Truth definitions without exponentiation and the Σ₁ collection scheme

We prove that: • If there is a model of I∆0 + ¬exp with cofinal Σ1-definable elements and a Σ1 truth definition for Σ1 sentences, then I∆0 + ¬exp + ¬BΣ1 is consistent, • there is a model of I∆0 + Ω1 + ¬exp with cofinal Σ1-definable elements, both a Σ2 and a Π2 truth definition for Σ1 sentences, and for each n ≥ 2, a Σn truth definition for Σn sentences. The latter result is obtained by constructing a model with a recursive truth-preserving translation of Σ1 sentences into boolean combinations of ∃Σ0 sentences. We also present an old but previously unpublished proof of the consistency of I∆0 + ¬exp + ¬BΣ1 under the assumption that the size parameter in Lessan’s ∆0 universal formula is optimal. We then discuss a possible reason why proving the consistency of I∆0 +¬exp+ ¬BΣ1 unconditionally has turned out to be so difficult. ∗Mathematical Institute of the Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland, [email protected]. Partially supported by grant N N201 382234 of the Polish Ministry of Science and Higher Education. †Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland, [email protected]. Partially supported by grant N N201 382234 of the Polish Ministry of Science and Higher Education. ‡School of Mathematics, University of Manchester, Manchester M13 9PL, U.K., [email protected].