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

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  • pakaran42
    Nov 4, 2003
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      --- 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 Gödel'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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