal arithmetic
8191:
8190:
8056:Ackermann ordinal
7328:Feferman, Solomon
7236:978-0-521-88439-6
7170:Inline references
6526:is called doubly
5830:{\displaystyle n}
5536:{\displaystyle L}
5425:{\displaystyle L}
5348:is equivalent to
5168:, then the least
5151:{\displaystyle T}
4802:Nonprojectibility
4764:{\displaystyle n}
4668:{\displaystyle n}
3879:, yields a model
3471:{\displaystyle X}
3308:{\displaystyle X}
3069:{\displaystyle K}
2963:{\displaystyle M}
2793:{\displaystyle I}
2737:{\displaystyle 1}
1710:More generally, Γ
1189:a limit ordinal,
594:
178:ordinal notations
148:Ordinal notations
54:ordinal notations
16:(Redirected from
8216:
8175:
8174:
8161:
8160:
8008:
8001:
7994:
7985:
7984:
7979:
7972:
7966:
7965:
7963:
7951:
7945:
7944:
7902:
7893:
7890:
7884:
7869:
7863:
7856:
7850:
7846:
7844:
7843:
7838:
7836:
7835:
7816:
7810:
7803:
7797:
7790:
7784:
7783:
7739:
7730:
7717:
7708:
7702:
7701:
7699:
7687:
7681:
7674:
7665:
7658:
7652:
7651:
7649:
7648:
7638:
7630:
7621:
7620:
7618:
7592:
7586:
7579:
7573:
7572:
7570:
7569:
7554:
7545:
7544:
7542:
7541:
7531:
7523:
7517:
7516:
7514:
7513:
7503:
7495:
7489:
7488:
7448:
7442:
7435:
7429:
7428:
7426:
7425:
7415:
7407:
7362:
7361:
7323:
7317:
7314:
7312:
7311:
7306:
7304:
7303:
7288:
7287:
7274:
7269:
7247:
7241:
7240:
7220:
7214:
7213:
7203:
7179:
7136:
7127:
7115:
7008:, Springer 1977
6972:, Springer 1989
6947:
6945:
6944:
6939:
6927:
6925:
6924:
6919:
6908:
6893:
6891:
6890:
6885:
6873:
6871:
6870:
6865:
6838:
6830:
6803:
6801:
6800:
6795:
6793:
6792:
6768:
6767:
6755:
6754:
6737:Skolem functions
6724:
6722:
6721:
6716:
6713:
6712:
6703:
6667:
6665:
6664:
6659:
6657:
6656:
6647:
6646:
6645:
6644:
6630:
6629:
6613:
6611:
6610:
6605:
6588:-stable ordinal
6587:
6585:
6584:
6579:
6554:
6552:
6551:
6546:
6525:
6523:
6522:
6517:
6502:
6500:
6499:
6494:
6482:
6480:
6479:
6474:
6462:
6460:
6459:
6454:
6452:
6451:
6442:
6441:
6440:
6439:
6425:
6424:
6405:
6403:
6402:
6397:
6382:
6380:
6379:
6374:
6362:
6360:
6359:
6354:
6342:
6340:
6339:
6334:
6332:
6331:
6322:
6321:
6320:
6319:
6305:
6304:
6285:
6283:
6282:
6277:
6262:
6260:
6259:
6254:
6238:
6236:
6235:
6230:
6218:
6216:
6215:
6210:
6208:
6207:
6198:
6197:
6196:
6195:
6181:
6180:
6161:
6159:
6158:
6153:
6148:
6147:
6128:
6126:
6125:
6120:
6105:
6103:
6102:
6097:
6081:
6079:
6078:
6073:
6061:
6059:
6058:
6053:
6051:
6050:
6041:
6040:
6039:
6038:
6024:
6023:
6004:
6002:
6001:
5996:
5991:
5990:
5974:
5972:
5971:
5966:
5952:
5950:
5949:
5944:
5942:
5941:
5926:
5925:
5924:
5923:
5909:
5908:
5889:
5887:
5886:
5881:
5860:
5858:
5857:
5852:
5836:
5834:
5833:
5828:
5816:
5814:
5813:
5808:
5805:
5800:
5784:
5782:
5781:
5776:
5742:
5740:
5739:
5734:
5722:
5720:
5719:
5714:
5711:
5706:
5690:
5688:
5687:
5682:
5680:
5659:
5657:
5656:
5651:
5649:
5648:
5632:
5630:
5629:
5624:
5621:
5616:
5600:
5598:
5597:
5592:
5590:
5569:
5567:
5566:
5561:
5559:
5558:
5542:
5540:
5539:
5534:
5522:
5520:
5519:
5514:
5512:
5511:
5491:
5489:
5488:
5483:
5478:
5477:
5468:
5467:
5451:
5449:
5448:
5443:
5431:
5429:
5428:
5423:
5408:
5406:
5405:
5400:
5398:
5397:
5396:
5395:
5381:
5380:
5379:
5378:
5364:
5363:
5347:
5345:
5344:
5339:
5327:
5325:
5324:
5319:
5291:
5289:
5288:
5283:
5281:
5280:
5264:
5262:
5261:
5256:
5232:
5230:
5229:
5224:
5207:
5206:
5187:
5185:
5184:
5179:
5157:
5155:
5154:
5149:
5134:
5132:
5131:
5126:
5124:
5123:
5107:
5105:
5104:
5099:
5086:transitive model
5058:
5056:
5055:
5050:
5035:
5033:
5032:
5027:
5025:
5024:
5008:
5006:
5005:
5000:
4985:
4983:
4982:
4977:
4959:
4957:
4956:
4951:
4949:
4948:
4932:
4930:
4929:
4924:
4922:
4921:
4905:
4903:
4902:
4897:
4895:
4894:
4869:
4867:
4866:
4861:
4849:
4847:
4846:
4841:
4825:
4823:
4822:
4817:
4797:
4795:
4794:
4789:
4787:
4786:
4770:
4768:
4767:
4762:
4749:Solomon Feferman
4738:
4736:
4735:
4730:
4728:
4727:
4701:
4699:
4698:
4693:
4691:
4690:
4674:
4672:
4671:
4666:
4650:
4648:
4647:
4642:
4640:
4639:
4615:
4613:
4612:
4607:
4605:
4604:
4576:
4574:
4573:
4568:
4556:
4554:
4553:
4548:
4536:
4534:
4533:
4528:
4526:
4525:
4507:
4505:
4504:
4499:
4486:
4484:
4483:
4478:
4466:
4464:
4463:
4458:
4436:regular cardinal
4426:
4424:
4423:
4418:
4406:
4404:
4403:
4398:
4387:
4386:
4377:
4376:
4360:
4358:
4357:
4352:
4350:
4349:
4340:
4339:
4334:
4320:
4318:
4317:
4312:
4300:
4298:
4297:
4292:
4281:such that every
4280:
4278:
4277:
4272:
4261:: these are the
4248:
4246:
4245:
4240:
4237:
4236:
4235:
4225:
4205:
4203:
4202:
4197:
4185:
4183:
4182:
4177:
4163:-comprehension.
