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From: Archimedes Plutonium on 30 Jun 2010 16:06


Archimedes Plutonium wrote:
(snipped all else)
>
>
> INDIRECT (contradiction) Method, Long-form; Infinitude of Primes Proof
> and
> the numbering is different to show the reductio ad absurdum structure
> as
> given by Thomason and Fitch in Symbolic Logic book.
>
>
> (1) Definition of prime as a positive integer divisible
> only by itself and 1.
>
> (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
> Reason: definition of primes
>
> (3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
> with p_n the largest prime Reason: this is the supposition step
>
> (3.1) Set S are the only primes that exist Reason: from step (3.0)
>
> (3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
> form a new number
>
> (3.3) Divide W+1 successively by each prime of
> 2,3,5,7,11,..pn and they all leave a remainder of 1.
> Reason: can always operate

No, the reason for that is the Unique Prime Factorization theorem

>
> (3.4) W+1 is necessarily prime. Reason: definition of prime, step
> (1).
>
> (3.5) Contradiction Reason: pn was supposed the largest prime yet we
> constructed a new prime, W+1, larger than pn
>
> (3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
>
> (4) Set of primes are infinite Reason: steps (1) through (3.6)
>

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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