Hi Everyone, (is that a well-defined word?)
First of all, I have not forgotten about an un-replied post from Bruno but I
didn't want to quip something away without thinking about it and crafting
something decent.
But in terms of why should Everything be well defined, let me say a few
things. Firstly, although my opinion should be irrelevant, I am not
convinced it should be well defined or even well definable perhaps by others
in some future. However, if one attributes some form of all inclusivity in
their Physics definition of Universe then it is really easy to "prove" the
universe +does not exist+ using Russell-like arguments. My basic intuition
on why I think Everything should be well definable, as rough and illogical
as this connection might seem to anyone outside my mind is this: the
universe (whatever its actual contents and laws and nature might be) exists
implies Everything is definable.
I do not assume the universe exists but I think that is something we can
agree that its existence goes beyond reasonable doubt. I loathe such
statements as "it would be absurd to conjecture that the universe does not
exist" as proof that it does. I want something deeper and maybe someday
that deeper reason for the existence and perhaps proof of its existence will
come. The former discussion (why it exists) +might+ have light shed on it
by an actual -proof- that it does exist. My guess: that will be as
un/satisfying as the question 'why do groups exist.'
Incidentally, a proof that the universe exists would prove unquestionably
that some forms of theism are correct such as pantheism and panentheism (I
am of the latter camp). Assuming all these conjectures such as the MUH work
out, of course.
In my analysis, and please provide us with -your- (your = Everyone, which is
also hopefully well defined!!) input, the conclusion I make from Russell's
argument is something like this:
[Thm 1 - Russell]
If one uses the context of classical (binary) logic, the axiom of foundation
implies that any 'set' theory with a universal set and subsets axiom is
inconsistent relative to classical logic.
proof: slight modification of Russell's argument.
{Thm 1, rephrased}
If one uses the context of classical logic, then the following three axioms
can't be simultaneously consistent whilst maintaining overall consistency
relative to binary, classical logic:
1. The Foundation Axiom --
http://en.wikipedia.org/wiki/Foundation_axiom
2. A universal set axiom - ie - there is an x such that for all y, y 'is
in' x
3. The subsets axiom --
http://en.wikipedia.org/wiki/Comprehension_axiom
Intuitively, the foundation axiom is meant to disallow sets that can be
elements of themselves and what are known as 'vicious circles'. The
Foundation axiom is what I think is similar to Euclid's parallel postulate
in its malleability. Modifying the FA and dropping it altogether has been
investigated for a few decades now by others.
The question then becomes why should this particular theorem, with its
assumptions, apply in cases for which the hypotheses are not assumable (such
as if they are known to be wrong)?
To suggest that this theorem by Russell automatically applies in all more
general cases, that it can automatically be made any stronger at all,
-might- end up being correct but this is a fundamental error in the
investigation of mathematics in my opinion. To suggest that de facto,
Russell's theorem applies in all cases when the hypotheses are relaxed is no
more valid than, for example, suggesting that the intermediate value theorem
is true for all non-continuous functions. Sure, the conclusion can be true
for some non-continuous functions but when that continuity hypothesis is
relaxed (such as, in the analogy, if we departed classical logic and went to
a logic based on [0,1] for example, or perhaps something that is far more
abstract than that where the truth set is the underlying set of an
MV-algebra), the conclusion (that the universal set does not exist, in the
analogy), is not correct. I am tentatively assuming that the three axioms
above are inconsistent relative to -the context of logic one is using- for
any finite logic but in the case with infinitely many truth values, my
intuition about classical logic begins to fail me completely and I have my
doubts as to whether this theorem is going to be true there, too.
Hence, this is why I think it is possible that in some sense a universal set
can exist and Everything would just be defined to be it, thus providing a
good definition for Everything, thus proving Everything is definable... Not
just some sense, but some interesting sense without seemingly needing to, at
random, sacrifice the axiom of choice as well (such as in some NF approaches
inspired by Quine (sp?)).
