Concept

Admissible ordinal

Overview

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. The term was coined by Richard Platek in 1966.

The first two admissible ordinals are ω and

{\displaystyle \omega _{1}^{\mathrm {CK} }}

(the least nonrecursive ordinal, also called the Church–Kleene ordinal). 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. One sometimes writes

{\displaystyle \omega _{\alpha }^{\mathrm {CK} }}

for the

{\displaystyle \alpha }

-th ordinal that is either admissible or a limit of admissibles; an ordinal that is both is called recursively inaccessible. 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). 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.

{\displaystyle \alpha }

is an admissible ordinal if and only if there is a standard model

{\displaystyle M}

of KP whose set of ordinals is

{\displaystyle \alpha }

, in fact this may be take as the definition of admissibility. The

{\displaystyle \alpha }

th admissible ordinal is sometimes denoted by

{\displaystyle \tau _{\alpha }}

p. 174 or

{\displaystyle \tau _{\alpha }^{0}}

The Friedman–Jensen–Sacks theorem states that a countable

{\displaystyle \alpha }

is admissible if and only if there exists some

⊆

{\displaystyle A\subseteq \omega }

such that

{\displaystyle \alpha }

is the least ordinal not recursive in

{\displaystyle A}

. Equivalently, for any countable admissible

{\displaystyle \alpha }

, there is an

⊆

{\displaystyle A\subseteq \mathbb {N} }

making

{\displaystyle \alpha }

minimal such that

⟨

∈

⟩

{\displaystyle \langle L_{\alpha },\in ,A\rangle }

is an admissible structure.p. 264

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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Another kind of the indivisible is outside of the whole genus of the continuous; and in this way incorporeal substances, like God, angel and soul, are called indivisible. Such a kind of indivisible does not belong to the continuous, as a part of it, but as touching it by its power; hence, according as its power can extend itself to one or to many, to a small thing, or to a great one, in…
Aquinas Thomas, Summa Theologica, Part I
States what Admissible ordinal is.
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We may _define_ the "inductive" numbers as _all those that possess all inductive properties_; they will be the same as what are called the "natural" numbers, _i.e._ the ordinary finite whole numbers. To all such numbers, proofs by mathematical induction can be validly applied. They are those numbers, we may loosely say, which can be reached from 0 by successive additions of 1; in other words, they are all the…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
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Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Admissible ordinal is.
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Philosophers who discussed Admissible ordinal

Saul Kripke

Related ideas

Place in the tradition

Part of American philosophy.

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