Marchal wrote: "Only a very tiny part of the possible mathematical
structures can be
formalised or axiomatized.
There is no possible formalisation of Arithmetical Truth.
This is a consequence of Godel's Theorem."
Strong disagreement. You should have said: Only a very tiny part of the
possible mathematical structures can be formalised or axiomatized in a
purely finitistic way. There is no possible FINITE formalisation of
Arithmetical Truth which is both complete and consistent. A full
formalization of number theory is denumerably transfinite.
Being able to go to explicitly transfinite fomalizations and transfinite
computations should be possible...and probably much sooner than anyone
would guess.
As I mentioned in a previous post, by manipulating space-time as a
"building material," it may be possible in the not very far future to
create regions of Malament-Hogarth space-time that can function as
computing engines. These would be devices with an explicitly
transfinite computing capacity, giving the results of such computation
in a finite (short) time, as explained by Earman and Norton in their
paper, `Forever Is a Day: Supertasks in Pitowsky and Malament-Hogarth
Spacetimes',
Philosophy of Science, 60, pp. 22-42, 1993.
As a trivial and totally
impractical (from a computing standpoint) illustration of a
Malament-Hogarth spacetime, consider this commentary by Jon Pérez
Laraudogoitia:
"An example of such a spacetime is an electrically charged black hole
(the Reissner-Nordstrom spacetime). A well known property of black holes
is that, in the view of those who remain outside, unfortunates who fall
in appear to freeze in time as they approach the event horizon of the
black hole. Indeed those who remain outside could spend an infinite
lifetime with the unfortunate who fell in frozen near the event horizon.
If we just redescribe this process from the point of view of the
observer who does fall in to the black hole, we discover that we have a
bifurcated supertask. The observer falling in perceives no slowing down
of time in his own processes. He sees himself reaching the event horizon
quite quickly. But if he looks back at those who remain behind, he sees
their processes sped up indefinitely. By the time he reaches the event
horizon, those who remain outside will have completed infinite proper
time on their world-lines."
Obviously, nobody is going to go down a black hole and be torn apart by
tidal forces to do a computation, but the illustration makes the point
that such space-times, relying ONLY on classical general relativity, can
actually "do' a denumerably transfinite computation in what for an
appropriately placed observer is a finite time. I will refer to
Lucian Wischik's 1997 paper "Non-finite computation in Malament-Hogarth
spacetimes," available from
http://www.wischik.com/lu/researcher.html,
which gives more of the theory on this.
Steve Price, MD
Received on Fri Jul 16 1999 - 17:15:31 PDT