In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and Lα ⊧ Σ0-collection.[1][2] The term was coined by Richard Platek in 1966.[3]
The first two admissible ordinals are ω and (the least nonrecursive ordinal, also called the Church–Kleene ordinal).[2] Any regular uncountable cardinal is an admissible ordinal.
By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal, but for Turing machines with oracles.[1] One sometimes writes for the -th ordinal that is either admissible or a limit of admissibles; an ordinal that is both is called recursively inaccessible.[4] There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals (one can define recursively Mahlo ordinals, for example).[5] But all these ordinals are still countable. Therefore, admissible ordinals seem to be the recursive analogue of regular cardinal numbers.
Notice that α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ < α for which there is a Σ1(Lα) mapping from γ onto α.[6] is an admissible ordinal iff there is a standard model of KP whose set of ordinals is , in fact this may be take as the definition of admissibility.[7][8] The th admissible ordinal is sometimes denoted by [9][8]p. 174 or .[10]
The Friedman-Jensen-Sacks theorem states that countable is admissible iff there exists some such that is the least ordinal not recursive in .[11]
^ abFriedman, Sy D. (1985), "Fine structure theory and its applications", Recursion theory (Ithaca, N.Y., 1982), Proc. Sympos. Pure Math., vol. 42, Amer. Math. Soc., Providence, RI, pp. 259–269, doi:10.1090/pspum/042/791062, MR 0791062. See in particular p. 265.
^ abFitting, Melvin (1981), Fundamentals of generalized recursion theory, Studies in Logic and the Foundations of Mathematics, vol. 105, North-Holland Publishing Co., Amsterdam-New York, p. 238, ISBN 0-444-86171-8, MR 0644315.
^G. E. Sacks, Higher Recursion Theory (p.151). Association for Symbolic Logic, Perspectives in Logic
^Friedman, Sy D. (2010), "Constructibility and class forcing", Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, pp. 557–604, doi:10.1007/978-1-4020-5764-9_9, MR 2768687. See in particular p. 560.
^Kahle, Reinhard; Setzer, Anton (2010), "An extended predicative definition of the Mahlo universe", Ways of proof theory, Ontos Math. Log., vol. 2, Ontos Verlag, Heusenstamm, pp. 315–340, MR 2883363.
^K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.38). Accessed 2021-05-06.
^K. J. Devlin, Constructibility (1984), ch. 2, "The Constructible Universe, p.95. Perspectives in Mathematical Logic, Springer-Verlag.
^ abJ. Barwise, Admissible Sets and Structures (1976). Cambridge University Press
^P. G. Hinman, Recursion-Theoretic Hierarchies (1978), pp.419--420. Perspectives in Mathematical Logic, ISBN 3-540-07904-1.
^S. Kripke, "Transfinite Recursion, Constructible Sets, and Analogues of Cardinals" (1967), p.11. Accessed 2023-07-15.
^W. Marek, M. Srebrny, "Gaps in the Constructible Universe" (1973), pp.361--362. Annals of Mathematical Logic 6
and 25 Related for: Admissible ordinal information
In set theory, an ordinal number α is an admissibleordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other...
\beta } is the least admissibleordinal larger than an admissibleordinal larger than α {\displaystyle \alpha } . A countable ordinal α {\displaystyle \alpha...
hyperarithmetical, and the smallest admissibleordinal after ω {\displaystyle \omega } (an ordinal α {\displaystyle \alpha } is called admissible if L α ⊨ K P {\displaystyle...
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite...
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories...
generalisation of recursion theory to subsets of admissibleordinals α {\displaystyle \alpha } . An admissible set is closed under Σ 1 ( L α ) {\displaystyle...
least admissibleordinal greater than δ {\displaystyle \delta } . The cardinal δ {\displaystyle \delta } is said to be Woodin-in-the-next-admissible if for...
example of an admissible set is the set of hereditarily finite sets. Another example is the set of hereditarily countable sets. Admissibleordinal Barwise,...
hierarchy defined relative to some admissibleordinal α {\displaystyle \alpha } . For a given admissibleordinal α {\displaystyle \alpha } , define the...
best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement...
theory of truth. He has also contributed to recursion theory (see admissibleordinal and Kripke–Platek set theory). Two of Kripke's earlier works, "A Completeness...
least limit of admissibleordinals or the least limit of infinite cardinals and B O {\displaystyle {\mathsf {BO}}} is Buchholz's ordinal. ψ Ω ( ε Ω ω +...
of ω. admissible An admissible set is a model of Kripke–Platek set theory, and an admissibleordinal is an ordinal α such that Lα is an admissible set AH...
functions from ordinals to ordinals), introduced by Oswald Veblen in Veblen (1908). If φ0 is any normal function, then for any non-zero ordinal α, φα is the...
Richter, P. Aczel, Inductive Definitions and Reflecting Properties of AdmissibleOrdinals (1974, p.23). Accessed 30 August 2022. Devlin, Keith (1984). Constructibility...
truth. He has also made important contributions to set theory (see admissibleordinal and Kripke–Platek set theory) David Kellogg Lewis, another student...
Peter (1974). "Inductive Definitions and Reflecting Properties of AdmissibleOrdinals". Studies in Logic and the Foundations of Mathematics. 79: 301–381...
essentially the nonrecursive analog to the stability property for admissibleordinals. More generally, a cardinal number κ is called λ-Πm-shrewd if for...
it is admissible or derivable. A derivable rule is one whose conclusion can be derived from its premises using the other rules. An admissible rule is...
increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform...
Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated...
by α-recursion theory, which is the study of definable subsets of admissibleordinals. Hyperarithmetical theory is the special case in which α is ω 1 C...
von Neumann universe, V {\displaystyle V} . The stages are indexed by ordinals. In von Neumann's universe, at a successor stage, one takes V α + 1 {\displaystyle...
attempted to replace cardinal utility with the apparently-weaker concept of ordinal utility. Cardinal utility appears to impose the assumption that levels...
ability to cope with poor-quality input information. It can use non-numeric (ordinal), non-exact (interval) and non-complete expert information to solve multi-criteria...