# A Theory of Sets by Morse Anthony P.

By Morse Anthony P.

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A not unusual sort 'of parade is ' ( X c X' c X") ,. A less common sort is '(xu~'nnx"-tx"c~""3XF")~. Our theory of notation and subsequent mathematical definitions will make possible a unique interpretation of the two parades just mentioned as well as a host of others. 34 AGREEMENT. A is of power n if and only if A is a nexus in which some symbol of type n appears and no symbol of type less than n appears. For example, are of power 6. 35 D E F I N I T I O N A L SCHEMA. We accept as a definition each expression which can be obtained by replacing ' A ' by an expression of odd power in any one of the expressions: '((X A X' A X " ) SZ ( ( X A X ' ) A X " ) ) ', ' ( ( X A X ' A X " A X m ) E ( ( X h X ' A X " ) AXm))', etc.

If A is the expression ' ( x = x' E x"x " 3 XI'" u g " p )' then the complicate of A is ' ( x = x' E (X"X") 3 (X"" u x""x"")) '. 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.

If a is free in A then a is a variable and A is an expression. 2 R U L E . A variable is free in a form if and only if it occurs therein less than twice. 3 A G R E E M E N T . in A . A is a formula if and only if some variable is free Roughly speaking, a variable is free in a formula if and only if every occurrence is a free occurrence. 8 0. 4 RULE. If A is a formula, Cis a formula, B is different from A and is obtained from A either by replacing some free variable of A by C or by schematically replacing some schematic expression by C, then a variable is free in B if and only if it is free in both A and C.