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L(Q)-preservation theorems

Published online by Cambridge University Press:  12 March 2014

Jörg Flum
Jörg Flum*
Affiliation:
Mathematisches Institut der Albert-Ludwigs-Universität, 78 Freiburg, Germany
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Extract

Much effort has been spent to prove that the reduced product operation preserves, and sometimes even strengthens the L-equivalence of structures, where L is some infinitary language. A similar result is suggested by the following well-known fact:

Assume D is a nonprincipal ultrafilter on ω and, for n Є ω, Cn is a set. If the ultra-product ΠωCn/D is infinite, it has a cardinality ≥ Hence, by Łos' theorem, (i) if and φ(x) is a first-order formula, then iff where Q is the unary quantifier “there are many.”

We shall prove some generalizations of (i). In particular, we show

(ii) if D is a nonprincipal ultrafilter over I = ω, and then where L(Q) is the language obtained from the first-order language by adding the quantifier Q.

(ii) remains true, if D is an ω-regular or an atomless filter over a set I.

Lipner [7] proved that if is regular, then the L(Q)-equivalence is preserved under direct products. We show that the assumption “ is regular” is necessary.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 3 , September 1975 , pp. 410 - 418
Copyright
Copyright © Association for Symbolic Logic 1975

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References

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