A Theory of Sets by Anthony P. Morse

By Anthony P. Morse

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Metamathematics is made up of statements about expressions. Axioms, theorems, and definitions belong to mathematics. Rules and agreements belong to metamathematics. 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.

Our schemators are ‘uI, ‘vI, ‘wI, y, ‘v,7, ‘wl,, ‘Un’, ‘V”’, ‘w”’, _ _ _ ‘UNI,)‘”,,,,) ‘W,n’, ,, etc. - - LU - I,,, - Inference starts with dejinitions and axioms. Each definition and each axiom will be an expression explicitly described or explicitly introduced by an appropriate marginal label. The scope of such a marginal label will end just before the next heading, aside, or marginal notation. In addition, we have so constructed the definitions themselves that no definition can be an initial segment of a different definition.

From our definition ‘((P A 4 ) = - (P + 4))’ we learn that ‘ ( p A 4)’ is a definiendum and that ‘(x A y)’ is a form. However, ‘(x A x)’ is not a form. From our definitions ‘ ( V X ~E X- A X - 2 ~ ) ’ and ‘( VXEX’ vx - we learn that ‘ V y not a form. 11y’ and = v x ; (XEX’) vx)’ ‘ V t E y l t ’ are forms. However, ‘ V x EX’is 8 0. Language and Inference Forms are to be read as a whole. The individual constants are usually incidental. The expression ‘ ( p + q)’ has nothing to do with limits and the expression ‘(cx +A as x + a)’ has nothing to do with implication.

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