On the scheme of induction for bounded arithmetic formulas

Reflection principles and provability algebras in formal arithmetic

“…§4.1 below and [43,94]). In fact, modulo EA the axiom of EA + is equivalent to the cut-elimination theorem.…”

A note on the interpretability logic of finitely axiomatized theories

“…Wilkie and Paris [7] show that IA0 + ~1 is a completely adequate theory for arithmetizing syntax. E.g., if T is a theory satisfying the assumptions made above, we can formalize in IA0 + ~1 (as an R+-formula) ProofT(x,y), which represents the relation 'z is a proof of the formula y from T'.…”

“…We assume that the theories T we consider are given by an R+-formula aT(Z) having just ~ free plus the relevant information on what the set of natural numbers of T is; aT gives the set of codes of non-logical axioms of the theory (cf. [7]). We also assume that the numbers of T satisfy IA0 + ~1, and that T is finitely axiomatized and sequential.…”

“…Two last remarks: we assume that the reader is familiar with the discussion of systems and arithmetization in [7]; he or she is also advised to keep a copy of Visser's [6] at hand. 1.…”

“…To be able to do so, we only consider arithmetical theories that satisfy a number of conditions to be given now. (Details about the notions used below may be looked up in [2], [5], [6] and [7].) Officially we will be working in a relational version of the language of arithmetic, in which successor, addition and multiplication are (2-, 3-and 3-place) relation symbols.…”