By P. B Andrews
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Extra info for A Transfinite Type Theory with Type Variables
If one wishes not only to formalize the semantics of finite type theory in Q, but to prove in Q the consistency of a formulation of finite type theory which has an Axiom of Infinity, then one must evidently add to Q an Axiom of Infinity for individuals of type 1. ) Alternatively, one might in some appropriate way extend Q to higher transfinite types, and then interpret the individuals of the finite type theory in %I2 rather than in %I1 when formalizingthe semantics of the finite type theory. The following lemmas about wffs will be useful: - L a m 9: There are no type symbols a,p and y in K : such that CL > @ and p > y.
118 If 2 I- Bo then 2 I- Si; Bo, provided that (1) y 2 tc and all free occurrences of xu in Bo are free for all free variables of A,; (2) xy is the only x-variable free in Bo; (3) no x-variable occurs free in 2. l. 119 Rule of Simultaneous Substitution (Rule Sub) : 1 If A? x" ... , (2) yi 2 aiand all free occurrences of xii in Bo are free for all free variables of A:, ; (3) no &variable other than x+ioccurs free in B,; (4) no xi-variable occurs free in a member of X . , A:, or in members of 2".
8. , is free in [VbE,] or [VbEb] but Note that each of that the occurrence of [VbE,] in C, is not free in C, for any of these xp", (where k 2 0) be a complete k variables. ,yyk, xi,=', , list of those distinct variables which occur free in [VbE,] or PbEbJ but for which the occurrence of [WE,] is not free in Co. 9 54 [I1 BASIC LOGIC IN Q V U " V X ~-.. 11 I- VU' 2 v$;' -*- . ) vxym[Aa = B,] *.. [VbEOl = W'bEbl vyy, 1 --. -,or yk-variable can occur free in Vul - * . 13 I- Vyt, - 0 . Co = DO by inductive hypothesis, since Dois obtained from Coby replacing the occurrence of [VbE,] by an occurrence of [VbEb], and this occurrence of pibEO] is free in Cofor all type variables.
A Transfinite Type Theory with Type Variables by P. B Andrews