Some properties of invariant sets

In [1] two interesting invariance notions were introduced: the notions of a set of godel numbers being invariant to automorphisms of the structures ( ω , ·) and ( ω , E ) respectively. Here, · and E are defined by n · m ≃ φ n ( m ) and nEm if and only if n Є W m , where { φ n } and { W n } are acceptable enumerations of the partial recursive functions and r.e. sets respectively. In this paper we continue the study of the invariant sets, and especially the invariant r.e. sets, of gödel numbers. We start off with an easy result which characterizes the Turing degrees containing invariant sets. We then take a closer look at r.e. sets invariant with respect to automorphisms of ( ω , E ). Using the characterization [1, Theorem 4.2] of such sets, we will derive a somewhat different characterization (which was stated, but not proved, in [1, Proposition 4.4]) and, using it as a tool for constructing invariant sets, prove that the r.e. sets invariant with respect to automorphisms of ( ω , E ) cannot be effectively enumerated. We will next discuss representations of r.e. sets invariant with respect to automorphisms of ( ω , ·). Although these sets do not have as nice a characterization as the r.e. sets invariant with respect to automorphisms of ( ω , E ) do, the techniques of [1] can still profitably be used to investigate their structure. In particular, if f is a partial recursive function whose graph is invariant with respect to automorphisms of ( ω , ·), then for every a in the domain of f , there is a term t(a) built up from a and · only such that f(a) ≃ t(a) . This is an analog to [1, Corollary 4.3]. We will also prove an analog to a result mentioned in the previous paragraph: the r.e. sets invariant with respect to automorphisms of ( ω , ·) cannot be effectively enumerated.