On Apr 19, 3:46 pm, Günther Greindl <[EMAIL PROTECTED]>
> Dear Nichomachus,
> > decision. If she measures the particle's spin as positive, she will
> > elect to switch cases, and if she measures it with a negative spin she
> > will keep the one she has. This is because she wants to be sure that,
> > having gotten to this point in the game, there will be at least some
> > branches of her existence where she experiences winning the grand
> > prize. She is not convinced that, were she to decide what to do using
> > only the processes available to her mind, she would guarantee that
> > same result since it is just possible that all of the mutiple versions
> > of herself confronted with the dilemma may make the same bad guess.
> I have also thought along these lines some time ago (to use a qubit to
> ensure that all outcomes are chosen, because one should not rely on
> one's mind decohering into all possible decisions).
> The essential question is this: what worlds exist? All possible worlds.
> But which worlds are possible? We have, on the one hand, physical
> possibility (this also includes other physical constants etc, but no
> totally unphysical scenarios).
> I have long adhered to this "everything physically possible", but this
> does break down under closer scrutiny: first of all, physical relations
> are, when things come down to it, mathematical relations.
> So we could conclude with Max Tegmark: all possible mathematical
> structures exist; this is ill defined (but then, why should the
> Everything be well defined?)
> Alastair argues in his paper that everything logically possible exists
> (with his non arbitrariness principle) but, while initially appealing,
> it leads to the question: what is logically possible? In what logic?
> Classical/Intuitionist/Deviant logics etc etc...then we are back at
> Max's all possible structures.
> For all this, I am beginning very much to appreciate Bruno's position
> with the Sigma_1 sentences; but I still have to do more reading and
> catch up on some logic/recursion theory for a final verdict ;-))
> One objection comes to mind immediately (already written above): why
> should the Everything be well defined?
> To go back to your original question: to consider if both variants are
> chosen by the player of the game by herself (without qubit) seems to
> depend on which kind of Everything you choose. And that, I think, is the
> crux of the matter.
> Cheers,
> Günther

Thank you for your illuminating comments, Günther. And though
Tegmark's ensemble may be less than well-defined right now, there are
other ensembles that are. My understanding of the Universal Dovetailer
is that it will generate the output of every possible computer
program, which, assuming that our universe is computable, implies that
it contains ours and every other possible version of our universe. And
unless there are any mathematical entities or structures in Max
Tegmark's ensemble that are not computable, then Tegmark's enseble
should be a subset of Schmidhuber's. On this note I can't do any
better than Russell's discussion in section 3.2 of Theory of Nothing,
which says that Schmidhuber's plentitude should properly be considered
a subset of Tegmark's ensemble. Are there any "Mathematical
Structures" that are not computable? Surely any finite axiom system,
if consistent, would have a finite number of non-trivial theorems. It
is said that a program could be written to generate all theorems of
any consistent axiom system, so that would seem to imply
computability. (Although Goedel's theorem indicates that any system of
sufficient complexity cannot be both consistent and complete, so it
follows that consistent axiom systems of sufficient complexity will
allow for the existence of undecidable propositions. But what bearing
this has on the present discussion about the computability of these
systems is sort of unclear to me.) What would constitute an
uncomputable mathematical structure? I don't know, but I admit that my
ignorance on the subject doesn't demonstrate their non-existence.

And yes, Günther, I agree with your wholeheartedly that "physical
relations are mathematical relations" at their core. However, simply
because a mathematical expression may model a given physical process
or relationship leaves us in the dark as to the reason why this
equation models this particular phenomenon. Feynman gives as example
in his book The Charater of Physical Law of an equation used in
electrolysis that relates the current, the time exposed, and the
concentration of the solution to the amount of a metal that is
deposited. But the relationship so expressed is clearly seen to be a
result of physical processes and not to be a consequence of more
general principles. Saying that X is physically possible must be
equivalent to saying that X necessarily stands in an allowable
relationship to the fundamental physical process of the world.

For example: imagine a simplified physics, say like a cellular
automaton, that is able to support living things, yet that will never
allow for it to emerge spontaneously. Would Tegmark's ensemble include
configurations of it exhibiting living things? Schmidhuber's evidently
would. I realize that this example assumes a great deal, and therefore
may be worthless. But we know from Stephen Wolfram that there are even
one dimensional CAs that are universal in the sense that they may be
used to instantiate a universal turing machine. It's a short hop to
the assumption that perhaps some definite level of complexity is
required for life, though it may be less than that required to have it
develop on its own.

So regarding the possibility of possible worlds: Could it be the case
that there exist worlds where John McCain has been President of the
USA since 2000, and yet no worlds where John Howard is still the Prime
Minister of Australia?

It seems like a cop-out to point out that if our universe is infinite
and exhibits uniform distribution of matter that there exists all
configurations of matter somewhere in it. Definitely less meaningful
than saying exactly which branches are possible are which are not.

I have more to say about this but I am afraid to ramble on any more
than I already have since I fear that what I have already written is
more confused than I am.  :)

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