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

Abstract

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.

Proof-theoretic conservations of weak weak intuitionistic constructive set theories

“…(see above informal description) as well as other, more familiar principles (see [19]) which are less important from proof theoretic viewpoint.…”

“…To address these questions we recall Sato's basic results Theorem 1. (See [19].) |Basic + Ext| = ε 0 , |Basic + Ext + Δ 0 -Sep| = Γ 0 .…”

“…It is thus natural to investigate proof-theoretic strengths of Sato's weak weak intuitionistic constructive set theories and following [19] ask whether they are conservative extensions of the underlying arithmetical intuitionistic formalisms. To address these questions we recall Sato's basic results Theorem 1.…”

“…The proof techniques are the same as those used in [13,14]. In the sequel we adopt basic notations and abbreviations used in [19] and often use superscript (i) to emphasize that underlying formal system is based on the intuitionistic logic.…”

“…Working in classical logic K. Sato [19] introduced several weak refinements of basic weak set theoryboth intensional and extensional -and determined their proof-theoretic ordinals. Notably Sato's weak weak classical set theories are less expressive than Myhill-Friedman's weak intuitionistic formalisms that can (arguably) mimic constructive essence of entire ZF.…”