On Tue, Apr 20, 2004 at 12:42:16PM -0700, Hal Finney wrote:
> 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.)

First express the natural numbers as sets in a standard way:

0 = {}, a+1 = {a}

Then express the state of the i-th cell of the CA at iteration t as

CA(i, t) = { { {}, {i} }, { {{}}, {{t}} }, 0 } if off,
           { { {}, {i} }, { {{}}, {{t}} }, 1 } if on

Then the CA's entire history can be expressed as the union of CA(i, t) for all i, t.

> 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?  

The answers are yes, yes, a set. 

> Basically the Tegmarkian concept, except where he
> speaks of axixomatizable mathematical structures, you are generalizing
> even beyond that?

It's an attempt to simplify the Tegmarkian concept. Certainly in the class
of all sets there are lots of sets that are not mathematical structures as
defined by Tegmark. It doesn't hurt the argument to leave them in, and
this way I don't have to spend several pages explaining what mathematical
structures are.

> 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?


> 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?

Because after changing your prior for the actual world being the class of
all set to 1, you can always find a new measure to adopt (expressing how
much you care about each part of the world) that would ensure that your
behavior is exactly the same as before.

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