ONT Re: Set Theory
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Note 5
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| Elementary Set Theory
|
| The Classification Axiom Scheme (cont.)
|
| A precise statement of the classification axiom scheme
| requires a description of formulae. It is agreed that: †
|
| a. The result of replacing "!a!" and "!b!"
| by variables is, for each of the following,
| a formula.
|
| !a! = !b!
|
| !a! in !b!
|
| b. The result of replacing "!a!" and "!b!" by variables
| and "A" and "B" by formulae is, for each of the following,
| a formula.
|
| if A, then B
|
| A iff B
|
| it is false that A
|
| A and B
|
| A or B
|
| for every !a!, A
|
| for some !a!, A
|
| !b! in {!a! : A}
|
| {!a! : A} in !b!
|
| {!a! : A} in {!b! : B}
|
| Formulae are constructed recursively, beginning with
| the primitive formulae of (a) and proceeding via the
| constructions permitted by (b).
|
| II. Classification Axiom-Scheme.
|
| An axiom results if in the following "!a!" and "!b!"
|
| are replaced by variables, "A" by a formula $A$,
|
| and "B" by the formula obtained from $A$
|
| by replacing each occurrence
|
| of the variable which replaced !a!
|
| by the variable which replaced !b!:
|
| For each !b!, !b! in {!a! : A} if and only if !b! is a set and B.
|
| † This circuitous sort of language is unfortunately necessary.
| Using the convention of quotation marks for names, for example,
| "Boston" is the name of Boston, if $A$ is a formula and $B$ is
| a formula then "$A$ => $B$" is not a formula. For example, if
| $A$ is "x = y" and $B$ is "y = z", then '"x = y" => "y = z"'
| is not a formula. Formulae (for example "x = y") contain no
| quotation marks. Instead of "$A$ => $B$" we want to discuss
| the result of replacing "!a!" by $A$ and "!b!" by $B$ in
| "!a! => !b!". This sort of circumlocution can be avoided by
| using Quine's corner convention.
|
| JLK, Gen Top, page 253.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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