Recursively invariant β-recursion theory

We introduce recursively invariant β-recursion theory as a new approach towards recursion theory on an arbitraty limit ordinal β. We follow Friedman and Sacks and call a subset of β β-recursively enumerable if it is Σ₁-definable over L_β. Since Friedman-Sacks' notion of a β-finite set is not invariant under β-recursive permutations of β we turn to a different notion. Under all possible invariant generalizations of finite there is a canonical one which we call i-finite. We consider further in the inadmissible case those criteria for the adequacy of generalizations of finite which have earlier been developed by Kreisel, Moschovakis and others. We look at infinitary languages over inadmissible sets L_β and the compactness theorem for these languages, the characterization of the basic notions of β-recursion theory in terms of model theoretic invariance, the definition of β-recursion theory via an equation calculus and axioma for computation theories. In turns out that in all these approaches the i-finite sets are those subsets of β respectively L_β which behave like finite sets.

Invariant β-recursion theory contains classical recursion theory and α-recursion theory as special cases. We start in the second half of this paper the systematic development of invariant β-recursion theory for all limit ordinals β. We study in particular i-degrees, which generalize Turing degrees and α-degrees. Besides 0 (the degree of β-recursive sets) and 0′ (the largest β-r.e. degree) there exist incomparable β-r.e. i-degrees for every limit ordinal β. Similar as the step from ω to α gave rise to the introduction of regular respectively hyperregular sets we arrive in invariant β-recursion theory at the new notion of an i-absolute β-r.e sets. This notion is useful in order to describe a difference among hyperregular β-r.e. sets which occurs exclusively in the inadmissible case. The study of i-degrees is most difficult for those β which are strongly inadmissible (i.e. σ1 cf β<β^∗). For those strongly inadmissible β where β^∗ is regular we give two new constructions which rely heavily in the combinatorial properties of regular cardinals (◊, closed unbounded sets and the Δ-System lemma). We construct a β-r.e. degree a > 0 such that no degree b⩽a contains a simple set and we prove a splitting theorem for simple β-r.e. sets. We base the definition of a simple set on the general notion of a 1-finite set.