4162:
4160:
4159:
4154:
4151:
4146:
4130:
4128:
4127:
4122:
4105:
4104:
4088:
4086:
4085:
4080:
4067:
4065:
4064:
4059:
4056:
4055:
4046:
4022:
4020:
4019:
4014:
4002:
4000:
3999:
3994:
3991:
3990:
3981:
3950:By a theorem of
3946:
3944:
3943:
3938:
3935:
3934:
3925:
3905:
3903:
3902:
3897:
3895:
3894:
3840:
3838:
3837:
3832:
3830:
3829:
3813:
3811:
3810:
3805:
3802:
3801:
3792:
3776:
3774:
3773:
3768:
3766:
3765:
3749:
3747:
3746:
3741:
3739:
3738:
3718:
3716:
3715:
3710:
3707:
3706:
3697:
3681:
3679:
3678:
3673:
3670:
3669:
3660:
3571:
3569:
3568:
3563:
3551:
3549:
3548:
3543:
3531:
3529:
3528:
3523:
3521:
3520:
3504:
3502:
3501:
3496:
3494:
3493:
3477:
3475:
3474:
3469:
3457:
3455:
3454:
3449:
3446:
3445:
3444:
3428:
3408:
3406:
3405:
3400:
3388:
3386:
3385:
3380:
3368:
3366:
3365:
3360:
3358:
3357:
3341:
3339:
3338:
3333:
3331:
3330:
3314:
3312:
3311:
3306:
3294:
3292:
3291:
3286:
3283:
3282:
3281:
3265:
3245:
3243:
3242:
3237:
3234:
3229:
3213:
3211:
3210:
3205:
3193:
3191:
3190:
3185:
3183:
3182:
3157:
3155:
3154:
3149:
3144:
3143:
3111:
3109:
3108:
3103:
3100:
3095:
3075:
3073:
3072:
3067:
3055:
3053:
3052:
3047:
3045:
3044:
3019:
3017:
3016:
3011:
3006:
3005:
2969:
2967:
2966:
2961:
2949:
2947:
2946:
2941:
2939:
2938:
2913:
2911:
2910:
2905:
2900:
2899:
2871:
2869:
2868:
2863:
2860:
2855:
2835:
2833:
2832:
2827:
2824:
2819:
2799:
2797:
2796:
2791:
2779:
2777:
2776:
2771:
2769:
2768:
2743:
2741:
2740:
2735:
2723:
2721:
2720:
2715:
2713:
2712:
2696:
2694:
2693:
2688:
2683:
2682:
2670:
2669:
2653:
2651:
2650:
2645:
2643:
2642:
2620:
2618:
2617:
2612:
2610:
2609:
2608:
2607:
2580:
2578:
2577:
2572:
2567:
2566:
2565:
2564:
2547:
2546:
2527:
2525:
2524:
2519:
2517:
2516:
2515:
2514:
2487:
2485:
2484:
2479:
2474:
2473:
2472:
2471:
2454:
2453:
2434:
2432:
2431:
2426:
2421:
2420:
2408:
2407:
2389:
2388:
2365:
2363:
2362:
2357:
2352:
2351:
2344:
2343:
2326:
2325:
2309:
2307:
2306:
2301:
2289:
2287:
2286:
2281:
2279:
2278:
2259:
2257:
2256:
2251:
2248:
2243:
2227:
2225:
2224:
2219:
2198:
2193:
2170:
2168:
2167:
2162:
2157:
2156:
2134:
2132:
2131:
2126:
2089:
2087:
2086:
2081:
2079:
2078:
2052:
2050:
2049:
2044:
2042:
2041:
2025:
2020:
2004:
2002:
2001:
1996:
1991:
1990:
1968:
1966:
1965:
1960:
1955:
1954:
1942:
1941:
1751:
1749:
1748:
1743:
1741:
1740:
1698:
1696:
1695:
1690:
1688:
1687:
1668:is known as the
1667:
1665:
1664:
1659:
1642:
1641:
1617:
1615:
1614:
1609:
1592:
1591:
1575:
1573:
1572:
1567:
1549:
1547:
1546:
1541:
1530:
1529:
1507:
1505:
1504:
1499:
1481:
1479:
1478:
1473:
1455:
1453:
1452:
1447:
1429:
1427:
1426:
1421:
1410:
1409:
1388:
1387:
1368:
1366:
1365:
1360:
1343:
1341:
1340:
1335:
1333:
1332:
1313:
1311:
1310:
1305:
1303:
1302:
1286:
1284:
1283:
1278:
1260:
1258:
1257:
1252:
1241:
1240:
1224:
1222:
1221:
1216:
1205:
1204:
1188:
1186:
1185:
1180:
1166:Veblen hierarchy
1162:
1160:
1159:
1154:
1136:
1134:
1133:
1128:
1126:
1125:
1109:
1107:
1106:
1101:
1089:
1087:
1086:
1081:
1070:
1069:
1053:
1051:
1050:
1045:
1033:
1031:
1030:
1025:
1023:
1022:
1001:
1000:
984:
982:
981:
976:
959:
958:
942:
940:
939:
934:
922:
920:
919:
914:
912:
911:
895:
893:
892:
887:
875:
873:
872:
867:
856:
855:
833:
831:
830:
825:
823:
822:
801:
800:
784:
782:
781:
776:
765:
764:
748:
746:
745:
740:
732:
731:
715:
713:
712:
707:
705:
704:
688:
686:
685:
680:
678:
677:
661:
659:
658:
653:
645:
644:
628:
626:
625:
620:
605:
603:
602:
597:
595:
592:
586:
585:
576:
575:
560:
559:
558:
557:
550:
549:
520:
519:
507:
506:
499:
498:
474:
473:
450:
448:
447:
442:
440:
439:
438:
437:
416:
414:
413:
408:
406:
405:
389:
387:
386:
381:
369:
367:
366:
361:
353:
352:
222:Peano arithmetic
154:Ordinal notation
113:
112:
96:
95:
58:ordinal analysis
21:
8224:
8223:
8219:
8218:
8217:
8215:
8214:
8213:
8204:Ordinal numbers
8194:
8193:
8192:
8187:
8173:
8170:
8169:
8168:
8159:
8156:
8155:
8154:
8140:
8138:
8117:
8111:
8098:
8052:
8043:
8035:Epsilon numbers
8017:
8012:
7982:
7973:
7969:
7952:
7948:
7933:
7903:
7896:
7891:
7887:
7870:
7866:
7857:
7853:
7831:
7827:
7825:
7822:
7821:
7817:
7813:
7804:
7800:
7791:
7787:
7772:
7737:
7731:
7720:
7709:
7705:
7688:
7684:
7675:
7668:
7659:
7655:
7646:
7644:
7636:
7632:
7631:
7624:
7593:
7589:
7580:
7576:
7567:
7565:
7555:
7548:
7539:
7537:
7529:
7525:
7524:
7520:
7511:
7509:
7501:
7497:
7496:
7492:
7449:
7445:
7436:
7432:
7423:
7421:
7413:
7409:
7408:
7365:
7350:
7324:
7320:
7293:
7292:
7277:
7276:
7270:
