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On the MP axiom of intensionality: connecting types and sets

This paper should be regarded as a sequel to our 2008 ASL Winter Meeting: we will summarize the most important theorems of the non-standard set theory MP, developed from Russell's assumptions in his Introduction to Mathematical Philosophy, and our main focus will be the clarification of the most notable feature of MP-the axiom of intensionality. We will show that stratification is not needed if we adopt a few simple propositions introducing several type-theoretical notions in this non-standard set theory. this should be regarded as a necessary step in the demonstration of the advantages of MP over ZFC, which does not discriminate the concepts of `set' and `predicate' as it is needed in other areas of logic. There is also a very important philosophical point to be made: Although technically a second order theory, MP constitutes a more ontologically innocent approach than ZFC, providing the same results while presuming significantly less.

Types; Sets; Second order set theory

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