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A new omitting types theorem for L(Q)

Published online by Cambridge University Press:  12 March 2014

Matt Kaufmann
Matt Kaufmann*
Affiliation:
Universtiy of Wisconsin, Madison, Wisconsin 53706 Purdue University, W. Lafayette, Indiana 47907
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For L a countable first-order language, let L(Q) be logic with the quantifier Qx which means “there exist uncountably many x”. We assume a little familiarity with Keisler's paper [8]. One finds there completeness and compactness theorems for L(Q), as well as an omitting types theorem: a syntactic condition is given for a consistent countable theory to have a model satisfying ∀x⋁Σ(x), where Σ is a countable set of formulas of L(Q). (See also Chang and Keisler [3] for the first-order omitting types theorem, due to Henkin and Orey.) An analogous theorem is proved in Barwise, Kaufmann, and Makkai [1] and in Kaufmann [6] for stationary logic. However, a more general theorem than just an anlaogue to Keisler's is proved there. Conditions are given which are sufficient for a theory T to have models satisfying sentences such as aas1aas2 … aasn⋁Σ(s1, … sn), ∀xaas ∨ Σ(x, s), and so forth. Bruce [2] had asked whether such a theorem can be proved for L(Q). with “aa” replaced by “Q*”, where Q* is ¬Q¬ (“for all but countably many”).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 4 , December 1979 , pp. 507 - 521
Copyright
Copyright © Association for Symbolic Logic 1979

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References

REFERENCES

[1]Barwise, J., Kaufmann, M. and Makkai, M., Stationary logic, Annals of Mathematical Logic, vol. 13 (1978), pp. 171–224.CrossRefGoogle Scholar
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