Primitive recursive ordinal functions with added constants

Abstract

One of the basic differences between the primitive recursive functions on the natural numbers and the primitive recursive ordinal functions (PR) is the nearly complete absence of constant functions in PR. Since ω is closed under all of the functions in PR, if α is any infinite ordinal, then λξ·α is not in PR. It is easily seen, however, that if one adds to the initial functions of PR the constant function λξ·ω, then all of the ordinals up to ω#, the next largest PR-closed ordinal, have their constant functions in this class. Since, however, such collections of functions are always countable, it is also the case that if one adds to the initial functions of PR the function λξ. α for uncountable α, then there are ordinals β < α whose constant functions are not in this collection. Because of this, the following objects are of interest:Definition. For all α,(i)PR(α) is the collection of functions obtained by adding to the initial primitive recursive ordinal functions, the function λξ· α;(ii) PRsp(α), the primitive recursive spectrum of α, is the set {β < α ∣ λξβ ∈ PR(α);(iii) Λ (α)= μρ(ρ∉ PRsp(α)).