The Axiom of Choice and Combinatory Logic (preliminary Draft)

We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice. §1. The problem. It is well-known that combinatory logic is inconsistent with the choice schema ACV for operations: (∀x)(∃y)A(x, y)→ (∃f)(∀x)A(x, fx) (1) where A is an arbitrary formula. Indeed, by classical logic, if K is the constantfunction combinator, (∀x)(x = K ∨ ¬x = K) (2) Since K 6= S (otherwise the underlying combinatory algebra would be trivial), one can conclude with a standard argument by cases that (∀x)(∃y)(x 6= y) (3) An application of ACV 3 to (3) immediately provides an operation f without fixed points, against the well-known (purely constructive) fixed point theorem (∀f)(∃x)(fx = x) It is also known that (2), and hence classical logic, is essential for getting the contradiction. In fact, if CL+Ext is the equational theory of combinatory logic with extensionality, it is shown by Barendregt in [2]: 1991 Mathematics Subject Classification. 03B40, 03B55, 03F50, 03F25, 03F05. Research supported by MIUR and Università di Firenze. Presented at the AILA-SILFS Scuola Estiva di Logica, Cesena 23-27 Settembre 2002. Thanks are due to AILA, SILFS and to the Director of the School, prof. Silvio Ghilardi, for the invitation. 1According to Barendregt [2], the observation goes back to Scott’s lecture at LMPS’71 in Bucharest. 2Of course K and S stand for the basic constants of combinatory logic CL; see [3], ch. 7 or section 2 below. 3Since (2) implies (∀x)(∃!y)((x = K ∧ y = S) ∨ (x 6= K ∧ y = K)). the inconsistency could also be derived with the weaker axiom of unique choice AC!: (∀x)(∃!y)A(x, y)→ (∃f)(∀x)A(x, fx) For additional results of Friedman and Feferman on AC! in the domain of partial applicative theories, see [4, p. 228].