By Morse Anthony P.

ISBN-10: 1114312681

ISBN-13: 9781114312685

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Extra resources for A Theory of Sets

Sample text

The only reason for making the remainder of the definitions is to lift certain important forms. 3 ORIENTING DEFINITIONS. 6 ( y x = vx) . 17 (_u "xx'x"x =-u Mxx'x''x ") etc. 111 Through rules and agreements we shall try to make clear when a given variable isfree, indicial, or accepted in an expression. * We shall also try to make clear just what expressions are formulas, and we shall give rules of inference for establishing theorems. Theorems, of course, are of particular interest to us. Our rules of inference enable us, step by step, to use theorems already known to us to discover new theorems.

28, replacing the words ‘ is accepted ’ by ‘ does not appear ’. T o assess the consequences of this proposed simplification let us adopt the unordered summation notation ‘ C x E A ux’. - 0. Language and Inference 12 I t is then natural to expect that (EnE 4 n = 6 = E m E 4 m). Also it seems clear that ‘ E x E y x’ is a formula. Inasmuch as ‘ (Y AxY) ’ is a formula in which ‘y’ is free it seems inescapable that -+ ‘ ( x -+ A x x ) ’ is a formula, but not, we hope, a theorem. Thus, since ‘y’ is free in ‘ Z x E y x’ we feel compelled to admit that ‘Ex Ex x’ is also a formula.

A formula is fundamental if and only if it is either a simple prefix or an expression of the kind (Act) where A is a simple prefix and ct is the initial symbol of A . 15 RULE. I f F is a strict formula devoid of schemators, and A is a form obtained from F by replacing variables by schematic expressions, then: F is a simple formula, every variable which appears in A also appears in F, a is free in F if and only if ci is free in A , and A can be obtained from F by replacing variables which do not appear in A by schematic expressions.