2 Mar 2015
Proof that the Arithmetical System is incomplete .
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
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 .
To be expected from Godel .