**From:** Marc Geddes (*m_j_geddes@yahoo.com.au*)

**Date:** Fri Feb 17 2006 - 00:04:00 MST

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*>Eliezer has posted a job notice at the SIAI website,
*

*>looking for research partners to tackle the problem
*

*>of rigorously ensuring AI goal stability under
*

self->enhancement transformations.

*>I would like to see this problem (or perhaps a more
*

*>refined one) stated in the rigorous terms of
*

*>theoretical computer science; and
*

*>I'd like to see this list try to generate such a
*

*>formulation.
*

Fascinating, fascinating.

I thought as a final post I'd better try to say

something actually intelligable, so I shall take one

crack at actually pointing to a solution :D

O.K...

Is there such a thing as

'a probability of a probability' ?

See a new paper by Robin Hanson arguing for a new

Bayesian framework wherein probabilities can be

assigned to priors:

http://hanson.gmu.edu/prior.pdf

Also see blog entry by Ben Goertzel:

*The management of uncertainty in the human brain: new

experimental insights*

"In other words, some of us maverick AI theorists have

been saying for a while that using just ONE number

(typically probability) to measure uncertainty is not

enough. Two numbers -- e.g. a probability and another

number measuring the "weight of evidence" in favor of

this probability (or to put it differently, the

"confidence" one has in the probability) -- are needed

to make a cognitively meaningful algebra of

uncertainty."

Link:

http://www.post-interesting.com/

I assume Bayesian probability theory could be

reformulated in terms of some kind of fuzzy set

theory. Then the notion of 'a probability of a

probability' would be referring to fuzzy sets

containing other fuzzy sets. The problem of sets

containing other sets has never been fully solved.

According to Roger Penrose:

'In fact, the way that mathematicians have come to

terms with this apparently paradoxical situation is to

imagine that some kind of distinction has been made

between 'sets' and 'classes'...Roughly speaking, any

collection of sets whatever could be allowed to be

considered as a whole, and such a collection would be

called a *class*. Some classes are respectable enough

to be considered as sets themselves, but other classes

would be considered to be 'too big' or 'too untidy' to

be counted as sets. We are not neccessarily allowed

to collect *classes* together, on the other hand, to

form larger entities. Thus 'the set of all sets' is

not allowed...but the 'class of all sets' is

considered to be legitimate...

There is something unsatisfactory about all

this...This procedure might be reasponable if there

were a clear-cut criterion telling us when a class

actually qualifies as being a set. However the

'distinction' appears often to be made in a very

circular way."

-Roger Penrose. 'The Road To Reality' , Page 373

(Hard-back version)

Paper on possible extensions to set theory:

http://web.mit.edu/dmytro/www/NewSetTheory.htm

"Abstract: We discuss the problems of incompleteness

and inexpressibility. We introduce almost

self-referential formulas, use them to extend set

theory, and relate their expressive power to that of

infinitary logic. We discuss the nature of proper

classes. Finally, we introduce and axiomatize a

powerful extension to set theory."

O.K, so... did I solve it? Are any of these ideas of

relevence?

"Till shade is gone, till water is gone, into the shadow with teeth bared, screaming defiance with the last breath, to spit in Sightblinder’s eye on the last day”

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