A Transfinite Type Theory with Type Variables by P. B Andrews

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General comments on Q Various extensions of the system Q are of course possible. Among the simplest of these are systems obtained by adding additional axioms to the list of axioms for Q. For example, it will for certain purposes be desirable to add one or more of the following as axioms or axiom schemata: (11) Poaxa 2 ~oa[~a(oa)~oal* (1 1') 3ia(oa, *VPoa *3xaPoaXa 3 ~ o a [ i a ( o a ) ~ o a l . yz = xu. (12') VaVx4pO2x4= V Y ~ P O Z Y ~ . [$;A01 3 ISL,AoII 3 VCAO, provided this is a wff, a and b are distinct, and all free occurrences of c in A .

Note that no trouble arises when A . Qoaay,] y, were derivable in Q . I. Qoaaya] ya would also be derivable, so by Rule R and the axiom Ja(oa)[Qoaay,]2 C, (an instance of 4,) we would obtain C, & y, as a theorem of Q'. It is clear that Theorem 7 applies to Q' with respect to any principal interpretation of Q, in particular with respect to a principal interpretation in which the domain 5Dl of the model (and hence each domain) contains at least two elements. However for such an interpretation WJC, 3 ya] is not always t, since we can always choose x so that W,ya is not the same as W,C,.

3. 157 t-VaAo = Vb[StAol,provided that (1) VaA, is a wff; (2) 6 does not occur free in A,,; (3) all free occurrences of a in A. are free for 6. 4 by Rule P. 3. %Ao = VaBo,provided this is a wff. 2 151 151. 6 by Rule P. , xn are distinct primitive variables; (2) for each i (n 2 i 2 l), no xi-variable other than xi, occurs free in A, or B,. The proof is by induction on n. For n = 1, the theorem has already 1. been proved as Theorem 159. 1, 159. --, a", xpl, nrm is a complete list of distinct type variables and wff-variables which are free in A , or B, but for which the occurrence of A , in Cois not free in Co.

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