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Re: [PrimeNumbers] Digest Number 1129

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  • Décio Luiz Gazzoni Filho
    ... Hash: SHA1 ... If you assume some standard physics assumptions (Heisenberg Uncertainty Principle and finiteness of matter), then you can t build a machine
    Message 1 of 3 , Nov 2, 2003
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      On Sunday 02 November 2003 16:08, chasag@... wrote:
      > Imagine a counting machine counting up and running for an infinite time
      > ,After an eternity, a infinite large number n is reached and a very large
      > number of primes have been logged.

      If you assume some standard physics assumptions (Heisenberg Uncertainty
      Principle and finiteness of matter), then you can't build a machine that
      counts past as certain threshold: there's just not enough information storage
      available. Although of course one would expect the universe to end far sooner
      than the time it'd take to reach this threshold.

      But I'd rather point out that there's no such thing as an infinite large
      number n. Sure we can distinguish the relative sizes of aleph_n's, but not in
      the sense that you meant. And tell me how an ``infinite large number'' is
      reached and yet the machine only has logged ``a very large number of
      primes''? Yes, I know the set of primes has asymptotic density zero, but it
      isn't any less or more infinite than the set of naturals (with cardinality
      aleph_0), as the bijection n |-> p_n shows.

      > The counting machine also counts the number (n - number of primes). Thus the
      > number of primes is n -(n - number of primes).This is a difference of two
      > infinities leaving a finite result.

      This must rank as some of the biggest pieces of rubbish I've ever read. First
      realize that your assumption is flawed: your machine can't ``count to
      infinity.'' Now suppose that indeed your machine could somehow count all
      integers, and consequently all primes. By Euclid's proof, there's an infinite
      number of primes, so the result is infinite. Pretty simple. Your circular
      logic is flawed by assuming that there's a finite number of primes, which has
      been shown to be false 2000 years ago. Please, learn the very very basics
      before posting such rubbish to the list.

      > The number of primes is not infinite but on the verge of infinity.

      THE NUMBER OF PRIMES IS INFINITE. How many times will the list have to repeat
      that? I can understand the Goldbach provers, in the sense of Bruce Schneier's
      assertion ``anybody can come up with a cipher he can't break,'' but someone
      who is outsmarted by the simplest proof in mathematics, and a 2000+ year old
      one at that? Please.

      > This is where a prime gap might exist. The prime gap is incalculably high.

      Prime gaps can be incalculably high, indeed, in the sense that I argued in the
      first paragraph. That doesn't make them infinite though.

      > The verge of infinity is a suborder of infinity. I know the logic of this is
      > fuzzy, but somebody much more clever than me could develop a new kind of
      > arithmetic.

      The logic isn't fuzzy, the logic is lacking.

      Décio
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