ONT Re: Set Theory
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Note 4
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| Elementary Set Theory
|
| The Classification Axiom Scheme (cont.)
|
| The first definition assigns a special name to those classes which
| are themselves members of classes. The reason for this dichotomy
| among classes is discussed a little later.
|
| 1. Definition. x is a 'set' iff for some y, x in y.
|
| The next task is to describe the use of the classifier. The first blank
| in the classifier constant is to be occupied by a variable and the second
| by a formula, for example {x : x in y}. We accept as an axiom the statement:
| u in {x : x in y} iff u is a set and u in y. More generally, each statement
| of the following form is supposed to be an axiom:
|
| u in {x : ··· x ···} iff u is a set and ··· u ···.
|
| Here "··· x ···" is supposed to be a formula and "··· u ···" is supposed to be the
| formula which is obtained from it by replacing every occurrence of "x" by "u".
| Thus u in {x : x in y and z in x} iff u is a set and u in y and z in u.
|
| This axiom scheme is precisely the usual intuitive construction of classes except
| for the requirement "u is a set". This requirement is very evidently unnatural
| and is intuitively quite undesirable. However, without it a contradiction may
| be constructed simply on the basis of the axiom of extent. (See theorem 39
| and the discussion preceding it.) This complication, which necessitates a
| good bit of technical work on the existence of sets, is simply the price
| paid to avoid obvious inconsistencies. Less obvious inconsistencies
| may very possibly remain.
|
| JLK, Gen Top, page 252.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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