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From: Wei Dai <weidai.domain.name.hidden>

Date: Sun, 22 Sep 2002 00:58:28 -0400

On Sat, Sep 21, 2002 at 09:20:26PM -0400, Vikee1.domain.name.hidden wrote:

*> For those of you who are familiar with Max Tegmark's TOE, could someone tell
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*> me whether Georg Cantor's " Absolute Infinity, Absolute Maximum or Absolute
*

*> Infinite Collections" represent "mathematical structures" and, therefore have
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*> "physical existence".
*

I'm not sure what "Absolute Maximum" and "Absolute Infinite Collections"

refer to (a search for "Absolute Infinite Collections" on Google gave no

hits), but I'll take the question to mean whether proper classes (i.e.

collections that are bigger than any set, for example the class

of all sets) have "physical existence".

I think the answer is yes, or at least I don't see a reason to rule it

out. To make the statement meaningful, we need (at least) two things, (1)

a way to assign probabilities to proper classes so you can say "I am

observer-moment X with probability p" where X is a proper class, and (2)

a theory of consciousness of proper classes, so you can know what it feels

like to be X when X is a proper class.

(2) seems pretty hopeless right now. We don't even have a good theory of

consciousness for finite structures yet. Once we have that, we would still

have to go on to a theory of consciousness for countably infinite sets,

and then to uncountable sets, before we could think about what it feels

like to be proper classes. But still, it may not be impossible to work it

out eventually.

As to (1), Tegmark doesn't tell us how to assign probabilities to observer

moments. (He says to use a uniform distribution, but gives no proposal for

how to define one over all mathematical structures.) However, it does not

seem difficult to come up with a reasonable one that applies to proper

classes as well as sets.

Here's my proposal. Consider a sentence in set theory that has one unbound

variable. This sentence defines a class, namely the class of sets that

make the sentence true when substituted for the unbound variable. It may

be a proper class, or just a set. Call the classes that can be defined by

finite sentences of set theory describable classes. Any probability

distribution P over the sentences of set theory, translates to a

probability distribution Q over describable classes as follows:

Q(X) = Sum of P(s), over all s that define X

Take P to be the universal a priori probability distribution (see Li

and Vitanyi's book) over the sentences of set theory, and use the

resulting Q as the distribution over observer moments.

Of course this distribution is highly uncomputable, so in

practice one would have to use computable approximations to it. However,

computability is relative to one's resources. We have access to certain

computational resources now, but in the future we may have more. We may

even discover laws of physics that allow us to compute some non-recursive

functions, which in turn would allow us to better approximate this Q. The

point is that by using Q, instead of a more computable but less dominant

distribution (such as ones suggested by Schmidhuber), in our theory of

everything, we would not have to revise the theory, but only our

approximations, if we discover more computational resources.

Received on Sat Sep 21 2002 - 21:59:44 PDT

Date: Sun, 22 Sep 2002 00:58:28 -0400

On Sat, Sep 21, 2002 at 09:20:26PM -0400, Vikee1.domain.name.hidden wrote:

I'm not sure what "Absolute Maximum" and "Absolute Infinite Collections"

refer to (a search for "Absolute Infinite Collections" on Google gave no

hits), but I'll take the question to mean whether proper classes (i.e.

collections that are bigger than any set, for example the class

of all sets) have "physical existence".

I think the answer is yes, or at least I don't see a reason to rule it

out. To make the statement meaningful, we need (at least) two things, (1)

a way to assign probabilities to proper classes so you can say "I am

observer-moment X with probability p" where X is a proper class, and (2)

a theory of consciousness of proper classes, so you can know what it feels

like to be X when X is a proper class.

(2) seems pretty hopeless right now. We don't even have a good theory of

consciousness for finite structures yet. Once we have that, we would still

have to go on to a theory of consciousness for countably infinite sets,

and then to uncountable sets, before we could think about what it feels

like to be proper classes. But still, it may not be impossible to work it

out eventually.

As to (1), Tegmark doesn't tell us how to assign probabilities to observer

moments. (He says to use a uniform distribution, but gives no proposal for

how to define one over all mathematical structures.) However, it does not

seem difficult to come up with a reasonable one that applies to proper

classes as well as sets.

Here's my proposal. Consider a sentence in set theory that has one unbound

variable. This sentence defines a class, namely the class of sets that

make the sentence true when substituted for the unbound variable. It may

be a proper class, or just a set. Call the classes that can be defined by

finite sentences of set theory describable classes. Any probability

distribution P over the sentences of set theory, translates to a

probability distribution Q over describable classes as follows:

Q(X) = Sum of P(s), over all s that define X

Take P to be the universal a priori probability distribution (see Li

and Vitanyi's book) over the sentences of set theory, and use the

resulting Q as the distribution over observer moments.

Of course this distribution is highly uncomputable, so in

practice one would have to use computable approximations to it. However,

computability is relative to one's resources. We have access to certain

computational resources now, but in the future we may have more. We may

even discover laws of physics that allow us to compute some non-recursive

functions, which in turn would allow us to better approximate this Q. The

point is that by using Q, instead of a more computable but less dominant

distribution (such as ones suggested by Schmidhuber), in our theory of

everything, we would not have to revise the theory, but only our

approximations, if we discover more computational resources.

Received on Sat Sep 21 2002 - 21:59:44 PDT

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