The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations

Abstract

The primary purpose of this work is to analyze the constructive interpretations of classical theorems concerning the continuity of effective operations. These theorems are the “recursivizations” of the statements that all functions (on various spaces) are continuous. To discuss the question whether these statements are constructively valid, it would be necessary to analyze the notion of “constructively valid,” which is beyond the scope of this paper. We now proceed to describe our results. We deal with two kinds of effective operations: (i) partial operations on indices of partial functions, and (ii) total operations on indices of total functions. The two principal theorems of the subject assert that each such operation is continuous; for (i) this is due to Myhill-Shepherdson [7] and called MS here; for (ii) it is due to Kreisel, Lacombe, and Shoenfield [5] and called KLS. (Explicit formulations of these and other principles mentioned in the introduction will be given in §1 below.)Several interpretations (of, e.g., Heyting's arithmetic HA) in the literature satisfy Markov's principle MP, and for these MS and KLS are valid, since a close inspection of the classical proofs of MS and KLS shows that they are derivable in HA + MP. But we also find an interpretation in the literature under which MS and KLS are valid while MP is not (hence MS and KLS do not imply MP in systems for which this interpretation is sound). This work is in §2.