By P. B Andrews
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Extra info for A Transfinite Type Theory with Type Variables
However if W,f,, = b and W,gll = g, then 11 W,[lfil = g,,] = VxlLfilxl A gllxl]] is f, so this instance of Axiom Schema 3 cannot be derived in that system. Thus Axiom Schema 3 is independent. Of course one might still ask whether certain instances of the Axiom Schema are derivable from the other axioms, or from the other axioms and other instances of this Axiom Schema. However for the present we shall concern ourselves only with the question of whether certain axiom schemata could be dispensed with altogether.
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. We simply use Rule R and Theorems 100, 130, 131, 132, 133 and 134. 0 Now suppose A . is a tautology without free variables. If I- A . = F,, then by Theorem 7 we see that W,[Ao 2 FO] = t for every assignment x, so WnAo = WnFo= f, contradicting our assumption that A. is a tauto0 logy. Hence we must have I- [Ao = T o ],so by Rule T (116), t- A o .
The proof is essentially the same as the proof of 120; simply replace x p by xp, and use 121 in place of Axiom 3. 123 k P p A o ] = [Vyp[SG;Ao]], provided that (1) no x-variable other than xp is free in Ao; (2) no y-variable is free in Ao; (3) all free occurrences of xb in A . are free for y e . The proof is immediate from 100,122, and the definition of VxBAo. gooxo, provided g is distinct from x. The proof is similar to the proof of 121, with Axiom 1 being used in place of Axiom 3. 40 BASIC LOGIC IN Q 125 Rule of Conjunction (Rule C): If &' I- A .