7265:
7256:
7253:
7252:
7248:
7244:
7237:
7221:
7217:
7180:
7176:
7172:
7163:Postscript file
7159:Sets and Proofs
7147:Michael Rathjen
7143:
7124:
7101:
7048:Larry W. Miller
7020:Craig Smorynski
6966:Wolfram Pohlers
6962:
6954:
6933:
6930:
6929:
6904:
6899:
6896:
6895:
6879:
6876:
6875:
6831:
6826:
6820:
6817:
6816:
6807:
6788:
6784:
6763:
6759:
6750:
6746:
6744:
6741:
6740:
6733:
6705:
6704:
6699:
6693:
6690:
6689:
6682:
6652:
6648:
6640:
6636:
6635:
6631:
6625:
6621:
6619:
6616:
6615:
6593:
6590:
6589:
6564:
6561:
6560:
6531:
6528:
6527:
6511:
6508:
6507:
6488:
6485:
6484:
6468:
6465:
6464:
6447:
6443:
6435:
6431:
6430:
6426:
6420:
6416:
6414:
6411:
6410:
6391:
6388:
6387:
6368:
6365:
6364:
6348:
6345:
6344:
6327:
6323:
6315:
6311:
6310:
6306:
6300:
6296:
6294:
6291:
6290:
6271:
6268:
6267:
6248:
6245:
6244:
6224:
6221:
6220:
6203:
6199:
6191:
6187:
6186:
6182:
6176:
6172:
6170:
6167:
6166:
6140:
6136:
6134:
6131:
6130:
6114:
6111:
6110:
6091:
6088:
6087:
6067:
6064:
6063:
6046:
6042:
6034:
6030:
6029:
6025:
6019:
6015:
6013:
6010:
6009:
5986:
5982:
5980:
5977:
5976:
5960:
5957:
5956:
5931:
5927:
5919:
5915:
5914:
5910:
5904:
5900:
5898:
5895:
5894:
5866:
5863:
5862:
5846:
5843:
5842:
5822:
5819:
5818:
5801:
5796:
5790:
5787:
5786:
5761:
5758:
5757:
5754:
5728:
5725:
5724:
5707:
5702:
5696:
5693:
5692:
5676:
5668:
5665:
5664:
5644:
5640:
5638:
5635:
5634:
5617:
5612:
5606:
5603:
5602:
5586:
5578:
5575:
5574:
5554:
5550:
5548:
5545:
5544:
5528:
5525:
5524:
5507:
5503:
5501:
5498:
5497:
5473:
5469:
5463:
5459:
5457:
5454:
5453:
5437:
5434:
5433:
5417:
5414:
5413:
5391:
5387:
5386:
5382:
5374:
5370:
5369:
5365:
5359:
5355:
5353:
5350:
5349:
5333:
5330:
5329:
5328:, stability of
5313:
5310:
5309:
5297:
5276:
5272:
5270:
5267:
5266:
5250:
5247:
5246:
5243:stable ordinals
5239:
5237:Stable ordinals
5202:
5198:
5193:
5190:
5189:
5173:
5170:
5169:
5143:
5140:
5139:
5119:
5115:
5113:
5110:
5109:
5093:
5090:
5089:
5078:
5072:
5041:
5038:
5037:
5020:
5016:
5014:
5011:
5010:
4991:
4988:
4987:
4965:
4962:
4961:
4944:
4940:
4938:
4935:
4934:
4917:
4913:
4911:
4908:
4907:
4890:
4886:
4884:
4881:
4880:
4855:
4852:
4851:
4835:
4832:
4831:
4811:
4808:
4807:
4804:
4782:
4778:
4776:
4773:
4772:
4756:
4753:
4752:
4723:
4719:
4717:
4714:
4713:
4712:In particular,
4707:
4686:
4682:
4680:
4677:
4676:
4660:
4657:
4656:
4635:
4631:
4629:
4626:
4625:
4600:
4596:
4594:
4591:
4590:
4582:
4562:
4559:
4558:
4542:
4539:
4538:
4521:
4517:
4515:
4512:
4511:
4493:
4490:
4489:
4472:
4469:
4468:
4452:
4449:
4448:
4445:
4412:
4409:
4408:
4382:
4381:
4372:
4368:
4366:
4363:
4362:
4345:
4341:
4335:
4333:
4332:
4330:
4327:
4326:
4306:
4303:
4302:
4286:
4283:
4282:
4266:
4263:
4262:
4255:large cardinals
4231:
4227:
4226:
4221:
4215:
4212:
4211:
4191:
4188:
4187:
4171:
4168:
4167:
4147:
4142:
4136:
4133:
4132:
4100:
4096:
4094:
4091:
4090:
4074:
4071:
4070:
4048:
4047:
4042:
4036:
4033:
4032:
4029:
4008:
4005:
4004:
3983:
3982:
3977:
3971:
3968:
3967:
3927:
3926:
3921:
3915:
3912:
3911:
3890:
3886:
3884:
3881:
3880:
3853:
3847:
3825:
3821:
3819:
3816:
3815:
3794:
3793:
3788:
3782:
3779:
3778:
3761:
3757:
3755:
3752:
3751:
3734:
3730:
3728:
3725:
3724:
3699:
3698:
3693:
3687:
3684:
3683:
3662:
3661:
3656:
3650:
3647:
3646:
3631:
3626:
3600:
3557:
3554:
3553:
3537:
3534:
3533:
3516:
3512:
3510:
3507:
3506:
3489:
3485:
3483:
3480:
3479:
3463:
3460:
3459:
3434:
3430:
3429:
3424:
3418:
3415:
3414:
3394:
3391:
3390:
3374:
3371:
3370:
3353:
3349:
3347:
3344:
3343:
3326:
3322:
3320:
3317:
3316:
3300:
3297:
3296:
3271:
3267:
3266:
3261:
3255:
3252:
3251:
3230:
3225:
3219:
3216:
3215:
3199:
3196:
3195:
3172:
3168:
3166:
3163:
3162:
3133:
3129:
3121:
3118:
3117:
3096:
3091:
3085:
3082:
3081:
3061:
3058:
3057:
3034:
3030:
3028:
3025:
3024:
2995:
2991:
2983:
2980:
2979:
2955:
2952:
2951:
2928:
2924:
2922:
2919:
2918:
2889:
2885:
2877:
2874:
2873:
2856:
2851:
2845:
2842:
2841:
2820:
2815:
2809:
2806:
2805:
2785:
2782:
2781:
2758:
2754:
2752:
2749:
2748:
2729:
2726:
2725:
2708:
2704:
2702:
2699:
2698:
2678:
2674:
2665:
2661:
2659:
2656:
2655:
2635:
2631:
2626:
2623:
2622:
2603:
2599:
2595:
2591:
2586:
2583:
2582:
2560:
2556:
2555:
2551:
2542:
2538:
2536:
2533:
2532:
2510:
2506:
2502:
2498:
2493:
2490:
2489:
2467:
2463:
2462:
2458:
2449:
2445:
2443:
2440:
2439:
2416:
2412:
2397:
2393:
2384:
2380:
2378:
2375:
2374:
2339:
2335:
2334:
2330:
2321:
2317:
2315:
2312:
