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Not to be confused with ordinal arithmetic →Transfinitely many variables |
→Transfinitely many variables: No, it is ordinal exponentiation. "provided that all but a finite number of them are zero" |
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===Transfinitely many variables===
More generally, Veblen showed that φ can be defined even for a transfinite sequence of ordinals α<sub>β</sub>, provided that all but a finite number of them are zero. Notice that if such a sequence of ordinals is chosen from those less than an uncountable [[regular cardinal]] κ, then the sequence may be encoded as a single ordinal less than
The definition can be given as follows: let <u>α</u> be a transfinite sequence of ordinals (i.e., an ordinal function with finite support) ''which ends in zero'' (i.e., such that α<sub>0</sub>=0), and let <u>α</u>[γ@0] denote the same function where the final 0 has been replaced by γ. Then γ↦φ(<u>α</u>[γ@0]) is defined as the function enumerating the common fixed points of all functions ξ↦φ(<u>β</u>) where <u>β</u> ranges over all sequences which are obtained by decreasing the smallest-indexed nonzero value of <u>α</u> and replacing some smaller-indexed value with the indeterminate ξ (i.e., <u>β</u>=<u>α</u>[ζ@ι<sub>0</sub>,ξ@ι] meaning that for the smallest index ι<sub>0</sub> such that α<sub>ι<sub>0</sub></sub> is nonzero the latter has been replaced by some value ζ<α<sub>ι<sub>0</sub></sub> and that for some smaller index ι<ι<sub>0</sub>, the value α<sub>ι</sub>=0 has been replaced with ξ).
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