Logic Colloquium '78, Proceedings of the colloquium held in by Maurice Boffa, Dirkvan Dalen and Kenneth Mcaloon (Eds.)

By Maurice Boffa, Dirkvan Dalen and Kenneth Mcaloon (Eds.)

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Then let LZ'] an Rng(b') by defining z'=w' iff be the equivalence class of 2' under this re- lation; [ z ' ] can be formed using A -separation. Let S2 be the set of all [='I CONTINUITY I N INTUITIONISTIC SET THEORIES for z' i n R n g ( b ' ) (which e x i s t s by a b s t r a c t i o n ) and l e t S n i t e s u b s e t s of S 2 41 b e t h e s e t of a l l f i - ( u s i n g e x p o n e n t i a t i o n , u n i o n , and a b s t r a c t i o n ) . 0 s e t of a l l f u n c t i o n s from a ' t o S { < q , < w ', z ' > / q > : Ea' 1 be t h e 1 V x y [ z j c f ( q , w ' / q ) &[XI€ f ( q , w ' / q ) & V Z ' Let S I f f i s in S .

H a such that Then the functions be defined by substituted in, the conclusion of the dependent choices axiom is true and realized: that is, is true ind realized. First we show \ J n 6 w P(f(n),f(n+l)) is true and realized. The truth follows from the last clause in the formula we have to use the definition contains Now, w* P(f(n),f(n+l)) i s a pair then (s) = is true and realized; Fcn(f) -t y with and = w . Suppose p n nC w . y = w = f(n) w = y and w* Dom(f) = ui is realized since if y* = w* , we have y = h(n) w = ; and 'j,,> c.

We now t u r n t o t h e n o n - l o g i c a l axioms, b e g i n n i n g w i t h t h e axiom cCX, which has t h e f o l l o w i n g form when w r i t t e n o u t : v n t w g m € w ( < n , m > E a& m M n ) & V n , m ( < n , m > d a& < n , r > c a + m = r ) t/x€a]n,mEw(x=) & V n , m ( p (a ( n ) , a ( m ) )< l/(n+l)+l/(m+l)) where p i s some r e c u r s i v e f u n c t i o n and M F i r s t we show pIl-nEw+ i s s o m e r e c u r s i v e sequence ( s e e 8 2 ) . $ ? l l - v n ~ w 3 m ~ w ( < n , m >&E m E a &m

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