2311:
2295:
2292:
2291:
2274:
2270:
2265:
2262:
2261:
2244:
2239:
2233:
2230:
2229:
2194:
2189:
2183:
2180:
2179:
2152:
2148:
2140:
2137:
2136:
2135:corresponds to
2102:
2099:
2098:
2071:
2067:
2062:
2059:
2058:
2037:
2033:
2021:
2016:
2010:
2007:
2006:
1986:
1982:
1974:
1971:
1970:
1950:
1946:
1937:
1933:
1931:
1928:
1927:
1920:
1901:
1897:
1848:
1838:
1799:in 1950 (in an
1792:
1786:
1764:is obtained in
1736:
1732:
1724:
1721:
1720:
1715:
1683:
1679:
1677:
1674:
1673:
1637:
1633:
1631:
1628:
1627:
1624:
1587:
1583:
1581:
1578:
1577:
1555:
1552:
1551:
1525:
1521:
1513:
1510:
1509:
1487:
1484:
1483:
1461:
1458:
1457:
1435:
1432:
1431:
1405:
1401:
1383:
1379:
1377:
1374:
1373:
1354:
1351:
1350:
1346:Veblen function
1325:
1321:
1319:
1316:
1315:
1298:
1294:
1292:
1289:
1288:
1266:
1263:
1262:
1236:
1232:
1230:
1227:
1226:
1200:
1196:
1194:
1191:
1190:
1174:
1171:
1170:
1142:
1139:
1138:
1121:
1117:
1115:
1112:
1111:
1095:
1092:
1091:
1065:
1061:
1059:
1056:
1055:
1039:
1036:
1035:
1018:
1014:
996:
992:
990:
987:
986:
954:
950:
948:
945:
944:
928:
925:
924:
907:
903:
901:
898:
897:
881:
878:
877:
845:
841:
839:
836:
835:
818:
814:
796:
792:
790:
787:
786:
760:
756:
754:
751:
750:
727:
723:
721:
718:
717:
700:
696:
694:
691:
690:
673:
669:
667:
664:
663:
640:
636:
634:
631:
630:
614:
611:
610:
591:
581:
577:
571:
567:
545:
541:
540:
536:
535:
531:
515:
511:
494:
490:
489:
485:
469:
465:
463:
460:
459:
454:
433:
429:
428:
424:
422:
419:
418:
401:
397:
395:
392:
391:
375:
372:
371:
348:
344:
342:
339:
338:
335:
324:
322:Veblen function
318:
313:
291:
287:
263:
259:
254:
210:
156:
150:
145:
139:
131:large cardinals
111:
108:
107:
106:
100:
94:
91:
90:
89:
86:
72:
62:halting problem
28:
23:
22:
15:
12:
11:
5:
8222:
8212:
8211:
8206:
8189:
8188:
8186:
8185:
8176:
8171:
8162:
8157:
8148:
8142:
8134:
8132:
8119:
8113:
8109:
8100:
8096:
8083:
8073:
8063:
8053:
8050:
8044:
8041:
8032:
8022:
8019:
8018:
8011:
8010:
8003:
7996:
7988:
7981:
7980:
7967:
7946:
7931:
7894:
7885:
7875:in libraries (
7864:
7858:K. J. Devlin,
7851:
7834:
7830:
7811:
7805:K. J. Devlin,
7798:
7792:S. Feferman, "
7785:
7770:
7718:
7703:
7682:
7666:
7653:
7622:
7609:(2): 213–262.
7587:
7574:
7546:
7518:
7490:
7443:
7430:
7363:
7348:
7318:
7302:
7299:
7296:
7291:
7286:
7283:
7280:
7273:
7268:
7264:
7260:
7249:W. Buchholz, "
7242:
7235:
7215:
7173:
7171:
7168:
7167:
7166:
7142:
7139:
7138:
7137:
7128:
7122:
7100:
7097:
7096:
7095:
7085:
7072:Hilbert Levitz
7069:
7045:
7027:
7017:
6999:
6981:
6961:
6958:
6953:
6950:
6937:
6917:
6914:
6911:
6907:
6903:
6883:
6863:
6860:
6857:
6854:
6851:
6848:
6845:
6842:
6837:
6834:
6829:
6825:
6805:
6791:
6787:
6783:
6780:
6777:
6774:
6771:
6766:
6762:
6758:
6753:
6749:
6731:
6711:
6708:
6702:
6698:
6681:
6678:
6670:
6669:
6655:
6651:
6643:
6639:
6634:
6628:
6624:
6603:
6600:
6597:
6577:
6574:
6571:
6568:
6544:
6541:
6538:
6535:
6515:
6504:
6492:
6472:
6450:
6446:
6438:
6434:
6429:
6423:
6419:
6395:
6384:
6372:
6352:
6330:
6326:
6318:
6314:
6309:
6303:
6299:
6275:
6264:
6252:
6228:
6206:
6202:
6194:
6190:
6185:
6179:
6175:
6151:
6146:
6143:
6139:
6118:
6107:
6095:
6071:
6049:
6045:
6037:
6033:
6028:
6022:
6018:
5994:
5989:
5985:
5964:
5953:
5940:
5937:
5934:
5930:
5922:
5918:
5913:
5907:
5903:
5879:
5876:
5873:
5870:
5850:
5826:
5804:
5799:
5795:
5774:
5771:
5768:
5765:
5753:
5750:
5749:
5748:
5732:
5710:
5705:
5701:
5679:
5675:
5672:
5661:
5647:
5643:
5620:
5615:
5611:
5589:
5585:
5582:
5571:
5557:
5553:
5532:
5523:definition in
5510:
5506:
5481:
5476:
5472:
5466:
5462:
5441:
5421:
5394:
5390:
5385:
5377:
5373:
5368:
5362:
5358:
5337:
5317:
5295:
5279:
5275:
5254:
5238:
5235:
5222:
5219:
5216:
5213:
5210:
5205:
5201:
5197:
5177:
5147:
5122:
5118:
5097:
5071:
5068:
5048:
5045:
5023:
5019:
4998:
4995:
4975:
4972:
4969:
4947:
4943:
4920:
4916:
4893:
4889:
4873:Gödel universe
4859:
4839:
4828:nonprojectible
4815:
4803:
4800:
4785:
4781:
4760:
4726:
4722:
4705:
4689:
4685:
4664:
4638:
4634:
4603:
4599:
4580:
4566:
4546:
4524:
4520:
4497:
4476:
4456:
4444:
4441:
4416:
4396:
4393:
4390:
4385:
4380:
4375:
4371:
4348:
4344:
4338:
4323:Mahlo cardinal
4310:
4290:
4270:
4234:
4230:
4224:
4220:
4195:
4175:
4150:
4145:
4141:
4131:is a model of
4120:
4117:
4114:
4111:
4108:
4103:
4099:
4078:
4054:
4051:
4045:
4041:
4028:
4025:
4012:
3989:
3986:
3980:
3976:
3933:
3930:
3924:
3920:
3893:
3889:
3875:, up to stage
3869:Gödel universe
