## Choice and uniformity in weak applicative theories (2005)

Venue: | Logic Colloquium ’01 |

Citations: | 11 - 0 self |

### BibTeX

@INPROCEEDINGS{Cantini05choiceand,

author = {Andrea Cantini},

title = {Choice and uniformity in weak applicative theories},

booktitle = {Logic Colloquium ’01},

year = {2005}

}

### OpenURL

### Abstract

Abstract. We are concerned with first order theories of operations, based on combinatory logic and extended with the type W of binary words. The theories include forms of “positive ” and “bounded ” induction on W and naturally characterize primitive recursive and polytime functions (respectively). We prove that the recursive content of the theories under investigation (i.e. the associated class of provably total functions on W) is invariant under addition of 1. an axiom of choice for operations and a uniformity principle, restricted to positive conditions; 2. a (form of) self-referential truth, providing a fixed point theorem for predicates. As to the proof methods, we apply a kind of internal forcing semantics, non-standard variants of realizability and cut-elimination. §1. Introduction. In this paper, we deal with theories of abstract computable operations, underlying the so-called explicit mathematics, introduced by Feferman in the midseventies as a logical frame to formalize Bishop’s style constructive mathematics ([18], [19]). Following a common usage, these theories