Re: Cantor Confusion

Han.deBruijn@xxxxxxxxxxxxxx wrote:
stephen@xxxxxxxxxx schreef:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:

stephen@xxxxxxxxxx wrote:

But everything can be modelled as a set.

Define "everything" and prove that claim.

By "everything", I meant everything mathematical. Of course that is not 100% precise.
And no, I cannot prove it. But so far all the various objects of mathematics can be
modelled using set theory. That is what is meant by set theory being a foundation
for mathematics. If someone were to invent something "mathematical" (whatever that may
mean exactly) that could not be described in terms of set theory, then set theory would
no longer serve as a foundation. But given that the basics such as the real numbers,
functions, limits, calculus, etc. all can be founded in set theory, it would have to
be something strange indeed. Not that there is anything wrong with strange, but you
probably would like it less than set theory.

Correction. By "everything" you probably mean "everything according to
nowadays mainstream mathematics", which _is_, of course, "mathematics",
according to your probably rather limited view. But since you can not
really prove anything of the kind, I will rest my case.

It's not much of a case. You have not presented any evidence that there exists
any sort of mathematics not describable by set theory. Until such evidence
exists, the hypothesis that mathematics can be modelled with set theory has
not been falsified. And don't bother presenting something that uses limits,
functions, etc. as all of those things can be modelled with set theory.

Ah, now you are trying to put the burden on me. But that is false play,
of course. You said something like "all heat is phlogiston". I do not
have to argue that this is not so. The burden remains yours. I didn't
even say that set theory is useless within mathematics. I've only said
that there's more to mathematics than set theory.

All the mathematics I've ever seen in a math class or read in a math
book or journal or done myself can be done in ZFC. Admittedly, there are
large areas of mathematics that I know little or nothing about. Still,
that would seem to be quite a bit of evidence right there for the
statement that ZFC can be used as a foundation for mathematics. So, the
burden is now on you to show some mathematics that can't be done in ZFC.

And I find the claim
that set theory is the one and only foundation of mathematics merely
a manifestation of a narrow mind, if not irresponsible behaviour.

That is a straw man. Who has claimed that? The actual claim is that it
is an economical foundation that has been worked out in detail. As such,
it serves the purpose of providing a foundation.

One shouldn't narrow down a beautiful discipline to such a poor paradigm.

To those who are unable to see it, there is no evidence that there is
any sort of mathematics not describable by set theory. But this is a
vicious circle. Mainstream mathematics simply DOES NOT ALLOW any sort
of mathematics that violates set theory.

Quite a silly thing to say. Do you have any evidence for such an absurd
statement? I can give some contrary evidence: I recall reading an
article by Saunders MacLane (in the Notices of the AMS, I think) where
he asked the logicians to put more effort into developing category
theory as a foundation. He didn't feel that set theory was a natural
foundation for algebra.

That's what the whole thread
here is about. And, as you know, there are, and there have been, many
other such threads in 'sci.math'. Personally, I find that (infinitary)
set theory is full of inconsistencies and paradoxes.

"Personally"? Claims of inconsistency and paradox are a dime a dozen.
Please provide a specific inconsistency.

It's unbelievable
that it's still finds so much support. Must be on the wrong planet ..

Yes, you'd probably be happier on Mars. On this planet, we require facts
and evidence, not personal assertions.

Apart from this, Im currently in the process of posting some pieces of
mathematical theory on Chebyshev polynomials. I seldomly feel the need
to employ set theory in this work (I've done it accidentally, though).
The reason is that set theory, as such, contributes virtually nothing
to understanding. I can do quite well without it, most of the time.

Once again, you demonstrate that you don't know what the word
"foundation" means.

--
David Marcus
.

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