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→Transfinitely many variables: No, it is ordinal exponentiation. "provided that all but a finite number of them are zero" |
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The fundamental sequence for an ordinal with [[cofinality]] ω is a distinguished strictly increasing ω-sequence which has the ordinal as its limit. If one has fundamental sequences for α and all smaller limit ordinals, then one can create an explicit constructive bijection between ω and α, (i.e. one not using the axiom of choice). Here we will describe fundamental sequences for the Veblen hierarchy of ordinals. The image of ''n'' under the fundamental sequence for α will be indicated by α[''n''].
A variation of [[Ordinal arithmetic#Cantor normal form|Cantor normal form]] used in connection with the Veblen hierarchy is — every nonzero ordinal number α can be uniquely written as <math>\alpha = \varphi_{\beta_1}(\gamma_1) + \varphi_{\beta_2}(\gamma_2) + \cdots + \varphi_{\beta_k}(\gamma_k)</math>, where ''k''>0 is a natural number and each term after the first is less than or equal to the previous term, <math>\varphi_{\beta_m}(\gamma_m) \geq \varphi_{\beta_{m+1}}(\gamma_{m+1}) \,,</math> and each <math>\gamma_m < \varphi_{\beta_m}(\gamma_m) \,.</math> If a fundamental sequence can be provided for the last term, then that term can be replaced by such a sequence to get <math>\alpha [n] = \varphi_{\beta_1}(\gamma_1) + \cdots + \varphi_{\beta_{k-1}}(\gamma_{k-1}) + (\varphi_{\beta_k}(\gamma_k) [n]) \,.</math>
For any β, if γ is a limit with <math>\gamma < \varphi_{\beta} (\gamma) \,,</math> then let <math>\varphi_{\beta}(\gamma) [n] = \varphi_{\beta}(\gamma [n]) \,.</math>
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