John Myhill

Constructive Set Theory

This paper is the third in a series collectively entitled Formal systems of intuitionistic analysis . The first two are [4] and [5] in the bibliography; in them I attempted to codify Brouwers mathematical practice. In the present paper, which is independent of [4] and [5], I shall do the same for Bishops book [1]. There is a widespread current impression, due partly to Bishop himself (see [2]) and partly to Goodman and the author (see [3]) that the theory of Godel functionals, with quantifiers and choice, is the appropriate formalism for [1]. That this is not so is seen as soon as one really tries to formalize the mathematics of [1] in detail. Even so simple a matter as the definition of the partial function 1/ x on the nonzero reals is quite a headache, unless one is prepared either to distinguish nonzero reals from reals (a nonzero real being a pair consisting of a real x and an integer n with ∣ x ∣ > 1/ n ) or, to take the Dialectica interpretation seriously, by adjoining to the Godel system an axiom saying that every formula is equivalent to its Dialectica interpretation. (See [1, p. 19], [2, pp. 57–60] respectively for these two methods.) In more advanced mathematics the complexities become intolerable.

A Complete Theory of Natural, Rational, and Real Numbers

The purpose of the present paper is to construct a fragment of number theory not subject to Godel incompletability. Originally the system was designed as a metalanguage for classical mathematics (see section 10); but it now appears to the author worthwhile to present it as a mathematical system in its own right, to serve however rather as an instrument of computation than of proof. Its resources in the latter respect seem very extensive, sufficient apparently for the systematic tabulation of every function used in any but the most recondite physics. The author intends to pursue this topic in a later paper; the present one will simply present the system, along with proofs of consistency and completeness and a few metatheorems which will be used as lemmas for future research. Completeness is achieved by sacrificing the notions of negation and universal quantification customary in number-theoretic systems; the losses consequent upon this are made good in part by the use of the ancestral as a primitive idea. The general outlines of the system follow closely the pattern of Fitchs “basic logic”; however the latter system uses combinatory operators in place of the variables used in the present paper, and if variables are introduced into Fitchs system by definition their range of values will be found to be much more extensive than that of my variables. The present system K is thus a weaker form of Fitchs system. It is apparently not known whether or not Fitchs system is complete.