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Note on a problem of L. Henkin

Published online by Cambridge University Press:  12 March 2014

Abraham Robinson
Abraham Robinson*
Affiliation:
University of Toronto
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Extract

A statement X in the lower predicate calculus is said to be persistent with respect to the set of statements K ([2], [3]), if whenever X holds in a model M of K then X holds also in all extensions of M which are models of K. If X is persistent with respect to the empty set, then it may be said to be absolutely persistent.

A statement X is called existential, if it is in prenex normal form and does not contain any universal quantifiers. This includes the possibility that X does not contain any quantifiers at all.

Let E be the class of all existential statements. Then it is not difficult to see that E is quasi-disjunctive. That is to say, given statements Y1, Y2 in E, there exists a statement Y in E such that

is provable.

L. Henkin [1] has raised the question how to characterise the statements X which are persistent with respect to a given set K (e.g. a set of axioms for a field or a group) by a syntactical condition. He has shown that, in order that a statement X be absolutely persistent, it is necessary and sufficient that there exist a statement Y ϵ E such that

is provable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 21 , Issue 1 , March 1956 , pp. 33 - 35
Copyright
Copyright © Association for Symbolic Logic 1956

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References

REFERENCES

[1]Henkin, L., Two concepts from the theory of models, this Journal, vol. 21 (1956), pp. 28–32.Google Scholar
[2]Robinson, A., On the metamathematics of algebra, Studies in logic and the foundations of mathematics, Amsterdam, North Holland Pub. Co., 1951, IX + 195 pp.Google Scholar
[3]Robinson, A., On axiomatic systems which possess finite models, Methodos, 1951, pp. 140–149.Google Scholar
[4]Tarski, A., Contributions to the theory of models I, II, Indagationes Mathematicae, vol. 16 (1954), pp. 572–581, 582–588.CrossRefGoogle Scholar