Constructive Functions in “The Foundations of Intuttionistic Mathematics”

Abstract

The formal system of the Foundations of Intutionistic Mathematics (FIM) has been criticized on the ground that its symbolism lacks a distinction between constructive functions and free choice sequences. Unquestionably, this distinction is vital for intuitionism. The fact that it wasn't symbolized is not oversight. Indeed, the author were personally led into the study of intuitionism by asking the question whether the constructive operations, functions, etc. of intuitionism can be identified with general (or partial) recursive functions. So it was a natural step to propose to identify the “laws” in Brouwer's definition of “set” or “spread” with general recursive functions. FIM does not have a separate sort of variables specifically for constructive functions. Nor does it have a prime formula C(α), for which more or less in the way of properties might be postulated, to express that α is a constructive function. However, FIM does certainly have a composite formula GR( α ) expressing that α is a general recursive function, and thus, if Church's thesis is used, a constructive function.