Some models for intuitionistic finite type arithmetic with fan functional

Abstract

In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional (HAω+ MUC). The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+as defined there cannot be shown to have the required properties inEL+ QF-AC, the reason being that a change in the definition ofW12alone does not suffice—if one wishes to establish closure under the operations of HAωthe definitions ofW1σfor other σ have to be adopted as well. It is difficult to see how to do this directly in a uniform way — but we succeed via a detour, which is described in §2.For a proper understanding, we should perhaps note already here thaton the assumption of the fan theorem, ECF+as defined in [T4] and the new model of this note coincide (since then the definition ofW12[T4, p. 594] is equivalent to the definition forW12in the case of ECF); but inELit is impossible to prove this (and under assumption of Church's thesis the two models differ).