By Morse Anthony P.
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Additional resources for A Theory of Sets
To give a rough idea of the roles played by schematic expressions, free variables, indicial variables, and accepted variables, we say that in a theorem a free variable is replaceable by a wide variety of formulas, a schematic expression is replaceable by a still wider variety of formulas, and an indicial variable, such as an index of summation or a dummy variable of integration, is replaceable by accepted variables. 1 RULE. If a is free in A then a is a variable and A is an expression. 2 R U L E .
1 We accept as a definition each expression which can be obtained from a 1 stencil by replacing ‘p’ by a verbless march of order 3, ‘ t ’ by ‘ x , X I ,x”, ‘ q ’ by ‘ ~ “ x x ‘ x ”and ) , ‘r’ by ‘~“xx’x’’). etc. 61 DEFINITIONAL SCHEMAS. O We accept as a definition each expression which can be obtained from a 2 stencil by replacing p by a verbal march M of order 1, ‘5’ by a subject of M , ‘ q ’ by ‘ux), and ‘ r ’ by ‘vx’. 1 We accept as a definition each expression which can be obtained from a 2 stencil by replacing p by a verbal march M of order 2, ‘s’ byasubjectofM, ‘q’by‘u’xx”,and‘r’by‘v‘xx”.
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. Formalization describes with care an explicit process for arriving at theorems. Our rules are to be taken for granted although some of them can be derived from others. Our rules are akin to, but different from, axioms and theorems.