A Topological Model for Intuitionistic Analysis with Kripke's Scheme

Va 3!bA(a, b ) 3 32 Va 3y{t(~(y)) > 0 A V x [ z ( ~ ( x ) ) > 0 3 y = x] A A(a, z(&(y)) l)]. (In the usual form of Brouwer’s Principle there is Va 3bA(a, b ) in premise.) The demand of uniqueness means that Brouwer’s Principle may be applied only to decidable properties. In [ 7 ] there is shown that the complete Brouwer’s Principle for numbers does not hold in this model. Thus the consistency of MYHILL’S system [8] was called in question. Following [3], VAN DALEN [4] constructed a topological model for intuitionistic theory of species of natural numbers, where Kripke’s Scheme was also valid, but the Principle of Uniformity was weakened (VX 3 ! x A ( X , x ) 3 3% V X A ( X , x ) instead of VX 3 x A ( X , x) 3 3x V X A ( X , x ) ) . In this paper we construct a topological model for intuitionistic theory, containing choice sequences and species of natural numbers, where Kripke’s Scheme, Brouwer’s Principle and the Principle of Uniformity are valid without restrictions. Thus we obtain a classical consistency proof of this theory. At present the author has in his disposition the proof of disjunction property and existence property for numbers for intuitionistic analysis with Kripke’s Scheme.