Download e-book for kindle: A Theory of Sets by Morse Anthony P. By Morse Anthony P.

ISBN-10: 1114312681

ISBN-13: 9781114312685

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Additional resources for A Theory of Sets

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Since we accept formulas of this sort as theorems we should use care in making them. Presumably only by mistake would someone fashion a n axiom which is not a formula. 25 DETACHMENT. If a theorem is obtained from ‘ ( p + q) ’ by replacing ‘ p ’ by a theorem and ‘q’ by a formula T, then Tis a theorem. 26 SUBSTITUTION. 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.

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. 22 RULE. If A , B, and C are formulas with A different from B and B different from C, and ifB 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.

Thus among our theorems are : ‘ (St Z X fJ U’XfJ ‘ (St 2 X +tJ U’XtJ I VXvtJ(2 = ( X VXVfJ(Z I = (X tJ) A U’XtJ)) ’, +fJ) A U’XtJ))’. 59 AGREEMENTS. O A is a 1 stencil if and only if A can be obtained by replacing ‘ E’ by an expression of class 0, ‘V ’ by an expression of class 1 or 2 or 3, and ‘A’ by an expression of class 4 in any one of the expressions: ‘ (Ep ; qr = Ez st zp ( q A r ) )’, ‘ (Vp ; qr = V t ;st ztq st ztr) ’, ‘ (Apr 3 Ap ; (x = x ) r) ’. 1 A is a 2 stencil if and only if A can be obtained by replacing ‘A’ by a n expression of class 4 in any one of the expressions : ‘ (Ap ; qr = As ; ((PI A ‘ (Apr = As ; (b)r) ’.