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By Anne S. Troelstra

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Example text

Xu_1~ < > . ,Xu+v~. We put n~m =def ~ n ' ( n , n ' n ~m ~def n ~ m ( ~, ~ are used in different partial ordering for an arbitrary between = m) & n~m o meanings finite primitive in this volume : sequences, recursive I ° ) as the natural as just defined ; 2 ° ) as a symbol well-ordering, in the discussion of the 25 principle of transfinite for "reduces to". ,Xu_1> tl(n) (n)i=x i for (n)i=o for i~u° ("tail of n" i

Prime formulae are expres- is defined as usual by two inductive clauses : Fm I) Prime formulae b e l o n g to Fm Fm 2) If (A&B), A,B ~ F m , (VxeA), then also (A--B) , (~x=A) . 7. (A V B ) , conventions. A x i o m s and rules. (a) A x i o m s a~d rules for m a n y - s o r t e d (b) A x i o m s intuitionistic predicate logic. for equality : x = x x = y & z x =y ~z =y ~ x x = z , = z , X ( ~ ) T = y ( ~ ) T . X(~)TZ ~ = y ( ~ ) T Z 0 , and the usual s~

9. Theorem Let H be a system based on intuitionistic let M(x) (Addition of defined sorts of variables). be a formula of (many-sorted) predicate logic ; Fm(~) , containing free only x, and ~ ~M(x). I, with rules for term and formula construction also extended to the new variables, with the axiom schemata and rules of H (but where in an axiom or axiom schema involving quantified variables, the axiom or axiom schema is not to be generalized by replacing quantification over the original variables by quantifiers over the new variables), and with the new axiom and schemata and rules ~, Mt ~ ( V ~ A _ ~ A t ) , Mt ~ ( ~ t ~ A ~ ) Then H' Proof.