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Topic: § 534 Finis
Replies: 61   Last Post: Aug 13, 2014 3:18 PM

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 David Petry Posts: 1,104 Registered: 12/8/04
Re: § 534 Finis
Posted: Aug 9, 2014 4:52 AM

On Wednesday, August 6, 2014 3:32:00 AM UTC-7, muec...@rz.fh-augsburg.de wrote:
> Set theorists claim that all rational numbers can be indexed by all natural numbers. In § 533 I have shown not only that every natural number n fails but even that with increasing n the number of unit intervals of rationals without any rational indexed by a natural less than n increases without bound, i.e., infinitely. Since nothing but finite natural numbers are available for indexing, and provably all fail, this task cannot be accomplished.

Here is a bijection between positive integers and positive rational numbers.

Define a function F(n) as F(n) = n/2 if n is even, and F(n) = -(1+n)/2 if n is odd.

Let the prime factorization of N be product( p_i ^ a_i ) where each p_i is prime and each a_i is a positive integer.

Let R(N) = product( p_i ^ F(a_i) )

Then {R(n), n = 1,2,3,...} contains every positive rational number.

Does that change your opinion on anything?