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Re: [PrimeNumbers] Digest Number 1129

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  • chasag@aol.com
    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
    Message 1 of 3 , Nov 2, 2003
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      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. 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. The number of primes
      is not infinite but on the verge of infinity. This is where a prime gap might
      exist. The prime gap is incalculably high. 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.


      [Non-text portions of this message have been removed]
    • Jose Ramón Brox
      If we assume that you can effectively keep running your machine for infinite time, then you will have counted aleph_0 natural numbers and aleph_0 prime
      Message 2 of 3 , Nov 2, 2003
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        If we assume that you can effectively keep running your machine for infinite time, then you will have counted aleph_0 natural numbers and aleph_0 prime numbers, and you could put them in a biyection: no natural number will be out of it.

        And you can not do the "difference of two infinities" because the operation is not defined in any way. You must do your definition first and get the conclusions after that. Anyway, when we talk about cardinals of sets, we say that the intersection of two infinite aleph_0 sets can be finite or infinite (so you cant define the difference as the cardinal of the intersection to get the result you want). For example: the intersection of natural and even numbers are the even numbers, an infinite set. The intersection of even and odd numbers gives the empty set, with cardinal zero. The intersection of even numbers and prime numbers gives the set {2}, with cardinal 1.

        I can't understand your proposal yet.

        Jose Brox

        ----- Original Message -----
        From: chasag@...
        To: primenumbers@yahoogroups.com
        Sent: Sunday, November 02, 2003 7:08 PM
        Subject: Re: [PrimeNumbers] Digest Number 1129


        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. 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. The number of primes
        is not infinite but on the verge of infinity. This is where a prime gap might
        exist. The prime gap is incalculably high. 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.


        [Non-text portions of this message have been removed]


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        [Non-text portions of this message have been removed]
      • 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 3 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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