Logical Methods pp 572-592

# n-Recursive Linear Orders without (n+1)-Recursive Copies

• Michael Moses
Chapter
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)

## Abstract

An algebraic structure, with universe ω, is said to be recursive if, and only if, the quantifier-free formulae in its language uniformly denote recursive relations in the structure. This concept generalizes to that of an n-recursive structure, in which the Σn formulae uniformly denote recursive relations. I present a sequence of simply defined relations in the language of linear order and show that a recursive linear order of type ωk (with k < ω) is n-recursive if, and only if, it has the first n relations in the sequence recursive. I can then show that, for each k, and each n ≤ (2k − 2), ωk has an n-recursive copy that is not (n+1)-recursive. This means that, for each n, there is an n-recursive but not (n+1)-recursive linear order (of some type ωk). I show, in fact, that there is, for each n, a linear order (of some type ωk · τ) that is n-recursive but has no (n+1)-recursive copy.

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