3849:Main article:
3846:
3843:
3828:
3824:
3800:
3797:
3791:
3787:
3764:
3760:
3737:
3733:
3705:
3702:
3696:
3692:
3668:
3665:
3659:
3655:
3630:
3627:
3625:
3622:
3617:large cardinal
3599:
3596:
3595:
3594:
3587:
3584:
3561:
3541:
3519:
3515:
3492:
3488:
3467:
3443:
3440:
3437:
3433:
3427:
3423:
3398:
3378:
3356:
3352:
3329:
3325:
3304:
3280:
3277:
3274:
3270:
3264:
3260:
3233:
3228:
3224:
3203:
3181:
3178:
3175:
3171:
3147:
3142:
3139:
3136:
3132:
3128:
3125:
3099:
3094:
3090:
3078:weakly compact
3065:
3043:
3040:
3037:
3033:
3009:
3004:
3001:
2998:
2994:
2990:
2987:
2972:Mahlo cardinal
2959:
2937:
2934:
2931:
2927:
2903:
2898:
2895:
2892:
2888:
2884:
2881:
2859:
2854:
2850:
2823:
2818:
2814:
2789:
2767:
2764:
2761:
2757:
2733:
2711:
2707:
2686:
2681:
2677:
2673:
2668:
2664:
2641:
2638:
2634:
2630:
2606:
2602:
2598:
2594:
2590:
2570:
2563:
2559:
2554:
2550:
2545:
2541:
2513:
2509:
2505:
2501:
2497:
2477:
2470:
2466:
2461:
2457:
2452:
2448:
2424:
2419:
2415:
2411:
2406:
2403:
2400:
2396:
2392:
2387:
2383:
2355:
2350:
2347:
2342:
2338:
2333:
2329:
2324:
2320:
2299:
2277:
2273:
2269:
2247:
2242:
2238:
2217:
2214:
2211:
2208:
2205:
2202:
2197:
2192:
2188:
2160:
2155:
2151:
2147:
2144:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2077:
2074:
2070:
2066:
2040:
2036:
2032:
2029:
2024:
2019:
2015:
1994:
1989:
1985:
1981:
1978:
1958:
1953:
1949:
1945:
1940:
1936:
1919:
1916:
1899:
1895:
1892:Howard ordinal
1844:
1836:
1833:
1832:
1788:Main article:
1785:
1782:
1739:
1735:
1731:
1728:
1711:
1686:
1682:
1657:
1654:
1651:
1648:
1645:
1640:
1636:
1623:
1620:
1607:
1604:
1601:
1598:
1595:
1590:
1586:
1565:
1562:
1559:
1539:
1536:
1533:
1528:
1524:
1520:
1517:
1497:
1494:
1491:
1471:
1468:
1465:
1445:
1442:
1439:
1419:
1416:
1413:
1408:
1404:
1400:
1397:
1394:
1391:
1386:
1382:
1358:
1331:
1328:
1324:
1314:is called the
1301:
1297:
1276:
1273:
1270:
1250:
1247:
1244:
1239:
1235:
1214:
1211:
1208:
1203:
1199:
1178:
1152:
1149:
1146:
1124:
1120:
1099:
1079:
1076:
1073:
1068:
1064:
1043:
1021:
1017:
1013:
1010:
1007:
1004:
999:
995:
974:
971:
968:
965:
962:
957:
953:
932:
910:
906:
885:
865:
862:
859:
854:
851:
848:
844:
821:
817:
813:
810:
807:
804:
799:
795:
774:
771:
768:
763:
759:
738:
735:
730:
726:
703:
699:
676:
672:
651:
648:
643:
639:
618:
607:
606:
589:
584:
580:
574:
570:
566:
563:
556:
553:
548:
544:
539:
534:
529:
526:
523:
518:
514:
510:
505:
502:
497:
493:
488:
483:
480:
477:
472:
468:
452:
436:
432:
427:
404:
400:
379:
359:
356:
351:
347:
333:
330:) the ordinal
320:Main article:
317:
314:
312:
309:
289:
285:
261:
257:
252:
214:formal systems
209:
206:
167:Turing machine
152:Main article:
149:
146:
141:Main article:
138:
135:
118:ordinals (see
109:
98:
92:
84:
70:
26:
9:
6:
4:
3:
2:
8221:
8210:
8207:
8205:
8202:
8201:
8199:
8184:
8180:
8177:
8166:
8163:
8152:
8149:
8147:
8143:
8137:
8131:
8127:
8123:
8120:
8116:
8108:
8104:
8101:
8095:
8091:
8087:
8084:
8081:
8077:
8074:
8071:
8067:
8064:
8061:
8057:
8054:
8048:
8045:
8040:
8036:
8033:
8031:
8027:
8024:
8023:
8020:
8016:
8009:
8004:
8002:
7997:
7995:
7990:
7989:
7986:
7977:
7971:
7962:
7957:
7950:
7942:
7938:
7934:
7932:9780444851635
7928:
7924:
7920:
7916:
7912:
7908:
7901:
7899:
7889:
7882:
7878:
7874:
7873:Spectrum of L
7868:
7861:
7855:
7848:
7832:
7815:
7808:
7802:
7795:
7789:
7781:
7777:
7773:
7771:9780444105455
7767:
7763:
7759:
7755:
7751:
7747:
7743:
7736:
7729:
7727:
7725:
7723:
7715:
7714:
7707:
7698:
7693:
7686:
7679:
7673:
7671:
7663:
7657:
7642:
7635:
7629:
7627:
7617:
7612:
7608:
7604:
7603:
7598:
7591:
7584:
7578:
7564:
7560:
7553:
7551:
7535:
7528:
7522:
7507:
7500:
7494:
7486:
7482:
7478:
7474:
7470:
7466:
7462:
7458:
7454:
7447:
7440:
7437:W. Buchholz,
7434:
7419:
7412:
7406:
7404:
7402:
7400:
7398:
7396:
7394:
7392:
7390:
7388:
7386:
7384:
7382:
7380:
7378:
7376:
7374:
7372:
7370:
7368:
7359:
7355:
7351:
7349:3-540-11170-0
7345:
7341:
7337:
7333:
7329:
7322:
7315:
7271:
7266:
7246:
7238:
7232:
7228:
7227:
7219:
7211:
7207:
7202:
7197:
7193:
7189:
7185:
7178:
7174:
7164:
7160:
7156:
7152:
7148:
7145:
7144:
7134:
7129:
7125:
7123:3-540-07451-1
7119:
7114:
7113:
7107:
7103:
7102:
7093:
7089:
7086:
7083:
7079:
7078:
7073:
7070:
7067:
7063:
7059:
7058:
7053:
7049:
7046:
7043:
7042:0-262-68052-1
7039:
7035:
7031:
7028:
7025:
7021:
7018:
7015:
7014:0-387-07911-4
7011:
7007:
7003:
7000:
6997:
6996:0-444-10492-5
6993:
6989:
6985:
6984:Gaisi Takeuti
