Pure Mathematics

A Theory of Sets by Morse Anthony P.

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By Morse Anthony P.

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1, or a symbol fixed by the first formula in which it appears among those listed in the chapters which follow. 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,,

75. This rule is independent of the intervening Theory of Notation. Examples. By detachment we learn that if ‘ ( A x ( x +x ) + ( x +x ) ) ’ is a theorem and if ‘Ax(% + x ) ’ is a theorem, then ‘ ( x -+ x ) ’ is a theorem. By substitution we learn that if is a theorem then ‘((Y+t) +Ax(y+t))’ ‘((Y+Y) +AX(Y+Y))’ 14 0. Language and Inference is a theorem. However, replacing ‘ t ’ or ‘y’ by ‘ x ’ is not allowed by substitution. By schematic substitution we learn that if ‘ (Ax ux -+ yx) ’ is a theorem then ‘ (Ax(x -+ x ) -+ (x -+ x)) ’ is also a theorem.

22 0. Language and Inference If A is the expression ‘(x u u u x’)’ then the complicate of A is A . 49 DEFINITIONAL SCHEMA. We accept as a definition each expression which can be obtained from ‘ ( ( X *E 9) (x E 9 + A ZZ)) ’ by replacing ‘E’by a nexus different from ‘m’. 57. 50 AGREEMENTS. 1 ‘One’, ‘The’. Our expressions of class 1 are: ‘A’, ‘A’, ‘far R’, ‘large’, ‘small’, ‘big’, ‘alm # Mcp , ‘alm q’, ‘Alm ‘p’, ‘A1 cpB. 3 ‘Ad’, ‘C’. Our expressions of class 3 are: ‘sup’, ‘inf’, ‘ad’, ‘osc’. 4 Our expressions of class 4 are those expressions which are of either class 0 or class 1 or class 2 or class 3.

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