Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-14T17:16:23.953Z Has data issue: false hasContentIssue false
  • English
  • Français

On Post's canonical systems

Published online by Cambridge University Press:  12 March 2014

Raymond M. Smullyan
Raymond M. Smullyan*
Affiliation:
Princeton University
Get access Rights & Permissions [Opens in a new window]

Extract

We assume familiarity with the canonical languages of Post (cf. [1], [2]).

A set S of strings is said to be representable in a canonical system (F) if there is a string π such that for every string X, X ∈ S if and only if πX is provable in (F).

Suppose that K is a finite alphabet containing at least 2 symbols, and that W is a set of strings in (the symbols of) K. If W is representable in some canonical system, is it necessarily representable in a canonical system which uses only the symbols of K ? We answer this question affirmatively. Thus, e.g., it is possible to construct a 2-sign universal system over its own alphabet.2

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 27 , Issue 1 , March 1962 , pp. 55 - 57
Copyright
Copyright © Association for Symbolic Logic 1962

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]Post, E., Formal reductions of the general combinatorial decision problem, American journal of mathematics, vol. 65 (1943), pp. 197–215.CrossRefGoogle Scholar
[2]Rosenbloom, P. C., The elements of mathematical logic, New York (Dover), 1950, Chapter IV.Google Scholar
[3]Smullyan, R. M., Theory of formal systems, Doctoral Dissertation, Princeton05 1959; also issued as M. I. T. Lincoln Laboratory group report 54–5, April 1959. Accepted for publication by the Princeton University Press as an Annals of Mathematics Study.Google Scholar
[4]Myhill, J. R., Three contributions to recursive function theory, Actes du XI ème Congrès International de Philosophie, vol. 14, Volume complémentaire et communications du Colloque de Logique, Amsterdam (North-Holland) and Louvain (E. Nauwelaerts), 1953, pp. 50–59.Google Scholar