Monads of indiscernibles. (English) Zbl 0671.03033

Monads in basic equivalences being simultaneously classes of indiscernibles in the corresponding language, are investigated. (Every class of indiscernibles is a subclass of a monad but it need not be a subclass of any monad of indiscernibles.) The paper, in fact, continues other papers of the same authors devoted to basic equivalences. From the point of view of model theory, monads are classes of elements of the same type in the corresponding language. Hence they correspond to ultrafilters on the countable ring of classes definable by set-formulas (with the corresponding parameter). Thus, monads of indiscernibles are counterparts to ultrafilters with Ramsey property. The assertion that there may exist a minimal monad (in Rudin-Keisler ordering) not being a monad of indiscernibles, is, perhaps, the most interesting assertion of the paper, as it is in striking opposition to the classical result that every minimal ultrafilter on \(\omega\) is Ramsey. Unfortunately the word “no”, when printed, was changed to “a” in the assertion (Theorem 8) and hence the sense was changed to its negation. But the reader can easily find this misprint when reading the commentary before the theorem and (or) the proof.
Reviewer: K.Čuda


03E70 Nonclassical and second-order set theories
54J05 Nonstandard topology
03E05 Other combinatorial set theory
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