Re: Relation between sets and their elements


"Paul Holbach" <paulholbachSPAMBAN@xxxxxxxxxx> wrote in message
news:1114396517.314043.54210@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>> Nam Nguyen wrote:
>> > "Paul Holbach" <paulholbachSPAMBAN@xxxxxxxxxx> wrote:
>> > > "Paul Holbach" <paulholbachSPAMBAN@xxxxxxxxxx> wrote:
>
>> > > Can sets lose their elements and keep their identity as distinct
>> > > entities, i.e. as entities distinct from the empty set?
>> > > (I don't think so.)
>
>> But it also seems that mathematics is time invariant and
>> therefore sets
>> can _not_ loose their elements.
>
> But the elements of many sets are contingently existing concrete
> things, which can cease to exist entirely.
>
> For example, the set of dinosaurs does seem to have lost its elements,
> doesn't it?
>
>> > What happened to the set of dinosaurs when
>> > the dinosaurs had become extinct?
>
>> Accordingly, either we have a set of dinosaurs that once lived ...
>
> You mean that if the living dinosaurs h a d b e e n (identical with)
> the set of dinosaurs, then this set would have been alive as well, for
> if all parts of a whole are alive, then the whole is alive too. (?)
>
> On the other hand, sets really don't appear to have the properties of
> their elements.
> The set of male dogs does not pee on lamp posts, does it?
>
>> ... or an empty
>> to begin with; but not a set of dinosaurs that would
>> one day walk out of
>> the set into oblivion, I'd think.
>
> Do you mean that before the dinosaurs were brought forth by evolution
> there had already been an empty set called "the set of dinosaurs" since
> the beginning of time whcih had been patiently waiting for the
> dinosaurs to become its elements?
>
> The problem I see is that all possible empty sets are identical with
> each other, i.e. if there is an empty set at all, there can only be
> exactly one empty set.
> But isn't the empty set necessarily empty?
> For instance, the set of female popes is now (Herr Ratzinger, the new
> pope, hopes 'forever') empty, but contingently and not necessarily so.
>
> So it seems there are two kinds of empty sets:
> ones that are contingently empty and ones that are necessarily empty.
>
> But how could that be possible, since there is only one empty set,
> *the* empty set, which is necessarily empty, i.e., which couldn't have
> possibly had any members.
> And if the set of femal popes should be necessarily empty too, then
> there couldn't possibly be any female popes, which state of affairs is
> doubtless a logically possible fact.
>
> How can things be rendered coherent?
>
> I'll try:
> Before there were any dinosaurs, there had been no empty set of
> dinosaurs. When there eventually were dinosaurs, the set of dinosaurs
> began to exist at the very same time and ceased to exist exactly when
> the last dinosaur had kicked the bucket.
>
> I'm afraid that depiction doesn't really convince me.
>
> I'm somewhat confused ...
>
> Was there a contingently empty set called "the set of dinosaurs
> *before* the first dinosaur appeared on earth, and is there still a
> contingently empty set called "the set of dinosaurs" now that there
> aren't any anymore?
>
> Regards
> PH
>

Your set of dinosaurs is poorly defined.

If you tighten up your definition of the set of dinsoaurs, then issue
disappears. Your set of dinosaurs may be "all dinosaurs alive 100 million
years ago", "all types of dinosaurs alive 100 million years ago", "all
dinosaurs that ever lived", or "all dinosaurs alive today" (the empty set).
If you define exactly what the set is, the problem goes away.

If you want the set to have different membership depending upon the date,
then its not a "set" in the mathematical logic sense.
There is no concept of time in set theory. If something happens (like comets
hitting the earth) that changes set membership, it creates a new set.



.

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