D. van Dalen

The Use of Kripke's Schema as a Reduction Principle

The comprehension principle of second order arithmetic asserts the existence of certain species (sets) corresponding to properties of natural numbers. In the intuitionistic theory of sequences of natural numbers there is an analoguous principle, implicit in Brouwers writing and explicitly stated by Kripke, which asserts the existence of certain sequences corresponding to statements. The justification of this principle, Kripkes Schema , makes use of the concept of the so-called creative subject . For more information the reader is referred to Troelstra [5]. Kripkes Schema reads There is a weaker version The idea to reduce species to sequences via Kripkes schema occurred several years ago (cf. [2, p. 128], [5, p. 104]). In [1] the reduction technique was applied in the construction of a model for HAS . On second thought, however, I realized that there is a straightforward, simpler way to exploit Kripkes schema, avoiding models altogether. The idea to present this material separately was forced on the author by C. Smorynski. Consider a second order arithmetic with both species and sequence variables. By KS we have (for convenience we restrict ourselves in KS to 0-1-sequences). An application of AC-NF gives Of course ξ is not uniquely determined. This is the key to the reduction of full second order arithmetic, or HAS , to a theory of sequences. We now introduce a translation τ to eliminate species variables. It is no restriction to suppose that the formulae contain only the sequence variables ξ 1 , ξ 3 , ξ 5 , … Note that by virtue of the definition of τ the axiom of extensionality is automatically verified after translation. The translation τ eliminates the species variables and leaves formulae without species variables invariant.

The Continuum and First-Order Intuitionistic Logic

Ever since Cantor, we have known that the reals and the rationals are not isomorphic (as equality structures, i.e., sets). Logically speaking, however, they are not all that different; in first-order classical logic they are elementarily equivalent, since the theory of infinite sets is complete. The same holds for ℝ and ℚ as ordered sets; again the theory of dense linear order without end points is complete. From an intuitionistic point of view these matters are more complicated; e.g., the theory of equality of ℚ is decidable, whereas the one of ℝ patently is not. This, in a roundabout way, shows that ℚ and ℝ are not isomorphic; of course, there is no need for such a detour, as Cantors original proof [2] is intuitionistically correct, and Brouwers new proof [1] is another alternative intuitionistic argument. In view of the fact that ℚ and ℝ behave so strikingly differently with respect to first-order logic, one is easily tempted to look for elementary equivalences among the subsets of ℝ. Until quite recently most model theoretic investigations of intuitionistic theories made use of special (artificial) notions of “model”, e.g., Kripke models, sheaf models,…; but there is no prima facie reason why one should not practice model theory much the same way as traditional model theorists do. That is to say on the basis of a naive set theory, or, in our case, of naive intuitionistic mathematics. This paper uses the method of ( k, p )-isomorphisms of Fraisse, and it is briefly shown that one half of the Fraisse theorem holds intuitionistically.