By Morse Anthony P.
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Extra resources for A Theory of Sets
5 Our expression ofclass 5 is: ‘1’. 6 Our expression of class 6 is : ‘Al’. 51 AGREEMENT. A symbol is a notarian if and only if it is the initial symbol of some expression of some class. 52 A G R E E M E N T . A is a march if and only if some parade in which no binariate is adjacent to a binariate can be obtained from ‘ (p) ’ by replacing ‘p’ by A . Thus of the three expressions ‘xx’ EX“’, ‘XI 3x , and ‘x E X ’ c c x ” 3 x”” only the third is a march. I t will turn out that a march is never a formula.
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. 16 RULE. 3 if A is a prefix, B is a n expression, whose initial symbol is a, and Q is ( A B ) ,then Q is a formula. Thus ‘A(x -+ x)y’ is not a formula since otherwise we could learn that ‘ t’ is both free and not free in ‘At,’.
I n this connection we should like to point out that no symbol is ever fixed by an expression in which ‘ = ’ does nut appear. 70 A G R E E M E N T . 4 C is obtained from ‘Axy’ by replacing ‘ x ’ by t and ‘y’ by A . 71 A G R E E M E N T . S is a string if and only if S can be obtained from one of the expressions G X > , GXXI,,
A Theory of Sets by Morse Anthony P.