Bruno Marchal wrote:
> Le 17-avr.-06, à 19:53, Tom Caylor a écrit :
> > Along those line, I notice that Chaitin (referencing du Sautoy) says
> > that if it could be proved that the Riemann Hypothesis is undecidable
> > then it is true, since if it were false then it would be decidable by
> > finding a zero off of the critical line.
> Yes, by the very work of Riemann, Lehmer, ... Turing, ... Odlyzko. If a
> (non trivial) zero is not on the critical line it must be found by a
> searching program (perhaps in a googol years, or a googol^googol years,
> or later ...). So, if someone prove that RH (Riemann Hypothesis) is
> undecidable in any very weak theory (far simpler than a lobian theory)
> then RH is true. Indeed.
See my last comment below.
> > (http://maa.org/features/chaitin.html). But could it be that the
> > Riemann Hypothesis follows quantum indeterminacy in something like the
> > following way? Just role-playing: "The Riemann zeta function does
> > indeed have zeros which are off of the critical line (or even, it has
> > zeros having real parts taking on every real value between 0 and 1.)
> 1 is already excluded. Actually the width of the critical strip where
> non trivial zeros are yet allowed has been made shorter an shorter.
> Alas, not enough to prove RH.
I meant to exclude 0 and 1. I am not aware that there have been any
subsets of the critical strip 0 < Re(s) < 1 that have been found to be
zero-free, yet. I know that it has been proved that at least 40% of
the zeros are on the critical line.
> > This is the non-computable truth. However, whenever a zero of the
> > Riemann zeta function is actually computed (observed), it falls on the
> > critical line."
> I let you take the full responsibility of this statement :)
> Are you introducing a wave collapse in Platonia ???
No. In the sense that I'm NOT saying: Given a (fixed) value of y,
there might be a zero at s=x+iy off of the critical line... until you
actually compute it at which time you find that it is on the critical
But, yes, in a different sense, and here I do take full responsibility,
with tongue in cheek, but not too far lest I bite it. Perhaps
mathematicians have been committing an error in the realm of infinity,
perhaps associated with the Axiom of Choice. Perhaps there are zeros
off of the critical line, but they all lie beyond any chosen (observed)
finite number. Perhaps the off-critical-line zeros are non-computably
existent but affect us in tangible ways. Perhaps the Riemann
Hypothesis is even *outside* of the realm of undecidable. I know
perhaps this is saying that do-able math is limited in an even stronger
sense, but I think it's possible.
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