# Download Foundational Studies, Selected Works Vol II by Andrzej Mostowski PDF

By Andrzej Mostowski

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**Sample text**

531 we obtain (nw) (2) Let us take for short + e ( y )E T . 2 gives (4) A ( Y , q) -+ e ( y ) T. y)[ecy) (qey -+ + FEY)] & A ( Y , q) & q e +~ F E Y € T . However, one s$es from (3) that A(Y, q) -+ EYE T and A(Y, q) & FEY -+ q1dE E T . From ( 5 ) we thus deduce that { ( Y ) [ e ( Y ) ( ~ E Y 5eY)lJ8~A ( Y , q) -+ q l d t -+ + E T which, by the remark that (EY)A(Y,q) E T results from (3), gives ( Y ) [ e ( Y )+ (qeY -+ EEY)] q/dF E T . 12, and so the equivalence qIdE (Y)[O(Y)4 (qeY FEY)]E T -+ holds.

If X t # X;,Xi,Xi then a) Bi(Xt; X i , X i ) & B~(x:;X:, xi) + x j l d ~E; T; b) Bi(X2; X i , Xi)& Bi(X2; Xi, Xi) + XildX;' E T . These four lemmas which one proves by induction on i together express the deducibility from the axioms of logic of the following proposition: for each function f mapping the set of all individuals one-one onto itself and for each set p (of arbitrary type) there is a uniquely determined f-image of ,u and a uniquely determined finverse image of p. 667. LEMMA. )& S(X:)] -, S(Xkz) E T.

O,(pn(B)) E N(W. From the above discussion we infer that from the correctness of the expression (1) for a number k 2 1 follows its correctness for k 1. Hence by induction we deduce the correctness of the expression (1) for arbitrary k 2 1. d. ) we may confine ourselves to proving the corresponding condition for functions of the set M+L. About the functions in M we can say nothing until M is more narrowly prescribed. However, we can give conditions which imposed on the sequence 8 ensure that the property under consideration holds for functions in L.