By P. B Andrews
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Additional resources for A Transfinite Type Theory with Type Variables
2 If W,[xa = yb] = t, then from the definition of W,Qozzit is clear that WG,, = w,yb . Now W,xa is in BWna ,where W,a = 7ca is in K'. Similarly w,y, is in %W,b, where W,b = xb is in K'. The domains By,where y is in K', are disjoint, so evidently W,a = W,b. Hence W,[a = b] = t, so W,Po = t by Lemma 5. This completes the proof of Theorem 7. THEOREM 8: The system Q is consistent. Proof: We first remark that assignments do indeed exist"). Indeed, we can describe certain assignments explicitly. For example, we can describe an 'I) Of course by using the Axiom of Choice in our meta-languagewe can easily find assignments.
VxaAo 2 Bo, provided this is a wff. Proof: . 1, 146. VuAo 3 B,, provided this is a wff. The proof is similar to the proof of 148, with 147 used in place of 146. VxaA, 3 VxaBo,provided this is a wff. Vx,Ao 3 141. Vx,B, Theorem 150 is then obtained by Rule R. VuAo 3 VuB,, provided this is a wff. Theproof is similar to the proof of 150, with 144, 149 and 142 used in place of 143, 148 and 141. 3 B, then i%? I- VxaA, 3 VxaB,, provided that this is a wff and no x-variable occurs free in #. 152 If X' I- A, Proofi Use Rule Gen (117), 150, and Rule MP (128).
To show this in detail one would of course make use of Lemmas 2-5 also. ) It is also clear that there is an effective method for determining of any wff of type 0 whether or not it is tautologous. 135 If A . is a tautology, then I- A o . The proof is by induction on the number of wff-variables which have free occurrences in A o . (Of course all such variables must be of type 0 since A. ) We easily see by induction on the length of Bo that if Bo is any pro0 positional wff in which no variables are free, I- Bo = To or I- Bo g Fo.
A Transfinite Type Theory with Type Variables by P. B Andrews