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By Anthony P. Morse
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Extra info for A Theory of Sets
7 Introduction We agree that D Z@s A if and only if A is a formative expression and D is a definition which can be obtained from ‘(x = y)’ by replacing ‘x’ by A and ‘y’ by a parenthetical expression. We agree that D raises A if and only if A is a formative expression and D is a definition which can be obtained from ‘(x E y)’ by replacing ‘x’ by A and ‘y’ by a parenthetical expression different from A . Thus our definition ‘((P A 4 ) = - (P + 4))’ both lifts and raises ‘ ( p A 4)’. Our definition ‘((x + x’) = (x + x’))’ lifts but does not raise ‘(x -+ x’)’.
Taken together these last two sections of Chapter 0 are independent of the Theory of Notation and make quite explicit the notion of formula and the concept of theorem. Because of the importance we attach to definitions we formulate in Appendix A the rules we follow in making them. Earlier, S. LeSniewski worked painstakingly along these lines. We shall employ the terminology ‘point’ and ‘set’ instead of the more usual terminology ‘set’ and ‘class’. We feel justified in this since to us a point is a set capable of belonging to a set.
21 RULE. A variable is accepted in a form if and only if it occurs therein less than twice. Thus a variable is accepted in a form, as opposed to a formula, if and only if the variable is free in the form. On the other hand, a variable is indicial in a form if and only if it is not free therein. 22 RULE. If A, B, and C are formulas with A different from B and B different from C, and if B can be obtained from A by replacing a free and accepted variable of A by C , then a is accepted in B if and only if a is accepted in A , and a is indicial in B if and only if a is indicial in A and does not appear in C.