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From: SLP <SLP.domain.name.hidden>

Date: Fri, 16 Jul 1999 19:08:10 -0500

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

Date: Fri, 16 Jul 1999 19:08:10 -0500

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

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