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[Apologies if this duplicates previous responses]
Tony Orlow wrote:
> I am beginning to realize just how much trouble the axiom of
> extensionality is causing here. That is what you're using, here, no? The
> sets are "equal" because they contain the same elements.
Yes, the basic definition of set equality, the '=' set operator.
> That gives no
> measure of how the sets compare at any given point in their production.
This makes no sense. Sets are not "produced" or "generated".
Sets simply exist.
> Sets as sets are considered static and complete.
Correct.
> However, when talking
> about processes of adding and removing elements, the sets are not
> static, but changing with each event.
Incorrect. If we define set A as containing the elements a, b, and c,
then A = {a, b, c}. Period. If we then talk about adding elements d
and e to set A, we're not actually changing set A, but describing
another set, call it A2, that is the union of A and {d, e}, so
A2 = {a, b, c, d, e}.
Nothing is ever "added to" a set. Rather, we apply operations (union,
intersection, etc.) to existing sets to create new sets. We don't
change existing sets.
> When speaking about what is in the
> set at time t, use a function for that sum on t, assume t is continuous,
> and check the limit as t->0. Then you won't run into silly paradoxes and
> unicorns.
That would be using the wrong nomenclature. We don't talk about
what set A contains at any particular time t. We can talk about the
sequence of sets A_1, A_2, A_3, etc. We can also talk about the
union of the entire sequence of sets as set A, if we like. Or the
intersection, or whatever. But every set we're talking about is
"static", unchanging, once it (the members it contains) is defined.
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