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
Mobile: +44 750 3385279
matthew.west@xxxxxxxxxxxxxxxxxxxxxxxxx
http://www.matthew-west.org.uk/  (011)
This email originates from Information Junction Ltd. Registered in England
and Wales No. 6632177.
Registered office: 2 Brookside, Meadow Way, Letchworth Garden City,
Hertfordshire, SG6 3JE.  (012)
> (If you have such a method, please send me the denotation by
> email, but please don't cc it to the list.)
>
> John
>
>
> _________________________________________________________________
> Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
> Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
> Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
> Shared Files: http://ontolog.cim3.net/file/
> Community Wiki: http://ontolog.cim3.net/wiki/
> To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J
> To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx
>  (013) 
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ 
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ 
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/ 
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx  (014)