Re: Cantor Confusion

David Marcus <DavidMarcus@xxxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:
Lester Zick wrote:
On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus
<DavidMarcus@xxxxxxxxxxxxxx> wrote:

If you use ZFC (or something similar) as your foundation for
mathematics, then everything is a set. Of course, while solid
foundations are good to have, if you are living on an upper floor, you
may prefer to ignore what is going on in the basement.

So you're saying that set "theory" is all of mathematics? Of course
since what you say isn't necessarily true that's not exactly a ringing
endorsement of set "theory".

It's quite simple. Set Theory can not be the foundation for mathematics,
because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics, but
it's not a set. Set theory may be of limited use, but it's supremacy is
complete nonsense, and will be overruled in time.

But everything can be modelled as a set. You simply do not understand
what "foundation" means in this context. Any calculation can
be rewritten as a set theory problem. It would be long, cumbersome,
and impractical, but it could be done. Just as an computer program
can be transformed into a Turing machine.

Yes, but I think it is a bit simpler than that. The study of algorithms
involves the study of certain functions on certain number systems. Such
functions and numbers can be handled in ZFC. That's the real point of
the construction of the natural numbers from sets: to show that set
theory can be used as the foundation for arithmetic and hence analysis,
etc. However, if you are doing numerical analysis or calculus, you think
of N, Z, Q, R, C, etc. as primitive objects. You don't care that they
can be modelled as sets.

But writing down all the details using nothing but sets would
be cumbersome. Just saying that numbers can be represented as sets
is not the same as actually demonstrating how a calculation would
be represented using sets and nothing but sets. As I said, it
is like writing a Turing machine that computes a Fourier transform.
Anyone who understands the theory of computation knows that it
can be done, but nobody is likely to do it, especially just to
satisfy who has no understanding of the theory of computation.

Stephen
.

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