ONT Re: Set Theory
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Note 3
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| Elementary Set Theory
|
| The Classification Axiom Scheme
|
| Equality is always used in the sense of logical identity;
| "1 + 1 = 2" is to mean that "1 + 1" and "2" are names of
| the same object. Besides the usual axioms for equality an
| unrestricted substitution rule is assumed: in particular the
| result of changing a theorem by replacing an object by its equal
| is again a theorem.
|
| There are two primitive (undefined) constants besides "=" and the other
| logical constants. The first of these is "in" [epsilon], which is read
| "is a member of" or "belongs to". The second constant is denoted, rather
| strangely, "{·· : ···}" and is read "the class if all ·· such that ···".
| It is the 'classifier'.
|
| A remark on the use of the term "class" may clarify matters. The term
| does not appear in any axiom, definition, or theorem, but the primary
| interpretation † of these statements is as assertions about classes
| (aggregates, collections). Consequently the term "class" is used
| in the discussion to suggest this interpretation.
|
| Lower case Latin letters are (logical) variables. The difference between a
| constant and a variable lies entirely in the substitution rules. For example,
| the result of replacing a variable in a theorem by another variable which does
| not occur in the theorem is again a theorem, but there is no such substitution
| rule for constants.
|
| I. Axiom of Extent. ‡
|
| For each x and each y it is true that x = y if and only
| if for each z, z is in x when and only when z is in y.
|
| Thus two classes are identical iff only every member of each is a member of the
| other. We shall frequently omit "for each x" or "for each y" in the statement of
| a theorem or definition. If a variable, for example "x", occurs and is not preceded
| by "for each x" or "for some x" it is understood that "for each x" is to be prefixed
| to the theorem or definition in question.
|
| † Presumably other interpretations are also possible.
|
| ‡ One is tempted to make this the definition of equality, thus
| dispensing with one axiom and with all logical presuppositions
| about equality. This is perfectly feasible. However, there would
| be no unlimited substitution rule for equality and one would have
| to assume as an axiom: If x is in z and y = x, then y is in z.
|
| JLK, Gen Top, pages 251-252.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