I'm not sure I understand the argument, so here are some comments and

Wei Dai writes:
> Consider all possible worlds consistent with your memories and current 
> experiences. In other words, all possible worlds that contain at least one 
> observer with memories and current experiences exactly identical to yours. 
> Are there more than one such world?
> (yes)
> Is every one of these worlds isomorphic to some mathematical structure?
> (How do you define "mathematical structure"?)
> A set class.
> (then yes)

I think a set class is intended to be the most general mathematical
structure possible, right?  There is nothing more general?  And that is
the point of this proposed definition for the set of possible worlds,
that it would be the most general mathematical structure possible.

This line of reasoning is then more consistent with Tegmark's approach
than to Schmidhuber's, who suggests that universes are isomorphic to

One thing I've never understood about this approach is exactly how a
computation is considered to be a set.  Take a 1D cellular automaton
for example as a simple computational model, with a specified set of
rules and particular initial conditions.  How would its computational
trace be expressed as a set?  (Or use a different computational model
if even simpler.)

> What criteria would you use to decide which of these possible worlds is 
> actual, given that they are all consistent with your memories and current 
> experiences?
> (more observations, experiments)

First, what does it mean for a possible world to be actual?  Does it
mean, that is "the" world we live in?  Or could it include the notion
of actual worlds different from the ones we are experiencing?

For an example of the latter, consider the MWI where a binary measurement
has been made but I have not yet been made aware of the result.  Then
there are two universes which have split, both are actual, my mental
state is exactly the same in both, but I might consider that I am only
living in one of the two.

Second, what is the Bayesian answer getting at with more observations and
experiments?  You go on to critique the answer, but what was the thrust
of it in the first place?  What kind of observations and experiments
would he make, and how would that determine which were possible and
which were actual?

This relates to my first question as follows.  If you are asking which
world we "actually" live in, one source of uncertainty about possible
worlds is that we only have a finite amount of information about
the world.  For example, there are possible worlds where there is a
bird flying overhead right now, and some where there is not.  By going
outside I can narrow down the possibilities.

But if you are asking which worlds are "actual" and which are only
"possible", that is a philosophical question.  Some people believe
the other worlds of the MWI are actual and some are only possible, and
if this is ever decided it will be by very different means than going
outside and looking for birds.  It will be a philosophical argument,
one which ultimately will depend on observations (because that's all
we have to start with) but whose nature is quite different from going
outside and looking at the sky.

> Ok, but after every new observation or experiment, there will still be 
> more than one possible world that is consistent with the new emperical 
> result, right?
> (yes)
> So then what?
> (apply Bayes's rule)
> Where does the prior come from?
> (???)

When you say, so then what, what are you asking?  Simple observation lets
me narrow down the number of possible worlds.  There's no argument about
that, right?  I can rule out any world inconsistent with my observations.

But you, or the Bayesian, want to go further, and set up a probability
distribution over possible worlds, is that it?  That is the goal of
Bayes rule.  But to apply that you need a prior, and the Bayesian is
momentarily stumped.

> Do you assign a non-zero prior to the class of all sets being the actual 
> world?
> (yes)

So here the "actual" world is larger than just the one (apparent) world
that I observe, right?  Now you are raising the possibility that, in
effect, an infinite number of worlds exist?  In speaking of the class
of all sets being the actual world, you mean that there would be an
infinite number of apparent worlds, each one corresponding to, what,
a set?  A class?  Basically the Tegmarkian concept, except where he
speaks of axixomatizable mathematical structures, you are generalizing
even beyond that?

This philosophical proposal, while perhaps novel and initially unlikely
to the Bayesian, cannot be rejected out of hand, hence he has to assign
a non-zero prior probability to it, right?

> Pragmatically, how does that differ from assigning a prior of 1 for the 
> class of all sets?
> (What do you mean by "pragmatically"?)
> I mean are there any circumstances in which you'd act differently if you 
> assigned a prior of 1 instead?
> (no)

This is the part which really confuses me.  Assigning a prior of 1 would
mean that you are certain that the actual world is the class of all sets.
Assigning a prior of one in a billion would mean that you thought it
very unlikely.  Yet these two possibilites won't make any difference
in behavior?  Why not?

Which of these two things are you saying: (A) there is no difference
in behavior no matter what priors you hold; or (B) there is something
special about the prior probability of the class of all sets being the
actual world, that for that one special case it makes no difference how
you will behave whether its probability is 1 or 1/1000000000?

Neither of these is immediately plausible to me, but before breaking my
mind on one of them, I should know which one it is.

> So why not just assume that the actual world is the class of all sets?
> (My principles of reasoning do not allow me to do so.)
> If you go back and look at how those principles of reasoning were derived 
> or justified, it was on the basis of simplicity and avoiding absurd 
> actions ("absurd" being defined by intuition or common sense). The 
> assumption that the actual world is the class of all sets is equally 
> justified on the basis of avoiding absurd actions and is simpler than 
> having a prior over possible worlds, so why not?
> (...)

I can't speak to this part without understanding better why there is no
difference in behavior whether the class of all sets is assumed to be
the world or not.


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