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David R Tribble <david@xxxxxxxxxxx> wrote:
> [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.
Just like when we add 5 to 2 to get 7, we do not change the 5 or 2
to create a 7. Or when you celebrate a birthday, your age changes,
but the number that represented your age does not change. A different
number is used to represent your age, but the "old" number remains
as it always ways.
This idea of "changing" sets seems to be at the heart of a lot
of people's misconceptions about set theory.
Stephen
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