Alastair Malcolm writes:
> I don't understand this last sentence - why couldn't he use the 'Lewisian
> defence' if he wanted - it is the Anthropic Principle (or just logic) that
> necessitates SAS's (in a many worlds context): our existence in a world that
> is suitable for us is independent of the uncountability or otherwise of the
> sets of suitable and unsuitable worlds, it seems to me. (Granted he does use
> the 'm' word in talking about level 4 (and other level) universes, but I am
> asking why he needs it to provide 'anthropic content'.)

I think the problem is that we can exist even in worlds that are not
particularly suitable, certainly worlds that are not as simple and
regular as our own.  Imagine a world in which a human being just pops
into existence but all around him is chaos.  There would be many more
such worlds than worlds where the human evolved in a sensible way.

Likewise we can't derive Occam's Razor or even the reasonableness of
induction (that the future will be like the past) without some kind of
measure that favors simple universes.  We could imagine a world in which
rabbits suddenly start to fly; or one in which horses suddenly start
to fly; or one in which trees suddenly start to fly; etc.  There are
apparently many more such worlds than ones which go on as they always
have.  So again without a measure that somehow favors the predictable
ones, we can't explain why induction continues to work.

> There are hints that it may be worth exploring fundamentally different
> approaches to the White Rabbit problem when we consider that for Cantor the
> set of all integers is the same 'size' as that of all the evens (not too
> good on its own for deciding whether a randomly selected integer is likely
> to come out odd or even); similarly for comparing the set of all reals
> between 0 and 1000, and between 0 and 1. The standard response to this is
> that one *cannot* select a real (or integer) in such circumstances - but in
> the case of many worlds we *do* have a selection (the one we are in now), so
> maybe there is more to be said than that of applying the Cantor approach to
> real worlds, and also on random selection.

I agree that it is hard to see how we could be experiencing one world
out of an infinite number of them, if they all had a uniform measure,
just as we can't pick a truly random integer.  Two solutions are, first,
to assume that worlds have non-uniform measure, so that each possible
world has a non-infinitesimal measure.  Or second, even if worlds do have
infinitesimal measure, observers can be considered as finite machines
which each span an infinite number of worlds.  Since an observer is only
finite, he can occupy a fraction of the multiverse which has non-zero
measure even if the worlds themselves have zero measure.

> I use the simple 'limit to infinity' approach to provide a potential
> solution to the WR problem (see appendix of
> - Russell's paper is not
> too dissimilar in this area, I think. This approach seems to cover at least
> the 'countable' region (in Cantorian terms), and also addresses the above
> problems (ie odd/even type questions etc). The key point in my philosophy
> paper is that it is mathematics (and/or information theory) that is more
> likely to map the objective distribution of types of worlds, compared to the
> particular anthropic intuition that is implied by the WR challenge.

I should note that I wrote the above comments before reading your paper,
and I see that you are familiar with the problems of chaotic universes and
induction failure that I mentioned.

Your solution is essentially the one we often discuss here, which is that
universes with shorter descriptions should be of higher measure than
those with longer descriptions, simply because a higher percentage of
infinite strings have a particular short-string prefix than a particular
long-string prefix.  There are various potential objections to this but
it is an attractive principle and seems to get us a long way towards
where we want to go.

> A final musing on finite formal systems: I have always
> considered formal systems to be a provisional 'best guess' (or *maybe* 2nd
> best after the informational approach) for exploring the plenitude - but it
> occurs to me that non-finitary formal systems (which could inter alia
> encompass the reals) may match (say SAS-relevant) finite formal systems in
> simplicity terms, if the (infinite-length) axioms themselves could be
> algorithmically generated. This would lead to a kind of 'meta-formal-system'
> approach. Just a passing thought...

My guess is that any such system, where infinite axioms could be generated
from a finite starting point, would be equivalent to just using a finite
axiom set.

Hal Finney

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