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

Abstract

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 Basic (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. | Basic | = ω ω and | Basic + Ext | = ε 0 , 2. | Basic + Δ 0 - Sep | = ε 0 and | Basic + Δ 0 - Sep + Ext | = Γ 0 . We also show that neither Reg nor AC affects the proof-theoretic strength, i.e., | T | = | T + Reg | = | T + AC | = | T + Reg + AC | where T is Basic plus any combination of Ext and Δ 0 - Sep .