No abstract

A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA 0 , those in second order set theory extending NBG (as well as those in n+3-th order number theory extending the so-called Bernays Gödel expansion of full n+2-order number theory etc.) are. In this article, we establish the equivalence between ∆ -FP). Our proof also shows the equivalence between ID 1 and ID 1 , both of which are defined in the standard way but with the starting theory PA replaced by ZFC (or full n+2-th order number theory with global well-ordering).Keywords subsystems of Morse-Kelley set theory · von Neumann-Bernays-Gödel set theory · higher order number theory · well-foundedness · proof theoretic strength Mathematics Subject Classification (2000) (Primary) 03F35 · (Secondary) 03B15 · 03D65 · 03E70 · 03F25

MSC:primary 03F25 03F35 secondary 03B30 03E70 Keywords:Proof-theoretic ordinal Axiom of extensionality Intensional set theory a b s t r a c t 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 + ∆ 0 -Sep| = ε 0 and |Basic + ∆ 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 ∆ 0 -Sep.

We highlight that the connection of well-foundedness and recursive definitions is more than just convenience. While the consequences of making well-foundedness a sufficient condition for the existence of hierarchies (of various complexity) have been extensively studied, we point out that (if parameters are allowed) well-foundedness is a necessary condition for the existence of hierarchies e.g. that even in an intuitionistic setting (Π 0 1 -CA 0 ) α wf(α) where (Π 0 1 -CA 0 ) α stands for the iteration of Π 0 1 comprehension (with parameters) along some ordinal α and wf(α) stands for the well-foundedness of α.