## 13957Re: Infinite primes-> a Turing Machine prime sieve that never stops?

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• Nov 4 9:28 AM
--- In primenumbers@yahoogroups.com, Roger Bagula <tftn@e...> wrote:
> The concept of "infinity"
> is a Platonic ideal.
> It can't be proven.

I don't know if "the concept of infinity" can be proven, but
statements about transfinite numbers can be.

> It is a limiting axiomatic definition (asymptotic).
> 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... )

Which is irrelevant to the philosophical argument.

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.
> 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.

No, they're just plain false (and I believe that can be shown from
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
> metamathematics invented and outside ordinary mathematics.
> It has it's own axiom system and rules.

Not quite the case.

> The idea of an infinite prime would have to be
> in this context and not in the context of finite primes at all.

True.

> A lot of people don't like it pointed out that irrational numbers
depend on
> a statement like 3)
> being accepted as an axiomatic definition.
> Like Euclid's product proof,
> Cantor's proof of transfinites is also widely faulted in
> modern mathematics.

By whom? And which proof? The diagonal argument? Care to name
someone who faults it?

> The result is you get philosophical
> 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.

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