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Extra resources for A Transfinite Type Theory with Type Variables

Example text

SYA, ,provided that 2 > y and a11 free occurrences of u in A . are free for all type variables in y. 2. 139 If 2 I- A . then 2 I- Sd0,provided that (1) 2 > y and all free occurrences of a in A. are free for all type variables in y; 111 BASIC LOGIC IN Q 45 (2) u is not free in X ; (3) for each wff-variable xp such that xp occurs free in A. and u occurs in p, no x-variable occurs free in a member of H and no xvariable other than xp occurs free in A o . , xi, be a list of all wff-variables such that xLi occurs free in A .

We shall often refer to 2 as a set of hypotheses or prernisses. We shall write 2 I- A , as an abbreviation for the phrase “ A , is derivable from the set 2 of hypotheses”. We define this concept inductively as follows : (1) If Ho is a member of 2,then 2 F H , . - A , . X? X? and an X-variable occurs free in A , or B,, the occurrence of A , in Co is free for all X-variables. X? unless its being so follows from the above rules. By a proof from the hypotheses 2 we shall mean a pair of finite sequences of wffs of type 0, such that every member of the first sequence is either an axiom of Q or is inferred from previous members of the sequence by Rule R (so every member of the first sequence is a theorem of Q), and such that every member of the second sequence is either a member of &‘ or a member of the first sequence or is inferred from previous members of the sequence according to clause (3) above.

For a complete description of the model ‘B’and the interpretation of Q in W, simply replace the sentences “We first choose three distinct objects t, f and r which are not functions. ” in our treatment of Vt on page 9 by the sentences “We first choose four distinct objects t, f, u and r which are not functions. ” Reading through the rest of Section 3 with the new interpretation of Q in mind, we see that Lemmas 1-4 and 6 still apply without change, while Lemma 5 must be modified as follows: ((W, 3 ooo)z))l) = t if is f or and )l) are both t; otherwise ((W, =ooo)~))l) = f.