Some Results on Combinators in The System TRC

We investigate the system TRC of untyped illative combinatory logic that is equiconsistent with New Foundations. We prove that various unstratified combinators do not exist in TRC. Introduction. We prove some results in the axiomatic system TRC introduced in [3]. The system TRC (for ‘type-respecting combinators’) is an untyped system of combinatory logic, in the sense of [1], [2]. TRC is a first order theory of functions (combinators) with equality and is illative, i.e. capable of expressing notions of propositional logic. Moreover, it is combinatorially complete for stratified combinators. The main interest of TRC is that it is equiconsistent with the theory NF [6], Quine’s ‘New Foundations’. As the consistency of NF remains an open problem, so does the consistency of TRC. The objects of study of a combinatory logic are combinators. We denote xy the application of the combinator x to the combinator y, and adopt the convention that xyz = (xy)z. The language of TRC has (in addition to equality and the binary function xy) constants Abst, Eq, p1 and p2, and functions k(x) and 〈x, y〉. The axioms of TRC are the following: I. k(x)y = x. II. pi〈x1, x2〉 = xi for i = 1, 2. III. 〈p1x, p2x〉 = x. IV. 〈x, y〉z = 〈xy, xz〉 V. Abst x y z = x k(y)(yz). 1991 Mathematics Subject Classification. 03B40.