SUO: *Date 08 Mar 2002 -- Extensionality
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Extensionality
Nota Bene. For set-theoretic relations in this presentation,
I employ the spelling "in" for the "membership relation" and
the spelling "c" for the "contained in" or "subset relation".
I will experiment with "`A`" and "`E`" as quantifier markers,
but I also write out quantifications in the fashion that has
become customary in informal presentations on the chalkboard.
Version 1 (Skolem-Morse-Hilbert-Bernays-VonNeumann-Gödel)
| 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.
|
| ...
|
| 1. Axiom of extent. For each x and each y it is true that x = y
| if and only if for each z, z in x when and only when z in y.
|
| ...
|
| 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.
|
| 1.1. Definition. x is a set iff for some y, x in y.
|
| JLK, 'Gen Top', pages 251-252.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
Version 2 (Zermelo-Fraenkel)
| This theory is called "Zermelo-Fraenkel set theory",
| and is designated by "ZF".
|
| The only nonlogical symbol of ZF is the binary predicate symbol "in".
| We intend that the individuals of ZF shall be the (pure) sets, and that
| "x in y" shall mean that x is a member of y.
|
| The first nonlogical axiom of ZF states that if two sets have
| exactly the same members, then they are equal. This axiom,
| called the "extensionality axiom", is:
|
| `A`z(z in x <=> z in y) => x = y.
|
| [if, for all z, z in x iff z in y, then x = y].
|
| JRS, 'Mat Log', page 239.
|
| Joseph R. Shoenfield, 'Mathematical Logic',
| Addison-Wesley, Reading, MA, 1967.
Isn't it nice to seem `E` folks agree on `E` thing?
Well, almost. Obligatory exercise for the reader.
From now on, then, we will most conventionally,
if not most conveniently, index this notion as:
"Skolem.Morse.Hilbert.Bernays.VonNeumann.Gödel.Zermelo.Fraenkel.extension".
Fraenkely, I think "Supercallifragilistic Extensionality" would
sound more musical, but who wants to try and keep book on taste?
Jon Awbrey
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