Re: Cantor Confusion

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.

Han de Bruijn

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.

Nobody can prove the Church-Turing thesis, but that does not prevent
people from being confident that our notion of computability is accurate.
Nobody can prove anything in science, but that does not prevent people from
placing a lot of confidence in it. For example, there is no proof that gravity acts on
all masses. I am surprised that you do not understand something as basic as that.

Stephen
.

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