[FOM] Arithmetical soundness of ZFC

[Timothy Y. Chow]

On Wed, 27 May 2009, William Messing wrote:
> I am intrigued by the suggestion that ZFC might refute the RH.  Are 
> there individuals who are attempting to show this?

I suppose you could interpret all of Odlyzko's efforts to compute zeros as 
"attempted disproofs," though I don't think Odlyzko himself would describe 
them that way.  He is firmly agnostic about RH.  In Derbyshire's book 
"Prime Obsession," Chapter 22, we read:

  JD: Andrew, you have gazed on more non-trivial zeros of the Riemann
      zeta function than any person alive.  What do you think of this
      darn Hypothesis?  Is it true, or not?
  AO: Either it's true, or else it isn't.
  JD: Oh, come on, Andrew.  You must have some *feeling* for an answer.
      Give me a probability.  Eight percent it's true, twenty percent
      it's false?  Or what?
  AO: Either it's true, or else it isn't.

  I could get no more from him than that.  He simply would not commit 
  himself.

The fact that Odlyzko of all people is still agnostic should make anyone 
think twice before asserting that the computational evidence for RH is 
overwhelming.  (Of course you could argue that there are compelling 
reasons to believe RH besides the numerical evidence.)

> I recall that at the conference for Hodge's retirement from Pembroke 
> College in Cambridge in 1973, Gelfand on perhaps his first visit to the 
> West, in after dinner conversation expressed the view that the RH might 
> be independent of the axioms of ZFC.

As far as I know, nobody has the slightest idea how to even begin 
demonstrating such a thing.  On the other hand, there are surely people 
out there who have ideas about RH but are holding their cards close to the 
chest.

Tim