Re: [PrimeNumbers] Digest Number 1129
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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
The logic isn't fuzzy, the logic is lacking.
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