Logical Methods pp 572-592 | Cite as

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

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## 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 Σ_{n} 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 ω^{k} (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* ≤ (2*k* − 2), ω^{k} 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 ω^{k}). I show, in fact, that there is, for each n, a linear order (of some type ω^{k} · τ) that is *n*-recursive but *has no* (*n*+1)-*recursive copy*.

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