6982:
6979:
6978:0-387-51842-8
6975:
6971:
6967:
6964:
6963:
6957:
6949:
6935:
6912:
6909:
6881:
6861:
6858:
6852:
6849:
6846:
6840:
6835:
6832:
6827:
6823:
6813:
6811:
6789:
6785:
6781:
6778:
6775:
6772:
6769:
6764:
6760:
6756:
6751:
6747:
6738:
6734:
6726:
6700:
6696:
6687:
6677:
6675:
6653:
6649:
6641:
6632:
6626:
6622:
6601:
6598:
6595:
6572:
6569:
6558:
6539:
6536:
6513:
6505:
6490:
6470:
6448:
6444:
6436:
6427:
6421:
6417:
6409:
6393:
6385:
6370:
6350:
6328:
6324:
6316:
6307:
6301:
6297:
6289:
6273:
6265:
6250:
6242:
6239:is the least
6226:
6204:
6200:
6192:
6183:
6177:
6173:
6165:
6144:
6141:
6116:
6108:
6093:
6085:
6082:is the least
6069:
6047:
6043:
6035:
6026:
6020:
6016:
6008:
5987:
5962:
5954:
5938:
5935:
5932:
5928:
5920:
5911:
5905:
5901:
5893:
5874:
5871:
5848:
5840:
5839:
5838:
5824:
5802:
5797:
5769:
5766:
5746:
5730:
5708:
5703:
5673:
5670:
5662:
5645:
5641:
5618:
5613:
5583:
5580:
5572:
5555:
5551:
5530:
5508:
5495:
5494:
5493:
5479:
5474:
5470:
5464:
5460:
5439:
5419:
5410:
5392:
5388:
5383:
5375:
5366:
5360:
5356:
5335:
5315:
5307:
5303:
5299:
5277:
5273:
5252:
5244:
5234:
5220:
5217:
5211:
5208:
5203:
5199:
5175:
5167:
5166:
5162:
5145:
5136:
5120:
5116:
5095:
5087:
5083:
5077:
5067:
5065:
5060:
5046:
5043:
5021:
4996:
4993:
4973:
4970:
4967:
4945:
4918:
4891:
4887:
4878:
4874:
4857:
4837:
4829:
4813:
4799:
4798:-reflection.
4783:
4758:
4750:
4746:
4742:
4724:
4710:
4708:
4687:
4662:
4655:. For finite
4654:
4636:
4623:
4619:
4601:
4588:
4584:
4564:
4522:
4518:
4509:
4474:
4440:
4437:
4433:
4428:
4414:
4391:
4378:
4373:
4369:
4342:
4336:
4324:
4308:
4288:
4268:
4260:
4256:
4252:
4232:
4228:
4222:
4218:
4209:
4193:
4173:
4164:
4148:
4143:
4115:
4109:
4106:
4101:
4097:
4076:
4068:
4043:
4039:
4024:
4010:
3978:
3974:
3965:
3961:
3957:
3953:
3948:
3922:
3918:
3909:
3891:
3887:
3878:
3874:
3870:
3866:
3862:
3858:
3852:
3842:
3826:
3822:
3789:
3785:
3762:
3758:
3735:
3731:
3722:
3694:
3690:
3657:
3653:
3644:
3640:
3636:
3621:
3618:
3614:
3610:
3606:
3592:
3588:
3585:
3582:
3578:
3577:
3576:
3573:
3559:
3539:
3517:
3513:
3490:
3486:
3465:
3441:
3438:
3431:
3425:
3410:
3396:
3376:
3354:
3350:
3327:
3323:
3302:
3278:
3275:
3268:
3262:
3249:
3231:
3226:
3214:is the first
3179:
3176:
3169:
3159:
3140:
3137:
3134:
3130:
3115:
3097:
3092:
3079:
3076:is the first
3063:
3041:
3038:
3035:
3031:
3021:
3002:
2999:
2996:
2992:
2985:
2977:
2973:
2970:is the first
2957:
2935:
2932:
2929:
2925:
2915:
2896:
2893:
2890:
2886:
2879:
2857:
2852:
2839:
2821:
2816:
2803:
2800:is the first
2787:
2765:
2762:
2759:
2755:
2745:
2731:
2709:
2679:
2666:
2662:
2639:
2636:
2632:
2628:
2604:
2600:
2596:
2592:
2588:
2561:
2557:
2543:
2539:
2529:
2511:
2507:
2503:
2499:
2495:
2468:
2464:
2450:
2446:
2436:
2417:
2413:
2409:
2404:
2401:
2398:
2385:
2381:
2372:
2367:
2348:
2345:
2340:
2331:
2322:
2318:
2297:
2275:
2271:
2267:
2245:
2240:
2215:
2212:
2209:
2206:
2203:
2200:
2195:
2190:
2177:
2172:
2153:
2142:
2116:
2110:
2104:
2096:
2091:
2075:
2072:
2068:
2064:
2056:
2038:
2034:
2030:
2027:
2022:
2017:
1987:
1976:
1951:
1938:
1934:
1926:, defined as
1925:
1915:
1913:
1909:
1905:
1893:
1889:
1888:
1882:
1880:
1874:
1872:
1868:
1864:
1860:
1856:
1852:
1847:
1842:
1830:
1826:
1822:
1821:
1820:
1818:
1814:
1810:
1806:
1802:
1798:
1791:
1781:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1737:
1726:
1717:
1714:
1708:
1706:
1702:
1701:predicatively
1684:
1671:
1655:
1652:
1646:
1638:
1634:
1619:
1605:
1602:
1596:
1588:
1584:
1563:
1560:
1557:
1534:
1526:
1522:
1518:
1515:
1495:
1492:
1489:
1469:
1466:
1463:
1443:
1440:
1437:
1414:
1406:
1402:
1398:
1392:
1384:
1380:
1370:
1356:
1349:(to the base
1348:
1347:
1329:
1326:
1322:
1299:
1295:
1274:
1271:
1268:
1245:
1237:
1233:
1209:
1201:
1197:
1176:
1168:
1167:
1150:
1147:
1144:
1122:
1118:
1097:
1074:
1066:
1062:
1041:
1019:
1015:
1011:
1005:
997:
993:
972:
969:
963:
955:
951:
930:
908:
904:
883:
860:
852:
849:
846:
842:
819:
815:
811:
805:
797:
793:
769:
761:
757:
736:
733:
728:
724:
701:
697:
674:
670:
649:
646:
641:
637:
616:
587:
582:
572:
568:
561:
554:
551:
546:
542:
537:
532:
527:
524:
521:
516:
512:
508:
503:
500:
495:
491:
486:
481:
478:
475:
470:
466:
458:
457:
456:
434:
430:
425:
402:
398:
377:
357:
354:
349:
345:
336:
329:
323:
308:
306:
302:
297:
295:
292:measures the
283:
279:
275:
271:
267:
255:
248:
245:
240:
238:
234:
230:
225:
223:
219:
215:
205:
201:
200:(see below).
