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Towards a consistent set-theory

Published online by Cambridge University Press:  12 March 2014

John Myhill*
Affiliation:
Temple University

Extract

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.

It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1951

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References

1 Myhill, J., Report on some investigations concerning the consistency of the axiom of redacibilily, this Journal, vol. 16 (1951), pp. 3542Google Scholar.

2 Tarski, A., Einführung in die mathematische Logik, Vienna 1937, Ch. VIIICrossRefGoogle Scholar.

3 Bourbaki, N., Foundation of mathematics for the working mathematician, this Journal, vol. 14 (1949), pp. 18; especially pp. 7–8Google Scholar.

4 Quine, W. V., Mathematical logic, New York 1940, p. 88Google Scholar.

5 J. Myhill, loc. cit.

6 J. Myhill, loc. cit.