Re: Cantor Confusion

On Fri, 8 Dec 2006 13:07:04 +0000 (UTC), 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.

Han de Bruijn

But everything can be modelled as a set.

A set of what? Points? That's preposterous. Nobody doubts everything
not only "can be modeled" but actually is a set of predicates.But what
makes those predicates points? The problem is not sets as such but the
points modern mathematikers like to pretend constitute those sets.

You simply do not understand
what "foundation" means in this context. Any calculation can
be rewritten as a set theory problem.

So what? Can set "theory" be rewritten as a set "theory" problem? The
problem is how we get the sets in the first place and what the sets
are sets of and not what you contend we can do with set "theory" once
we get it.

It would be long, cumbersome,
and impractical, but it could be done. Just as an computer program
can be transformed into a Turing machine.

So if we allow that TvN mechanics is the paradigm for computable
numbers it thereby becomes the paradigm for all mathematics? I suspect
you have a considerable number of erroneous hidden assumptions in this
little piece of legerdemind.

I do not know what you mean by "supremacy". Do you think 486
assembly language is the "supreme" programming language? It
currently is sort of a de facto candidate for a foundational
programming language.

Macro assembler is a paradigmatic machine programming interface. But
there are also better techniques.

~v~~
.

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