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StrictRM012305

# StrictRM012305 - 1 STRICT REVERSE MATHEMATICS Draft by...

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1 STRICT REVERSE MATHEMATICS Draft by Harvey M. Friedman Department of Mathematics The Ohio State University www.math.ohio-state.edu/~friedman/ January 31, 2005 NOTE: This is an expanded version of my lecture at the special session on reverse mathematics, delivered at the Special Session on Reverse Mathematics held at the Atlanta AMS meeting, on January 6, 2005. TABLE OF CONTENTS 1. Introduction. 2. Underlying logic of SRM. 3. Conservative and exact extensions, coding. 4. Inevitability of logical strength. 5. A strictly mathematical exact conservative extension of RCA0. 6. Strictly mathematical exact conservative extensions of WKL 0 , ACA 0 , P 1 1 -CA 0 . 7. SRM over FSZ, FSZE, FSZEXP. 8. Finite set theory for SRM. 9. Weak infinite set theory for countable SRM. 10. 3RM - third order arithmetic comprehension as a base theory for RM. 11. 3RM - third order recursive comprehension as a base theory for RM. 12. Comprehensive conservative extensions of PA that supports coding free RM - the system ALPO. 13. Some exact conservative extensions of RCA 0 . 14. General SRM, and the interpretability conjecture. 1. INTRODUCTION. Reverse Mathematics (RM) is now a well developed area of mathematical logic based on a simple robust setting using the rather sparse (two sorted) language of second order arithmetic with standard model (N,P(N);<,=,0,1,+,•, ). [Si99] serves as the standard text for RM. Unfortunately, it is currently out of print.

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2 As a consequence of the sparse language, coding is needed to state almost every mathematical statement considered. Fortunately, a great variety of mathematical statements can be treated with a well controlled system of unproblematic coding mechanisms. Most of these coding mechanisms were already in extensive use in recursion theory. The coding mechanisms can be considered to be problematic in the context of analysis, as the objects are treated as uncountable in ordinary mathematics. They are problematic in terms of the overarching goal of RM - to analyze the logical structure of mathematics. Nevertheless, even in analysis, coding mechanisms are in place that support a robust development with a virtually unending supply of attractive problems, independently of wider foundational issues. By Strict Reverse Mathematics (SRM), we mean a form of RM relying on no coding mechanisms, where every statement considered must be strictly mathematical. In particular, there is no base theory. We will certainly want to single out certain basic sets of statements and work over them, but these statements must themselves be strictly mathematical. Around the time we founded RM with [Fr74] and [Fr76], we privately circulated the manuscripts [Fr75a], [Fr75b], and [Fr76a], whose aim was to found SRM. We did not publish these manuscripts because we rightly felt that they did not present a robust program the way [Fr74] and [Fr76] did, and thought it was best to focus attention on the development of RM.
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StrictRM012305 - 1 STRICT REVERSE MATHEMATICS Draft by...

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