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For example, the ordinal 42 is generally identified as the set {0, 1, 2, ..., 41}. Conversely, any set S of ordinals that is downward closed — meaning that for any ordinal α in S and any ordinal β < α, β is also in S — is (or can be identified with) an ordinal. This definition of ordinals in terms of sets allows for infinite ordinals.
This is a well-ordering and hence gives an ordinal number. The definition of exponentiation can also be given by transfinite recursion on the exponent β. When the exponent β = 0, ordinary exponentiation gives α 0 = 1 for any α. For β > 0, the value of α β is the smallest ordinal greater than or equal to α δ · α for all δ < β ...
In written languages, an ordinal indicator is a character, or group of characters, following a numeral denoting that it is an ordinal number, rather than a cardinal number. Historically these letters were "elevated terminals", that is to say the last few letters of the full word denoting the ordinal form of the number displayed as a superscript .
The proof-theoretic ordinal of such a theory is the supremum of the order types of all ordinal notations (necessarily recursive, see next section) that the theory can prove are well founded—the supremum of all ordinals for which there exists a notation in Kleene's sense such that proves that is an ordinal notation.
A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number.
Ordinal indicator – Character(s) following an ordinal number (used when writing ordinal numbers, such as a super-script) Ordinal number – Generalization of "n-th" to infinite cases (the related, but more formal and abstract, usage in mathematics) Ordinal data, in statistics; Ordinal date – Date written as number of days since first day of ...
(The fact that every computable ordinal has a notation follows from the closure of this system of ordinal notations under successor and effective limits.) < O {\displaystyle <_{\mathcal {O}}} is not computably enumerable , but there is a computably enumerable relation which agrees with < O {\displaystyle <_{\mathcal {O}}} precisely on members ...
Computable ordinals (or recursive ordinals) are certain countable ordinals: loosely speaking those represented by a computable function.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 ...