Recursive expansions

C. Ash; J. Knight

Fundamenta Mathematicae (1994)

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

Abstract

top
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.

How to cite

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.