Truth definitions without exponentiation and the Σ₁ collection scheme

Abstract

We prove that:

• if there is a model of IΔ₀ + ¬exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then IΔ₀ + ¬exp + ¬ BΣ₁ is consistent,

• there is a model of IΔ₀ + Ω₁ + ¬exp with cofinal Σ₁-definable elements, both a Σ₂ and a Π₂ truth definition for Σ₁ sentences, and for each n ≥ 2, a Σ_n truth definition for Σ_n sentences.

The latter result is obtained by constructing a model with a recursive truth-preserving translation of Σ₁ sentences into boolean combinations of ∃Σ^b₀ sentences.

We also present an old but previously unpublished proof of the consistency of IΔ₀ + ¬exp + ¬ BΣ₁ under the assumption that the size parameter in Lessan's Δ₀ universal formula is optimal. We then discuss a possible reason why proving the consistency of IΔ₀ + ¬exp + ¬ BΣ₁ unconditionally has turned out to be so difficult.