A universal embedding property of the RETs

Abstract

Recursive equivalence types are an effective or recursive analogue of cardinal numbers. They were introduced by Dekker in the early 1950's. The richness of various theories related to the recursive equivalence types is demonstrated in this paper by showing that the theory of any countable relational structure can be embedded in or interpreted in these theories. A more complete summary is presented in the last paragraph of this section.Let E = {0,1, 2, …} be the natural numbers. If α ⊆ E, β ⊆ E, and there is a 1-1 partial recursive function f such that the image under f of α is β, α and β are called recursively equivalent (see [3]). The recursive equivalence type or RET of α, denoted 〈α〉, is the class of all β recursively equivalent to α. Addition of RETs is defined by 〈α〉 + 〈β〉 = 〈{2x ∣ x ∈ α} ∪ 〈{2x + 1 ∣ x ∈ β}〉. The partial ordering ≤ is defined on the RETs by A ≤ B iff (EC)(A + C = B). An RET, X, is called an isol if X ≠ X + 1 or, equivalently, if no representative of X is recursively equivalent to a proper subset of itself. The isols are thus the recursive analogue of the Dedekind-finite cardinals.