Monotone Quantifiers and Admissible Sets

Abstract

The study of generalized quatifiers, quantifiers other than “for all” and “there exist” has been taken up in the recent years by two branches of logic: model theory and recursion theory. It would seem that the admissible set theory would be one area where the two approaches could work together and it is the topic taken up here. In model theory, it has been traditional to study quatifiers based on cardinality considerations. It was in recursion theory, that the importance of just plain monotonicity emerged, and we are concerned with these quantifiers. Everything was inspired by the results from the folklore of the model theory of generalized quantifiers. One can think of most of the precise mathematical quantifiers as attempts to make one of these quatifiers precise. It is interesting to speculate whether any of these are precise enough to already list some of the commonly accepted axioms about them.