199:
194:
191:
187:
184:greater than
183:
179:
176:'s system of
175:
170:
168:
164:
160:
155:
144:
134:
132:
128:
123:
121:
117:
104:
101:), described
87:
82:
81:Church–Kleene
77:
73:
65:
63:
59:
55:
52:
48:
44:
40:
37:
33:
19:
8209:Proof theory
8182:
8135:
8129:
8125:
8114:
8106:
8093:
8089:
8079:
8069:
8059:
8038:
8029:
8014:
7970:
7949:
7914:
7910:
7888:
7867:
7854:
7814:
7801:
7788:
7745:
7741:
7712:
7706:
7697:1907.17611v1
7685:
7677:
7656:
7645:. Retrieved
7643:. 2006-02-07
7640:
7606:
7600:
7590:
7577:
7566:. Retrieved
7562:
7538:. Retrieved
7536:. 1993-02-21
7533:
7521:
7510:. Retrieved
7505:
7493:
7463:(1): 35–55.
7460:
7456:
7446:
7433:
7422:. Retrieved
7420:. 2017-07-29
7417:
7331:
7321:
7245:
7225:
7218:
7191:
7187:
7177:
7158:
7151:S. B. Cooper
7132:
7111:
7106:Barwise, Jon
7075:
7055:
7051:
7033:
7023:
7006:Proof theory
7005:
7002:Kurt Schütte
6988:Proof theory
6987:
6970:Proof theory
6969:
6955:
6814:
6727:
6683:
6671:
6086:larger than
5755:
5496:A set has a
5411:
5301:
5242:
5240:
5164:
5160:
5137:
5079:
5061:
4876:
4827:
4805:
4744:
4711:
4675:, the least
4652:
4578:
4510:if the rank
4488:
4446:
4429:
4361:is equal to
4258:
4250:
4207:
4165:
4031:
4030:
3949:
3907:
3876:
3872:
3864:
3854:
3720:
3638:
3632:
3601:
3574:
3411:
3160:
3022:
2916:
2802:inaccessible
2746:
2654:is equal to
2530:
2437:
2368:
2174:Next is the
2173:
2092:
2054:
1921:
1910:, let alone
1891:
1885:
1883:
1875:
1870:
1866:
1862:
1858:
1854:
1850:
1845:
1840:
1834:
1828:
1824:
1800:
1793:
1769:
1765:
1761:
1757:
1753:
1718:
1712:
1709:
1625:
1371:
1344:
1164:
1034:), and when
608:
325:
298:
281:
247:Peano axioms
241:
232:
228:
226:
216:(containing
211:
202:
195:
189:
185:
181:
171:
157:
124:
115:
80:
79:
66:
47:proof theory
29:
7961:1104.1842v1
7917:: 355–390.
7847:-separation
7762:10852/44063
7748:: 301–381.
7194:: 195–207.
6684:Within the
6559:there is a
4508:-reflecting
3814:instead of
2724:represents
1843:such that ε
923:(i.e., the
244:first-order
49:still have
8198:Categories
7879:catalog) (
7647:2010-08-10
7568:2021-08-10
7540:2021-08-10
7512:2021-08-10
7424:2021-08-10
7082:PostScript
6952:References
6614:such that
6129:is called
5975:is called
5861:is called
5723:iff it is
5265:such that
5188:such that
5108:such that
5074:See also:
4826:is called
4487:is called
4443:Reflection
4089:such that
3908:admissible
3620:unclear.)
1835:Here Ω = ω
1811:, Bridge,
1760:such that
1372:Ordering:
662:is called
218:arithmetic
78:is called
51:computable
32:set theory
7974:W. Chan,
7941:0049-237X
7829:Σ
7780:0049-237X
7477:1432-0665
7316:. (1987)"
7279:−
7263:Π
7210:0168-0072
6936:ρ
6882:η
6862:ρ
6853:η
6841:×
6824:ω
6697:ω
6654:β
6638:Σ
6633:≺
6627:α
6602:α
6596:β
6514:α
6491:α
6471:β
6449:β
6433:Σ
6428:≺
6422:α
6394:α
6371:α
6351:β
6329:β
6313:Σ
6308:≺
6302:α
6274:α
6251:α
6227:β
6205:β
6189:Σ
6184:≺
6178:α
6117:α
6094:α
6070:β
6048:β
6032:Σ
6027:≺
6021:α
5963:α
5939:β
5933:α
5917:Σ
5912:≺
5906:α
5875:β
5849:α
5794:Π
5731:σ
5700:Σ
5674:⊆
5646:σ
5610:Δ
5584:⊆
5556:σ
5505:Σ
5471:≺
5465:σ
5440:σ
5389:ω
5372:Σ
5367:≺
5361:α
5336:α
5316:α
5278:α
5253:α
5218:⊨
5212:∈
5204:α
5176:α
5121:α
5096:α
5047:ω
5018:Σ
4942:Σ
4915:Σ
4892:α
4858:α
4838:α
4814:α
4780:Π
4721:Π
4684:Π
4633:Π
4598:Π
4579:KP+Π
4565:ϕ
4557:-formula
4545:Γ
4523:α
4496:Γ
4475:α
4455:Γ
4415:ρ
4392:ω
4379:∩
4374:ρ
4347:#
4309:α
4289:α
4269:α
4219:ω
4194:α
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4140:Π
4116:ω
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4102:α
4077:α
4044:ω
4040:ω
4011:α
3979:α
3975:ω
3919:ω
3892:α
3823:ω
3786:ω
3759:ω
3732:ω
3691:ω
3654:ω
3560:ϵ
3540:ϵ
3487:ω
3436:Υ
3432:ε
3422:Ψ
3397:ϵ
3377:ϵ
3324:ω
3273:Ξ
3269:ε
3259:Ψ
3223:Π
3202:Ξ
3174:Ξ
3170:ε
3131:ε
3124:Ψ
3089:Π
3032:ε
2993:ε
2986:ψ
2926:ε
2887:ε
2880:ψ
2849:Δ
2813:Π
2756:ε
2706:Ω
2680:ν
2676:Ω
2663:ψ
2640:ν
2601:ε
2558:ε
2553:Ω
2540:ψ
2512:ω
2508:ω
2469:ω
2465:ω
2460:Ω
2447:ψ
2414:ε
2410:⋅
2399:ω
2395:Ω
2382:ψ
2341:ω
2337:Ω
2332:ε
2319:ψ
2298:ω
2276:ω
2237:Π
2201:−
2187:Π
2154:ω
2150:Ω
2143:ψ
2117:ω
2076:ω
2028:−
2014:Π
1988:ω
1984:Ω
1977:ψ
1952:ω
1948:Ω
1935:ψ
1738:α
1734:Γ
1730:↦
1727:α
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1606:δ
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1234:φ
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1198:φ
1177:δ
1151:δ
1145:γ
1123:γ
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1098:α
1075:α
1067:δ
1063:φ
1042:δ
1020:β
1016:ε
1006:β
994:φ
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956:γ
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816:ω
806:β
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758:φ
737:α
729:α
725:ε
698:ζ
675:ι
671:ε
650:α
642:α
638:ω
617:ι
583:ω
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543:ε
538:ω
533:ω
525:ω
522:⋅
513:ε
492:ε
487:ω
467:ε
435:ω
431:ω
426:ω
403:ω
399:ω
378:ω
358:α
350:α
346:ω
268:), so by
116:recursive
36:countable
7877:WorldCat
7485:35012853
7157:(eds.):
7155:J. Truss
7108:(1976).
