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**1 - 5**of**5**### www.math.ohio-state.edu/~friedman/ TABLE OF CONTENTS

"... this paper as reaxiomatizations of set theory. A vital feature of the standard set theories associated with the axiomatizations presented here is that they are missing the axiom of choice. This is an essential feature. For instance, ZF does not prove the existence of a standard model of each theorem ..."

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this paper as reaxiomatizations of set theory. A vital feature of the standard set theories associated with the axiomatizations presented here is that they are missing the axiom of choice. This is an essential feature. For instance, ZF does not prove the existence of a standard model of each theorem of ZFC; in fact, ZF does not prove the existence of a standard model of Zermelo set theory with the axiom of choice. Thus in this paper, we relate our axiomatizations to extensions of ZF by large cardinal axioms. In each case, we have chosen an appropriate version of the large cardinal axiom so that if ZF is replaced by ZFC then the resulting system is equivalent to a system which is familiar in the set theory literature. But one would like to know the relationship between the system with ZF and the system with ZFC. This relationship cannot be gauged by considering standard models. {PAGE } The normal way of gauging this relationship is through

### 1 STRICT REVERSE MATHEMATICS Draft

, 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 ..."

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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