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Minimal ultrafilters and maximal endomorphic universes. (English) Zbl 0587.03040

Some classical techniques concerning ultrafilters on the set of natural numbers and their ultrapowers (especially with regard to Rudin-Keisler ordering and minimal elements of it) are adapted to the alternative set theory. There is a new and quite nice observation in the paper that if A is a maximal endomorphic universe with standard extension then \(=^{\circ}_{A} = Id\) (where Id denotes the identity and x \(=^{\circ}_{A} y\) iff for every set formula \(\phi\) (t) with parameters in A the equivalence \(\phi\) (x)\(\equiv \phi (y)\) holds). This result is in contrast to the countable case when the following holds: If B is countable and \(B=Def(B)\) (B is closed on definitions by set formulas) then \(B=\{x; \{x\}=(=^{\circ}_{B})''\{x\}\}\).
Reviewer: K.Čuda

MSC:

03E70 Nonclassical and second-order set theories
03C20 Ultraproducts and related constructions
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