6874:, where
6555:-stable
6463:, where
6343:, where
6219:, where
6162:-stable
6062:, where
6005:-stable
5890:-stable
5064:Jensen's
4906:of KP +
4407:, where
4003:for the
3952:Friedman
3682:. Thus,
3194:, where
3056:, where
2950:, where
2780:, where
1797:Bachmann
1137:for all
834:and let
114:are the
76:supremum
74:; their
39:ordinals
8049: Γ
7358:0655036
7066:2272243
4186:is the
3964:oracles
3910:, thus
3572:, 0).
3409:, 0).
1813:Schütte
1805:Takeuti
1776:" and "
1090:as the
876:be the
303:is the
266:Gentzen
8181:
8124:
8105:
8088:
8078:
8068:
8058:
8037:
8028:
7939:
7929:
7778:
7768:
7716:(1973)
7508:. 1990
7483:
7475:
7418:Madore
7356:
7346:
7233:
7208:
7120:
7064:
7040:
7012:
6994:
6976:
6928:, and
5663:A set
5573:A set
5084:has a
3958:, and
3956:Jensen
3721:define
3639:cannot
1801:ad hoc
1770:ad hoc
1550:) or (
1482:) or (
174:Kleene
7956:arXiv
7881:EuDML
7738:(PDF)
7692:arXiv
7637:(PDF)
7530:(PDF)
7502:(PDF)
7481:S2CID
7414:(PDF)
7062:JSTOR
5292:is a
4589:by a
3960:Sacks
3861:above
1809:Aczel
1778:large
1774:small
120:below
103:below
56:(see
7937:ISSN
7927:ISBN
7776:ISSN
7766:ISBN
7473:ISSN
7344:ISBN
7231:ISBN
7206:ISSN
7153:and
7118:ISBN
7038:ISBN
7010:ISBN
6992:ISBN
6974:ISBN
6913:<
6599:>
5044:>
4620:and
4583:-ref
2637:<
2597:<
2504:<
2073:<
2055:sets
1884:The
1754:that
1603:<
1576:and
1561:>
1519:<
1508:and
1493:<
1467:<
1456:and
1399:<
1272:<
1261:for
1148:<
593:etc.
88:or ω
8097:Ω+1
8082:(Ω)
8072:(Ω)
8062:(Ω)
7919:doi
7758:hdl
7750:doi
7611:doi
7465:doi
7336:doi
7196:doi
6806:⌈φ⌉
6730:ATR
6676:.
6557:iff
6408:iff
6288:iff
6164:iff
6007:iff
5892:iff
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5601:is
5300:of
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5082:ZFC
4709:.
4706:m+1
3591:ZFC
3478:= (
3315:= (
2528:.
1900:Ω+1
1894:, ψ
1707:".
1618:).
1369:).
282:can
276:on
224:).
8200::
8139:+1
8112:(Ω
7935:.
7925:.
7915:94
7913:.
7909:.
7897:^
7774:.
7764:.
7756:.
7746:79
7744:.
7740:.
7721:^
7669:^
7639:.
7625:^
7607:20
7605:.
7599:.
7561:.
7549:^
7532:.
7504:.
7479:.
7471:.
7461:33
7459:.
7455:.
7416:.
7366:^
7354:MR
7352:.
7342:.
7204:.
7192:32
7190:.
7186:.
7090:,
7074:,
7054:,
7050:,
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6986:,
6968:,
5837:.
5492::
5409:.
5059:.
4875:,
3954:,
3871:,
3841:.
3645:,
3611:,
3607:,
3552:,
3532:;
3505:;
3389:,
3369:;
3342:;
3080:(=
2804:(=
2435:.
2171:.
1898:(ε
1881:.
1857:)=
1823:ψ(
1819::
417:,
390:,
8183:Ω
8172:1
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8110:0
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8094:ε
8092:(
8090:ψ
8080:θ
8070:θ
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8000:t
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7964:.
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7943:.
7921::
7833:1
7782:.
7760::
7752::
7700:.
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7650:.
7619:.
7613::
7571:.
7543:.
7515:.
7487:.
7467::
7427:.
7360:.
7338::
7301:I
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7239:.
7212:.
7198::
7165:.
7126:.
7084:)
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6916:)
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6902:(
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6844:(
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6779:.
6776:.
6773:.
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5294:Σ
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5221:T
5215:)
5209:,
5200:L
5196:(
5165:L
5163:=
5161:V
5146:T
5117:L
5022:1
5009:+
4997:P
4994:K
4974:L
4971:=
4968:V
4946:1
4919:1
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4877:L
4784:n
4759:n
4725:3
4704:Π
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4637:3
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4389:(
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4370:L
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3704:K
3701:C
3695:1
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3664:C
3658:1
3593:.
3583:.
3518:0
3514:P
3491:+
3466:X
3442:1
3439:+
3426:X
3355:0
3351:P
3328:+
3303:X
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3227:0
3180:1
3177:+
3146:)
3141:1
3138:+
3135:K
3127:(
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3000:+
2997:M
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1980:(
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1944:(
1939:0
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1849:=
1846:σ
1841:σ
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562:=
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334:0
332:ε
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286:0
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251:ε
233:o
229:o
190:O
186:o
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110:1
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