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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.
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 ...
Also, the ordinal indicators should be distinguishable from superscript characters. The top of the ordinal indicators (i.e., the top of the elevated letter a and letter o) must be aligned [1] with the cap height of the font. The alignment of the top of superscripted letters a and o will depend on the font.
When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. For summarizing the syntax in the entry name, the symbol is used for representing the neighboring parts of a formula that contains the symbol. See § Brackets for examples of use. Most symbols have two printed versions.
There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity, the following are equivalent for a set x: x is a (von Neumann) ordinal, x is a transitive set, and set membership is trichotomous on x, x is a transitive set totally ordered by set inclusion, x is a transitive set of transitive sets.
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 ...
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In mathematics, the Feferman–Schütte ordinal (Γ 0) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion . It is named after Solomon Feferman and Kurt Schütte , the former of whom suggested the name Γ 0 .
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