Arithmetical Incompleteness .

Arithmetical Incompleteness

Andre Willers

2 Mar 2015

Synopsis :

Proof that the Arithmetical System is incomplete .

Discussion :

1.As expected , it occurs in counting factors .

2.Let A be a number with factors.

3.Then A =a(1)^b(1)  * a(2)^b(2) … a(n)^b(n) , where a,b,n are integers

4. M= b(1)+ (b2) + …(b(n) , where M>=n

5. The Trick :

6.Clump 2 factors  , then 3 then to M .

The number of ways to write it is 2^M = mC0+mC1+mC2 +…mCm

Where mCr= M!/(r!*(M-r)!)

7.But the number of ways to write it is also

M^n    … http://en.wikipedia.org/wiki/Multinomial_distribution

8. Therefore ,

2^M = M^n

9. If M is odd  , this leads to a contradiction .

10. Thus, The Standard Arithmetical System is incomplete .

11. Note the correlation with Mersenne primes .

12. See

http://andreswhy.blogspot.com/2009/08/inside-of-zero.html

http://andreswhy.blogspot.com/2007/09/topos-4.html

To be expected from Godel .

Andre

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