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On Post's canonical systems

Published online by Cambridge University Press:  12 March 2014

Raymond M. Smullyan
Raymond M. Smullyan*
Affiliation:
Princeton University
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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

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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