Foundational Studies, Selected Works Vol II by Andrzej Mostowski

By Andrzej Mostowski

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53. (@)) other than X," belongs to Fr(o*(p,,(a))). Hence from the remark ~k}. Hence setting (6) y(x:, x:, -**, xk) "sb:r l;j , . , Xi: respectively are free, becomes bound at any of these places. ;)& ... (a)). ,)& Bit(X,";xi;, X;;) & ... & Bt,(Xt; X;, A$) ... ',,, (9) E T for each arrangement of numbers r ' , . ,... , Xi;,. 63 the expression (9) then gives ( 10) Bk(X,4 ; X:, x;,,) & Ni (X,* ; Xj ;) & .. ,X i ) ] E T . Let us denote ( 1 1) 8 D! , X k ) ] . From (10) and ( I 1) by an easy calculation we get 6&NiI(X,4;A'ji)& ...

Pn(6) = (EX,‘, 1) {S ( X +3 2 ) [c(x:+ 2 ;xn+ 1 ) -b 2111 (2) for n = I , 2, ... 52 there exists a number m such that (3) p,(6) E So. for n 2 m . 2+,), S(X,2+2), cw:+z; - c+L ) E So. ”+i) -+ S(X,Z+2)1}E T . +1) {e(X:+ I ) & S ( X + I ) & (Xi’+ 2 ) [C(x,2+ 2 ;X i + 1) S(xn2+2)1) E T. 1 states that the right-hand side of this equivalence is the negation of a thesis. Thus the left-hand side must also be the negation of a thesis which means that o(pn(6)) E T which by (1) proves the lemma. 5. THEOREM.

In the preceding paragraphs we have given a concise description of the language which is the object of consideration in the later part of this Paper. We could not give all the theorems and definitions which have been introduced for the study of this language, the reader can find them in Tarski's Papers [2] and [3]. 1. ) (cf. Giidel [I], Def. 33, p. 184). The i-th increase in type of a propositional function a we will denote ai. 2. LEMMA. If a E T and i is a natural number then aiE T . The proof proceeds by induction on the rank of consequences of the set L.

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