# Recursive expansions

Fundamenta Mathematicae (1994)

• Volume: 145, Issue: 2, page 153-169
• ISSN: 0016-2736

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## Abstract

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Let A be a recursive structure, and let ψ be a recursive infinitary ${\Pi }_{2}$ sentence involving a new relation symbol. The main result of the paper gives syntactical conditions which are necessary and sufficient for every recursive copy of A to have a recursive expansion to a model of ψ, provided A satisfies certain decidability conditions. The decidability conditions involve a notion of rank. The main result is applied to prove some earlier results of Metakides-Nerode and Goncharov. In these applications, the ranks turn out to be low, but there are examples in which the rank takes arbitrary recursive ordinal values.

## How to cite

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Ash, C., and Knight, J.. "Recursive expansions." Fundamenta Mathematicae 145.2 (1994): 153-169. <http://eudml.org/doc/212040>.

@article{Ash1994,
abstract = {Let A be a recursive structure, and let ψ be a recursive infinitary $\{Π\}_2$ sentence involving a new relation symbol. The main result of the paper gives syntactical conditions which are necessary and sufficient for every recursive copy of A to have a recursive expansion to a model of ψ, provided A satisfies certain decidability conditions. The decidability conditions involve a notion of rank. The main result is applied to prove some earlier results of Metakides-Nerode and Goncharov. In these applications, the ranks turn out to be low, but there are examples in which the rank takes arbitrary recursive ordinal values.},
author = {Ash, C., Knight, J.},
journal = {Fundamenta Mathematicae},
keywords = {recursive structure; recursive expansion; decidability conditions; rank},
language = {eng},
number = {2},
pages = {153-169},
title = {Recursive expansions},
url = {http://eudml.org/doc/212040},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Ash, C.
AU - Knight, J.
TI - Recursive expansions
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 2
SP - 153
EP - 169
AB - Let A be a recursive structure, and let ψ be a recursive infinitary ${Π}_2$ sentence involving a new relation symbol. The main result of the paper gives syntactical conditions which are necessary and sufficient for every recursive copy of A to have a recursive expansion to a model of ψ, provided A satisfies certain decidability conditions. The decidability conditions involve a notion of rank. The main result is applied to prove some earlier results of Metakides-Nerode and Goncharov. In these applications, the ranks turn out to be low, but there are examples in which the rank takes arbitrary recursive ordinal values.
LA - eng
KW - recursive structure; recursive expansion; decidability conditions; rank
UR - http://eudml.org/doc/212040
ER -

## References

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1. [AJK] C. J. Ash, C. G. Jockusch and J. F. Knight, Jumps of orderings, Trans. Amer. Math. Soc. 319 (1990), 573-599. Zbl0705.03022
2. [AK] C. J. Ash and J. F. Knight, Relatively recursive expansions, Fund. Math. 140 (1992), 137-155.
3. [AKS] C. J. Ash, J. F. Knight and T. Slaman, Relatively recursive expansions II, ibid. 142 (1993), 147-161. Zbl0809.03024
4. [G] S. S. Goncharov, Autostability and computable families of constructivizations, Algebra i Logika 14 (1975), 647-680 (in Russian); English transl.: Algebra and Logic 14 (1975), 392-409.
5. [MN] G. Metakides and A. Nerode, Effective content of field theory, Ann. of Math. Logic 17 (1979), 289-320. Zbl0469.03028
6. [V] A. Vlach, Hyperarithmetical relations in expansions of recursive structures, Ph.D. thesis, Univ. of Notre Dame, 1993. Zbl0793.03039

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