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Two concepts from the theory of models1

Published online by Cambridge University Press:  12 March 2014

Leon Henkin
Leon Henkin*
Affiliation:
University of Amsterdam, University of California (Berkeley)
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Extract

In this note we point out that two concepts which have been introduced separately into the theory of models are extensionally equivalent. The basic ideas can be expressed conveniently in the terminology of Tarski's theory of arithmetical classes. It will be recalled that an arithmetical class is a class consisting of all algebraic structures which satisfy some fixed sentence of the first-order functional calculus. A more general concept is that of a quasi-arithmetical class, whose elements are all algebraic structures which satisfy some fixed set of first-order sentences.

Now where H and K are any two classes of algebraic structures, Abraham Robinson has defined the relation H is persistent with respect to K to mean that (i) H∩K is non-empty, and (ii) whenever M ϵ H∩K, M′ is an extension of M, and M′ ϵ K, then M′ ϵ H. Concerning this notion Robinson has established the following theorem: If K is the class of abelian groups (or of commutative fields) and H is an arithmetical class which is persistent with respect to K, then H∩K contains a group (or field) which is finite. This leads us to the following definition.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 21 , Issue 1 , March 1956 , pp. 28 - 32
Copyright
Copyright © Association for Symbolic Logic 1956

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Footnotes

1

This paper was constructed while the author was working in The Netherlands under a Fulbright grant awarded by the U.S. government.

References

REFERENCES

[1]Tarski, Alfred, Some notions and methods on the borderline of algebra and metamathematics, Proceedings of the International Congress of Mathematicians, 1950, vol. 1 (1952), pp. 705–720.Google Scholar
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[4]Robinson, Abraham, On the metamathematics of algebra, North-Holland Publishing Company, Amsterdam, 1951.Google Scholar
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