A Theory of Sets by Morse Anthony P.

By Morse Anthony P.

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If T is a theorem in which b is free and A is such a formula that each variable in it is free in T, then the expression obtained from T by replacing b by A is also a theorem. 27 SCHEMATIC SUBSTITUTION. If T is a theorem, S is a schematic expression, and A is such a formula that each variable in it is either free in T o r occurs explicitly in S, and T is a formula obtained from T by schematically replacing S by A , then T‘is a theorem. 28 I NDlClAL SUBSTITUTION. Ifq is free in Q, Tis a theorem obtained from Q by replacing q by a formula A in which CY is indicial, B is obtained from A by replacing cc by a variable which is accepted in A , and finally T is obtained from Q by replacing q by B, then T is a theorem.

A is aparade if and only ifA is such an expression in which some binarian appears that A can be obtained from one of the expressions ' ( X X ' ) ), ' ( X X ' X " ) ), ' (X#'X"X") ... ), by replacing each biniariate a which is different from ' x ' by some expression which either is a itself or is of the kind (ca)where c is a nexus. A not unusual sort 'of parade is ' ( X c X' c X") ,. A less common sort is '(xu~'nnx"-tx"c~""3XF")~. Our theory of notation and subsequent mathematical definitions will make possible a unique interpretation of the two parades just mentioned as well as a host of others.

48 we were primarily interested in parenthetical simplification. 62 we have been primarily interested in notational uniformity as well as brevity. 64 we shall again be interested primarily only in parenthetical simplification. 63 DEFINITIONAL SCHEMA. We accept as a definition each expression which can be obtained from “IPI = I(P)O’ by replacing I p by a march and ‘ ’ by an expression of class 5. 26 0. Language and Inference For example ‘(Ix + x’ + X”I = I ( x + x’ + x ” ) I ) ’ is among our definitions.