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Some contributions to definability theory for languages with generalized quantifiers

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin  and
Douglas E. Miller
John T. Baldwin
Affiliation:
University of Illinoisat Chicago Circle, Chicago, Illinois 60680
Douglas E. Miller
Affiliation:
University of Illinoisat Chicago Circle, Chicago, Illinois 60680
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Extract

One of the first results in model theory [12] asserts that a first-order sentence is preserved in extensions if and only if it is equivalent to an existential sentence.

In the first section of this paper, we analyze a natural program for extending this result to a class of languages extending first-order logic, notably including L(Q) and L(aa), respectively the languages with the quantifiers “there exist un-countably many” and “for almost all countable subsets”.

In the second section we answer a question of Bruce [3] by showing that this program cannot resolve the question for L(Q). We also consider whether the natural class of “generalized Σ-sentences” in L(Q) characterizes the class of sentences preserved in extensions, refuting the relativized version but leaving the unrestricted question open.

In the third section we show that the analogous class of L(aa)-sentences preserved in extensions does not include (up to elementary equivalence) all such sentences. This particular candidate class was nominated, rather tentatively, by Bruce [3].

In the fourth section we show that under rather general conditions, if L is a countably compact extension of first-order logic and T is an ℵ1-categorical first-order theory, then L is trivial relative to T.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 3 , September 1982 , pp. 572 - 586
Copyright
Copyright © Association for Symbolic Logic 1982

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References

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