- Nov 4, 2003--- In primenumbers@yahoogroups.com, Roger Bagula <tftn@e...> wrote:
> The concept of "infinity"

I don't know if "the concept of infinity" can be proven, but

> is a Platonic ideal.

> It can't be proven.

statements about transfinite numbers can be.

> It is a limiting axiomatic definition (asymptotic).

Which is irrelevant to the philosophical argument.

> So arguing if such an such types of infinity

> exist , makes sense only if the limiting

> case is assumed to exist in the first place.

> Mathematics can be defined as two cases:

> 1) numbers where infinity is defined

> 2) numbers where no infinity is defined or definable as

> being a non-operational definition

> ( it crashes Mathematica, ha, ha... )

The number 3^100 cannot be represented exactly in most versions of C,

and pi cannot be represented exactly in any language I know of without

taking shortcuts. Does this mean neither exists?

> It is a philosophical distinction outside of normal mathematics.

No, they're just plain false (and I believe that can be shown from

> Called since GĂ¶del's time "metamathematical" statements/ arguments

> after it was realized that a philosophical threshold

> existed in certain kinds of proofs.

> Statements like :

> 3) Aleph1=2^Aleph0

> are outside of ordinary mathematics.

ZFC, though I wouldn't want to try from scratch).

Assume Aleph1=2^Aleph0. This implies that there is a mapping from the

Aleph1 real numbers onto the base 2 logarithms of the Aleph0 integers.

However, there are only countably many such logarithms (since there

are only countably many positive integers for them to be logarithms

of). This is a contradiction (and I'm sure it can be stated much more

formally). I would agree that the integers are a proper subset of the

base-2 logs of integers, just as the primes, perfect squares, and

numbers evenly divisible by 100 are proper subsets of the integers.

However, all those sets are countably infinite.

> The algebra of transfinites is really a form of

Not quite the case.

> metamathematics invented and outside ordinary mathematics.

> It has it's own axiom system and rules.

> The idea of an infinite prime would have to be

True.

> in this context and not in the context of finite primes at all.

> A lot of people don't like it pointed out that irrational numbers

depend on

> a statement like 3)

By whom? And which proof? The diagonal argument? Care to name

> being accepted as an axiomatic definition.

> Like Euclid's product proof,

> Cantor's proof of transfinites is also widely faulted in

> modern mathematics.

someone who faults it?

> The result is you get philosophical

So if I asked you to design a plastic that would work in a world where

> and nearly religious in tone arguments

> in mathematical forms like these egroups.

> At first I didn't realize that it was that basic a

> "postulate" , "axiom"

> or

> "definition", but it has become clear

> that it is.

> I really don't like to get in such discussions,

> since math people seem to ignore any philosophical issues by defining

> them away.

> Axioms and definitions are an answer to all their thinking problems?

> As a physical scientist ( chemist, physical scientist)

> I'm not bound by those rules in my thinking.

the electromagnetic constant was 1/1000 of its present value, you

could, without starting from scratch? That's about the equivalent of

what you're asking us to do.

Regards,

Nathan - << Previous post in topic Next post in topic >>