Then assuming the MUH, this set in that logic could be, literally, the
universe. That logic would be, in some sense, the +actual+ logic of the
universe.
The thing about set theory is that it was not designed to talk about
Physics. That's why it is too rich for Physics. It was meant to be about
math. This to me is like having non-Euclidean geometries that are wholly
irrelevant to GR. There are a select few types of non-Euclidean Geometries
applicable to GR, and hence, the observed part of the universe. There might
be that set theory out there analogous to that right non-Euclidean Geometry
that is applicable to Tegmark's MUH. Personally, I'm a big fan of dropping
the FA altogether as that seems to have no Physical motivation. When there
are things such as twin particles and spin oddities, I think may be a
"particle" that can be literally in itself like an unending self-similar
fractal particle of fractional dimension, or something as equally
counter-macro-intuitive. (I do have my doubts about the dimension of the
universe being a natural number, of course, although that would fit Occam's
Razor well if the data all fit that...)
On Mon, Apr 21, 2008 at 8:07 AM, Bruno Marchal <marchal.domain.name.hidden> wrote:
>
>
> Le 19-avr.-08, à 22:46, Günther Greindl a écrit :
>
> >
> > 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 ;-))
>
>
> You are on the right track :)
>
>
>
> >
> > One objection comes to mind immediately (already written above): why
> > should the Everything be well defined?
>
>
> I think you should ask Brian Tenneson. Personally I follow Plotinus and
> many Buddhist, Israelites, Muslims, and Christians, (and more) on the
> fact that the Big Picture (Brian's Universal set, I think) is not a
> member of what belongs to the Big Picture, so the everything cannot be
> described, nor really belonging to the everything.
> Even restraining ourself to the mathematical universe (i.e. adopting
> mathematicalism), they are good reasons to believe that the whole of
> mathematics cannot be considered as a mathematical object. One reason
> is that all attempts to make the mathematical everything purely
> mathematical have failed. Set theories (like ZF) miss some mathematical
> object and also, in some sense, produce to much unwanted mathematical
> objects. Lawvere made an attempt to get the Big Mathematical Picture as
> a category, but he fails, although he (re)deiscovered the topos
> catehories in the process, which captures more the mathematician's mind
> than the mathematical reality (realities). The set theory NF does
> capture Universal Sets, but then I have doubt that it does it in a
> genuine sense, or if it is not in inconsistent theory.
> Another reason is a not yet clear generalization of Gödel's
> incompleteness phenomenon along the line of Grim (ref in my thesis).
> With comp, we have also a universal object (the universal machine), but
> again, like the toposes, it is more on the side of the "mathematician"
> than of the "mathematical reality". With comp (actually with the
> arithmetical comp, i.e. with the lobian interview) we have good reason
> to suspect that the notion of "truth (about) us" *is* the big picture
> (Plotinus' One), but we can prove, then, that IF comp is true, and IF
> we are self-referentially correct (and thus lobian) then such a notion
> of truth cannot be expressed by any means by us. So I do thing that the
> everything, curiously enough, perhaps, does not belong to any
> everything thing we can defined (despite its name, probably because of
> its name).
> I am not sure about the motivation of Brian for his search of a
> universal set. May be he could tell us a bit more?
>
>
> bruno
>
> PS I intended to comment some other posts but I have work to do, and I
> think this week I will be very buzy. I keep reading the list and
> absence or delay in comments does not reflect lack of interest but
> accumulation of time consuming jobs those days. Normally I will have
> more time next week. Best regards to all of you.
> Ah ... A colleague of mine (the one who actually suggested me to look
> at that movie) told me that "The Prestige" is the most
> anti-computationalist movie ever conceivable, and he gave me a very
> good argument! Cannot say more now without trigging the "spoiler alert"
> ... :)
>
>
> >
> > 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
> >
> > >
> >
> http://iridia.ulb.ac.be/~marchal/ <http://iridia.ulb.ac.be/%7Emarchal/>
>
>
> >
>
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Received on Mon Apr 21 2008 - 15:52:01 PDT