Dear John, (01)
We are I think agreed. (02)
> The identity conditions for sets ensure that equality is determined
> by their elements. That principle is independent of whether the
> elements are specified by extension or intension. (03)
[MW] Exactly.
>
> To emphasize the importance of intensional definitions, consider
> just sets of integers. First, note that a finite set can be
> specified by extension:
>
> S = {6, 9, 12, 15, 18, 21}
>
> The same set could also be specified by intension:
>
> S = { x | x is an integer, x>4, x<23, and x is divisible by 3} (04)
[MW] Indeed. I would probably to both in a case like this.
>
> But note that no infinite set can be specified by extension in
> any finite statement in any language with a finite alphabet.
> It's always *necessary* to state some rule or predicate that
> defines the set by intension. For example,
>
> 1. Let 6 be an element in the set S.
>
> 2. If x is an element of S, then so is x+3.
>
> 3. S is the smallest set that satisfies conditions #1 and #2. (05)
[MW] Yes. Though as you note, only having an intensional definition does
not stop it being a set, since the extension is unchanging.
>
> For many finite sets, such as the set of all cows or the set
> of all molecules on planet earth at a given time slice, a
> specification by extension is not possible by any known method.
>
> For any 4D region that includes the future, no specification
> by extension is possible even for much smaller sets. (06)
[MW] Only if you have a fully deterministic system - in practice a rarity. (07)
> I can,
> for example, specify such a set by intension:
>
> The winning numbers for next week's New York State Lottery.
>
> But there is no way to evaluate the denotation before the date. (08)
[MW] Quite. (09)
Regards (010)
Matthew West
Information Junction
Tel: +44 560 302 3685
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matthew.west@xxxxxxxxxxxxxxxxxxxxxxxxx
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> (If you have such a method, please send me the denotation by
> email, but please don't cc it to the list.)
>
> John
>
>
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