Conservations of First-Order Reflections

Abstract

The set theory KPΠN+1 for ΠN+1-reflecting universes is shown to be ΠN+1-conservative over iterations of ΠN -recursively Mahlo operations for each N ≥ 2.

Mahlo classes for first-order reflections

“…There is an A ∈ Γ such that A ≃ (A ι ) ι∈J , and for any ι ∈ J, there is an α(ι) such that α(ι) < α and P (ι) ⊢ Q) holds, where for sentences δ, δ (Q) denotes the result of restricting each unbounded quantifier ∃x, ∀x Proof. As in [2] we see by induction on rank(a) < K that P (a) ⊢…”

“…As in [2] KPω + (V ∈ M h k+1 ( α)) is embedded to an infinitary one-sided sequent calculus with inference rules (M h k+1 ( α, β)). In one-sided sequent calculi, formulas are generated from atomic formulas and their negations a ∈ b, a ∈ b by propositional connectives ∨, ∧ and quantifiers ∃, ∀.…”

“…By induction on ordinals a < ε K+1 (the next epsilon number above K) we see that M k+1 ⊂ a<ε K+1 M k (a; < ε ). In [2] it is shown that a set theory KPΠ N for Π N -reflecting universes is Π N -conservative over the theory…”

Proof Theory of Weak Compactness

“…As observed in [2,5], ordinal analyses of Π N +1 -reflection yield a prooftheoretic reduction of Π N +1 -reflection in terms of iterations of Π N -recursively Mahlo operations. Specifically we show the following Theorem 1.4 in [8]. Let KPω denote the Kripke-Platek set theory with the axiom of Infinity, Π N (a) a universal Π N -formula, and RM N (X ) the Π N -recursively Mahlo operation for classes of transitive sets X :…”

A simplified ordinal analysis of first-order reflection

“…It is not hard for us to show that the assumption that the universe is Π nreflecting is proof-theoretically reducible to iterabilities of the lower operation rM n−1 (and Mostowski collapsings), cf. [3].…”