# Fundamental problems

From: Marc Geddes (m_j_geddes@yahoo.com.au)
Date: Fri Feb 17 2006 - 00:04:00 MST

>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."

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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