Dear Matthew, (01)
To avoid distractions of 3D vs 4D ontologies, I'll focus on the
importance of intensional definitions even in set theory itself. (02)
JFS>> But note that even for finite sets, an intensional definition
>> is necessary. (03)
MW> I would challenge necessary, but I agree they are useful. There
> is nothing to prevent sets having intensional definitions, just
> as long as that definition does not lead to varying memberships. (04)
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. (05)
To emphasize the importance of intensional definitions, consider
just sets of integers. First, note that a finite set can be
specified by extension: (06)
S = {6, 9, 12, 15, 18, 21} (07)
The same set could also be specified by intension: (08)
S = { x  x is an integer, x>4, x<23, and x is divisible by 3} (09)
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, (010)
1. Let 6 be an element in the set S. (011)
2. If x is an element of S, then so is x+3. (012)
3. S is the smallest set that satisfies conditions #1 and #2. (013)
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. (014)
For any 4D region that includes the future, no specification
by extension is possible even for much smaller sets. I can,
for example, specify such a set by intension: (015)
The winning numbers for next week's New York State Lottery. (016)
But there is no way to evaluate the denotation before the date.
(If you have such a method, please send me the denotation by
email, but please don't cc it to the list.) (017)
John (018)
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