[FOM] CH and mathematics
joeshipman at aol.com
Mon Jan 21 16:17:07 EST 2008
> How about "we'll never know whether busy-beaver(googol), expressed as
> base=10 integer, has an even or odd number of digits?"
>Why not Goldbach's Conjecture (GC), or Twin Primes? These are
>questions that may turn out not to be knowable at all.
But a proof of GC or TPC is much more conceivable than a proof settling
my question above.
For GC and TPC absence of a proof is our only evidence they are hard,
but we have much stronger theoretical reasons for believing that
questions about busy-beaver(googol) are unknowable (see Chaitin's work
showing that no consistent proof systems of size X can decide the value
of more than approximately X bits of his Omega number (which codes up
the halting of TMs)).
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