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

AbstractWe 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|=ωω and |Basic+Ext|=ε0,2.|Basic+Δ0-Sep|=ε0 and |Basic+Δ0-Sep+Ext|=Γ0. 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 Δ0-Sep