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From: Jesse Mazer <lasermazer.domain.name.hidden>

Date: Sat, 30 Nov 2002 13:59:03 -0500

Hal Finney wrote:

*>That would be true IF you include descriptions that are infinitely long.
*

*>Then the set of all descriptions would be of cardinality c. If your
*

*>definition of a description implies that each one must be finite, then the
*

*>set of all of them would have cardinality aleph-zero.
*

What about descriptions that are finite but would require an oracle machine

to actually "compute"? We can define Omega, whose digits can be known only

if you know whether each and every turing machine halts or not, in a finite

number of words, but it can't be computed by any turing machine. If I'm

remembering correctly there are also 2nd-order oracle machines that

"compute" things 1st-order oracle machines cannot, and 3rd-order oracle

machines, and so on, with a new level of oracle for each new ordinal

(although I'm not sure if it makes sense to talk about aleph-one-order

oracles or if it's just for the countable ordinals). This would suggest that

no countable list of "finite descriptions" which includes descriptions for

arbitrary oracle machines could ever be complete--as long as you have some

finite rule for generating the list, you can use diagonalization to come up

with a new number which has a finite description for some oracle machine.

I think that numbers like Omega really "exist" in the same sense as pi (in

Platonia, if you will). I am not so sure about "random reals" which have no

finite description of any kind. I don't know what implications this

philosophy would have for a TOE...even a number like Omega can in some sense

be computed "in the limit" by a turing machine (by initially assuming each

turing machine fails to halt, then simulating each one and revising the

number each time a particular machine is found to halt), even if the value

of a particular digit cannot be decided in a finite time as with pi or other

computable numbers. I assume this would probably be true for any number with

a finite description. Doesn't the UDA argument in some sense depend on the

idea of computing "in the limit" too?

--Jesse Mazer

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Received on Sat Nov 30 2002 - 14:00:30 PST

Date: Sat, 30 Nov 2002 13:59:03 -0500

Hal Finney wrote:

What about descriptions that are finite but would require an oracle machine

to actually "compute"? We can define Omega, whose digits can be known only

if you know whether each and every turing machine halts or not, in a finite

number of words, but it can't be computed by any turing machine. If I'm

remembering correctly there are also 2nd-order oracle machines that

"compute" things 1st-order oracle machines cannot, and 3rd-order oracle

machines, and so on, with a new level of oracle for each new ordinal

(although I'm not sure if it makes sense to talk about aleph-one-order

oracles or if it's just for the countable ordinals). This would suggest that

no countable list of "finite descriptions" which includes descriptions for

arbitrary oracle machines could ever be complete--as long as you have some

finite rule for generating the list, you can use diagonalization to come up

with a new number which has a finite description for some oracle machine.

I think that numbers like Omega really "exist" in the same sense as pi (in

Platonia, if you will). I am not so sure about "random reals" which have no

finite description of any kind. I don't know what implications this

philosophy would have for a TOE...even a number like Omega can in some sense

be computed "in the limit" by a turing machine (by initially assuming each

turing machine fails to halt, then simulating each one and revising the

number each time a particular machine is found to halt), even if the value

of a particular digit cannot be decided in a finite time as with pi or other

computable numbers. I assume this would probably be true for any number with

a finite description. Doesn't the UDA argument in some sense depend on the

idea of computing "in the limit" too?

--Jesse Mazer

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Received on Sat Nov 30 2002 - 14:00:30 PST

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