The Meta-R.E. sets, but not the Π₁¹ sets, can be enumerated without repetition

Abstract

In this paper we investigate the possibility of extending Friedberg's enumeration of the recursively enumerable (r.e.) sets without duplication [1, p. 312] to meta-recursion theory. It turns out that all of our proposed extensions are impossible save one: the metarecursively enumerable (meta-r.e.) sets can be enumerated without duplication, but only if all the recursive ordinals are used as indices (Theorems 1 and 2). The sets cannot be so enumerated, even if the index set is all recursive ordinals (Theorems 3 and 4). As a corollary, one proves there is no predicate P(n, x) with the property that for each set A there is exactly one integer n for which A = {x ∣ P(n, x)}. We also discuss enumerations of nonempty, infinite, and coinfinite and meta-r.e. sets.

On a positive set theory with inequality

“…In order to have full extensionality, one would need an enumeration without repetitions. Unfortunately, such an enumeration cannot exist, see [15]. However, [15] does give an enumeration without repetitions of the meta-r.e.…”

“…Unfortunately, such an enumeration cannot exist, see [15]. However, [15] does give an enumeration without repetitions of the meta-r.e. sets of recursive ordinals.…”

“…sets of recursive ordinals. Our idea is to modify slightly the construction of [15] so to enumerate Π 1 1 sets of natural numbers without repetitions up to atoms and coatoms.…”

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We introduce a quite natural Frege-style set theory, which we call Strong-Frege-2 (SF2 ), a sort of simplification of the theory considered in [13] (under the name Strong-Frege-3) and [1] (under the name F2). We give a model of a weaker variant of SF2 , called SF2 AC, where atoms and coatoms are allowed.To construct the model we use an enumeration "almost without repetitions" of the Π 1 1 sets of natural numbers; such an enumeration can be obtained via a classical priority argument much in the style of [5] and [15].

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“…To construct the model we use an enumeration "almost without repetitions" of the Π 1 1 sets of natural numbers; such an enumeration can be obtained via a classical priority argument much in the style of [5] and [15].…”