The strength of extensionality I — weak weak set theories with infinity

We measure, in the presence of the axiom of infinity, the proof-theoretic strength of the axioms of set theory which make the theory look really like a “theory of sets”, namely, the axiom of extensionality Ext, separation axioms and the axiom of regularity Reg (and the axiom of choice AC). We first introduce a weak weak set theory (which has the axioms of infinity and of collapsing) as a base over which to clarify the strength of these axioms. We then prove the following results about proof-theoretic ordinals: 1. and ,2. and . We also show that neither Reg nor affects the proof-theoretic strength, i.e., where T is Basic plus any combination of Ext and .