1
STRICT REVERSE MATHEMATICS
Draft
by
Harvey M. Friedman
Department of Mathematics
The Ohio State University
www.math.ohiostate.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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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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 Fall '08
 JOSHUA
 Math, Set Theory, Natural number, Finite set, RCA0

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