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 Cantor's original proof [2] is intuitionistically correct, and Brouwer's 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 Fraïssé, and it is briefly shown that one half of the Fraïssé theorem holds intuitionistically.
