Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T18:53:57.015Z Has data issue: false hasContentIssue false
  • English
  • Français

Axiomatizable theories with few axiomatizable extensions

Published online by Cambridge University Press:  12 March 2014

D. A. Martin  and
M. B. Pour-El
D. A. Martin
Affiliation:
Rockefeller University
M. B. Pour-El
Affiliation:
University of Minnesota
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we prove two theorems. They answer questions raised by Myhill in 1956. (We recall the well-known fact that Myhill's invention of the maximal set in 1956 [2] stemmed from his attempt to prove I below.)

I. There exists an axiomatizable, essentially undecidable theory in standard formalization such that all axiomatizable extensions of are finite extensions.

II. There exists an axiomatizable but undecidable theory in standard formalization such that

(a) has a consistent, complete, decidable extension ,

(b) If is an axiomatizable extension of then either

(i) is a finite extension of , or

(ii) is a finite extension of .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 2 , June 1970 , pp. 205 - 209
Copyright
Copyright © Association for Symbolic Logic 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Janiczak, A., Undecidability of some simple formalized theories, Fundamenta Mathematicae, vol. 40 (1953), pp. 131–139.CrossRefGoogle Scholar
[2]Myhill, J., Problem 9, this Journal, vol. 21 (1956).Google Scholar