Largest provable BB(N)?

From: Russell Wallace (wallacethinmintr_at_eircom.net)
Date: 10/28/04 

Date: Thu, 28 Oct 2004 16:17:51 GMT

Hi all,

Does anyone know whether there's a reasonable guess as to the largest
value for which the value of the busy-beaver function BB(N) is
provable?

Specifically: This would require a proof of non-halting for all the
N-state Turing machines that don't halt; so presumably there's an N
such that there's an N+1-state Turing machine whose non-halting cannot
be proven.

And by "provable" I mean within the scope of known mathematics... is
Zermelo-Fraenkel set theory the most powerful axiomatic system we
have? Is there any prospect that something more powerful might be
discovered in the future?

I imagine the actual value of N isn't known, but has anyone made an
informed guess as to whether it might be 10, 20, 50 or so?

Thanks, 

-- 
"Always look on the bright side of life."
To reply by email, remove the small snack from address.