Unfolding Feasible Arithmetic andWeak Truth

Abstract

In this paper we continue Feferman’s unfolding program initiated in (Feferman, vol. 6 of Lecture Notes in Logic, 1996) which uses the concept of the unfolding of a schematic system in order to describe those operations, predicates and principles concerning them, which are implicit in the acceptance of. The program has been carried through for a schematic system of non-finitist arithmetic in Feferman and Strahm (Ann Pure Appl Log, 104(1–3):75–96, 2000) and for a system (with and without Bar rule) in Feferman and Strahm (Rev Symb Log, 3(4):665–689, 2010). The present contribution elucidates the concept of unfolding for a basic schematic system of feasible arithmetic. Apart from the operational unfolding, we study two full unfolding notions, namely the predicate unfolding and a more general truth unfolding of, the latter making use of a truth predicate added to the language of the operational unfolding. The main results obtained are that the provably convergent functions on binary words for all three unfolding systems are precisely those being computable in polynomial time. The upper bound computations make essential use of a specific theory of truth over combinatory logic, which has recently been introduced in Eberhard and Strahm (Bull Symb Log, 18(3):474–475, 2012) and Eberhard (A feasible theory of truth over combinatory logic, 2014) and whose involved proof-theoretic analysis is due to Eberhard (A feasible theory of truth over combinatory logic, 2014). The results of this paper were first announced in (Eberhard and Strahm, Bull Symb Log 18(3):474–475, 2012).