Re: Svara that Goldbach's Conjecture is Unprovable

From: Arthur J. O'Dwyer (ajo_at_nospam.andrew.cmu.edu)
Date: 07/12/04 

Date: Mon, 12 Jul 2004 14:22:09 -0400 (EDT)

On Mon, 12 Jul 2004, Michael N. Christoff wrote:
>
> "Craig Feinstein" <cafeinst@msn.com> wrote...
> > The argument is as follows. Its very similar to a previous post that I
> > made which argues that Twin Primes is unprovable. I'll let you judge
> > whether this is a rigorous proof or not. I'm not claiming that it is
> > or that it is not - It's just for fun:
[...]
> > Therefore, the two conditions which define the two sets are
> > independent from one another, meaning that the only way to determine
> > whether the intersection of the two sets, S and T, is nonempty is to
> > directly calculate the elements in the intersection of S and T, i.e.,
> > test if there is a q such that q and 2n-q are both prime. But
> > this would take an infinite amount of time, since one would have to
> > test this for each natural number n. Therefore, Goldbach's Conjecture
> > is unprovable. QED

> Hi Craig. This sounds similar to your proof idea for p!=np. In that proof,
> if I recall correctly, you claimed (without adequate proof) that a single
> solution to the problem existed (in that case the problem was subset-sum),
> and then claimed that this single solution required at least a
> super-polynomial amount of computation to solve (hence subset-sum must be in
> NP-P so P != NP). In this proposition you state:
>
> "the only way to determine whether the intersection of the two sets, S and
> T, is nonempty is to directly calculate the elements in the intersection of
> S and T"
>
> You still need to prove that this truly is 'the only way'.

  There's a more basic flaw in the logic, of course. Craig states that
to prove or disprove Goldbach's conjecture would take an infinite amount
of time, since you'd have to test every even number greater than 2.
But he forgets that *if Goldbach's conjecture is false*, then you
only need to test N different integers, where N is a number such that
(2*N)+2 is not the sum of a pair of primes. And if you only have to
test a finite number of numbers, well, that certainly shouldn't take
an infinite amount of time!

  So Craig has shown that *if* GC is true, [and *if* certain other
implicit assumptions about prime numbers hold,] *then* GC is
unprovable. But he hasn't ruled out the possibility that it might
be false --- and if it's false, then it's most definitely provably
so!

-Arthur

Relevant Pages

  • Re: i have just proved golbach conjecture
    ... > assume there are a finite amount of primes ... > this is not divisible by any primes.. ... therefore there are an infinite amount of primes.. ... Could you state what you think Goldbach's Conjecture is? ...
    (sci.math)
  • Re: goldbachs conjecture
    ... Curiously according to this, are just the primes, ... Whether or not Goldbach's conjecture is true or not, ... Popper called the Goldbach Conjecture true if, ... Popper's context of computational falsifiability) ...
    (sci.math)
  • Re: Space between prime numbers
    ... > to be prime, except perhaps for choosing Mersenne numbers, ... It is not known if there are infinitely Mersenne primes, ... There are generalizations of this conjecture with the polynomials x-a_i ... A/sqrt= infinity. ...
    (sci.math)
  • Re: goldbachs conjecture
    ... Curiously according to this, are just the primes, ... Whether or not Goldbach's conjecture is true or not, ... Popper called the Goldbach Conjecture true if, ... when it comes on a counterexample. ...
    (sci.math)
  • Re: representation
    ... To say that every positive integer is a sum of 4 squares is not the ... of primes. ... Another way to disprove the conjecture without producing an explicit ...
    (sci.math)