February 11, 2018

A Theory of Sets - download pdf or read online

By Morse Anthony P.

ISBN-10: 1114312681

ISBN-13: 9781114312685

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By replacing variables by formulas. 2 To illustrate the flexibility of our theory of notation we have included among our binarians and notarians a good many symbols seldom used in elementary set theory. 0 . Language and Inference 28 I n other words, a string is a linear arrangement or concatenation of formulas. 9 it can be seen that if S is a string and T is a n expression then ( S T )is a string if and only if T is a string. Also if S is a string which is not a formula then there are a unique string T and a unique formula A such that S is ( TA).

Suppose F is a form which is neither a schematic form nor a variable. Variables free in F are of course accepted in F and all other variables are indicial in F. Now i f F is obtained from F by simultaneously replacing free variables which appear in F by formulas and schematically replacing schematic expressions which appear in F by Indicia1 and Accepted Variables 11 formulas then : a variable is indicial in F if and only if it is indicial in F and does not appear in any of the formulas replacing free variables, and a variable is accepted in F‘ if and only if it is accepted in F and does not appear in any of the formulas schematically replacing schematic expressions.

IfAisaparade thenweaccept as a definition the expression obtained from ‘ ( x ~ y ) ’by replacing ‘ x ’ by A and ‘y’ by the complicate of A. 39 we learn that ‘((X C X‘ = X I 3 X ” ) ( ( X c X’ =X”) A (X” 3 X ” ) ) ) ’ is a definition. If A is the rather weird expression ‘(x + X‘X” -+- x “ < n u x””)’ then the bisegments of A are ‘ +’, and ‘ < n u ’; ‘ +’, and the parentheses are of prime importance in A; ‘ x ” and ‘ x ” ’ are left in A ; ‘ x ” ’ and ‘x””’ are right in A; and the complicate of A is ‘+a’, ‘+a’, ‘ ( x -+ (x‘x”) -P (x“ < n u x””)) ’.

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A Theory of Sets by Morse Anthony P.

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