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The order in which the enumeration values are given matters. An enumerated type is an ordinal type, and the pred and succ functions will give the prior or next value of the enumeration, and ord can convert enumeration values to their integer representation. Standard Pascal does not offer a conversion from arithmetic types to enumerations, however.
In set theory, there is a more general notion of an enumeration than the characterization requiring the domain of the listing function to be an initial segment of the Natural numbers where the domain of the enumerating function can assume any ordinal. Under this definition, an enumeration of a set S is any surjection from an ordinal α onto S ...
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy, or a Schwichtenberg-Wainer hierarchy) [1] is an ordinal-indexed family of rapidly increasing functions f α: N → N (where N is the set of natural numbers {0, 1, ...}, and α ranges up to some large countable ordinal).
A primitive recursive ordinal function is defined in the same way, except that the initial function F(x, y) = x ∪ {y} is replaced by F(x) = x ∪ {x} (the successor of x). The primitive recursive ordinal functions are the same as the primitive recursive set functions that map ordinals to ordinals.
In mathematics, Rathjen's psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals M {\displaystyle M} to generate large countable ordinals . [ 1 ]
In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in Veblen (1908). If φ 0 is any normal function, then for any non-zero ordinal α, φ α is the function enumerating the common fixed points of φ β for β<α. These ...
The third function might be defined as one that maps each ordinal to the smallest ordinal that cannot yet be described with the above two functions and previous values of this function. This would map β to ω·β except when β is a fixed point of that function plus a finite number in which case one uses ω·(β+1).
A fixed point of a normal function is an ordinal such that () =. The fixed point lemma states that the class of fixed points of any normal function is nonempty and in fact is unbounded: given any ordinal α {\displaystyle \alpha } , there exists an ordinal β {\displaystyle \beta } such that β ≥ α {\displaystyle \beta \geq \alpha } and f ...