Stability of recursive structures in arithmetical degrees

Ash, C. J.

doi:10.1016/0168-0072(86)90048-5

Cited by 48 publications

(34 citation statements)

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confidence: 77%

“…We show that, under certain assumptions of recursiveness in 21, the recursive structure 21 is A^-stable for a < wfK if and only if there is an enumeration of 21 using a E^ set of recursive EQ infinitary formulae and finitely many parameters from 21. This extends the results of [1].To do this, we first obtain results concerning A^ paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a A^ function from the previous node and its label.…”

mentioning

confidence: 76%

“…We shall show that, under certain assumptions, 21 is A°-stable iff there exists a "formally A° enumeration of 21. "The basic outline of our argument is as in [1], where such a result was obtained for finite a. However, certain technicalities prevent the generalization to infinite a from being as straightforward as might be expected.…”

mentioning

confidence: 99%

“…To do this, we first obtain results concerning A^ paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a A^ function from the previous node and its label.…”

mentioning

confidence: 99%

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Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees

Ash¹

1986

Trans. Amer. Math. Soc.

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ABSTRACT. We show that, under certain assumptions of recursiveness in 21, the recursive structure 21 is A^-stable for a < wfK if and only if there is an enumeration of 21 using a E^ set of recursive EQ infinitary formulae and finitely many parameters from 21. This extends the results of [1].To do this, we first obtain results concerning A^ paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a A^ function from the previous node and its label. Introduction.We say that a recursive structure 21 is A\\-stable if, for every recursive structure 03 = 21, every isomorphism from 03 to 21 is A° in Kleene's hyperarithmetical hierarchy. We shall show that, under certain assumptions, 21 is A°-stable iff there exists a "formally A° enumeration of 21."The basic outline of our argument is as in [1], where such a result was obtained for finite a. However, certain technicalities prevent the generalization to infinite a from being as straightforward as might be expected. The argument seems most easily described in terms of recursive labelling systems, which were introduced in [1] for a similar purpose.The two basic results for a-systems are obtained in §1 and applied in the succeeding sections to the question of A°-stability.These basic results are also used in the related topic of A°-categoricity [2]. In §1 we define the notion of a recursively a-guided recursive labelling system, or a-system, and in Proposition 1 state the desired result similar to that of [lj involving the existence of r.e. points of 2N and of labellings of A^ paths in an

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supporting

confidence: 77%

mentioning

confidence: 76%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees

Ash¹

1986

Trans. Amer. Math. Soc.

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“…We may also assume that the least and greatest elements of L are in Ln • We will construct L = \JS Ls, and h -limshs in stages using a recursive construction. At the initial stage s = 0 we map the least [greatest] element of L to 0 [ 1 ] with highest priority and never change h on these arguments. At stage s, we are allowed to add new elements to L between the current values of h(a) and h(b) (a < b) only if there exists c e Ls with a < c < b.…”

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confidence: 99%

Every Low Boolean Algebra is Isomorphic to a Recursive One

Downey¹,

Jockusch²

1994

Proceedings of the American Mathematical Society

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Abstract. It is shown that every (countable) Boolean algebra with a presentation of low Turing degree is isomorphic to a recursive Boolean algebra. This contrasts with a result of Feiner (1967) that there is a Boolean algebra with a presentation of degree <0' which is not isomorphic to a recursive Boolean algebra. It is also shown that for each n there is a finitely axiomatizable theory T" such that every low« model of T" is isomorphic to a recursive structure but there is a low"+1 model of T" which is not isomorphic to any recursive structure. In addition, we show that n + 2 is the Turing ordinal of the same theory T" , where, very roughly, the Turing ordinal of a theory describes the number of jumps needed to recover nontrivial information from models of the theory. These are the first known examples of theories with Turing ordinal a for 3 < a < w.

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The Work of Anil Nerode: A Retrospective

Remmel

Crossley

1993

Logical Methods

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Stability of recursive structures in arithmetical degrees

Cited by 48 publications

References 1 publication

Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees

Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees

Every Low Boolean Algebra is Isomorphic to a Recursive One

The Work of Anil Nerode: A Retrospective

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