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Some models for intuitionistic finite type arithmetic with fan functional
Published online by Cambridge University Press: 12 March 2014
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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 in EL + QF-AC, the reason being that a change in the definition of W12 alone does not suffice—if one wishes to establish closure under the operations of HAω the definitions of W1σ for 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 that on the assumption of the fan theorem, ECF+ as defined in [T4] and the new model of this note coincide (since then the definition of W12 [T4, p. 594] is equivalent to the definition for W12 in the case of ECF); but in EL it is impossible to prove this (and under assumption of Church's thesis the two models differ).
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- Copyright © Association for Symbolic Logic 1977
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