Re: ... cosmology? KNIGHT KNAVE
George,
At 21:17 20/07/04 -0700, George Levy wrote:
Bruno, John, Russell
I am half-way through Smullyan's book.
Nice! You will see how easy it will be to state precisely the main result
and the open problems in my thesis once you grasp the whole FU.
Of course, to really appreciate, there will be a need to study a little bit
two non-classical logics: intuitionist logic and quantum logic ('course).
It is an entertaining book for someone motivated enough to do all these
puzzles, but I think that what is missing is a metalevel discussion of
what all this means.
Sure! I see that now that we are going back to the basic you are
impatiently anticipating the things! That's makes me happy!
Mathematical fireworks occur because we are dealing with self-referential
systems. In the old days they may have called them
reflexive. Reflection is, I think, an essential component of conscious
thought.
Yes.
The type of reflection I have encountered so far in the book involves
infinite reflections which lead to paradoxes.
For example if someone says I am a knave then obviously we have a paradox.
I disagree!!!
I'm a little bit sorry because I have not resisted
putting a little trap in the problems yesterday.
Actually there is no paradox here!
(More below).
The human mind, however, does not have the capacity to deal with an
infinite number of reflections. (I think that you think that I think that
you think.).
You anticipate a little bit too much. In a sense
not only humans can deal with infinite
reflexion but machine can do aswell...I will not try to explain
now. In FU this is explained in the heart of
the matter section 25. To be honest Smullyan's
explanation is rather short. When we will arrive
at that stage it will be good to (re)read the diagonalisation
posts (accessible from my url).
There are two really fundamental theorems: their official
name are
1) The second recursion theorem of Kleene (2-REC)
2) The diagonalisation lemma (DL)
The first one can aptly be called (with the comp hyp) the
fundamental theorem of theoretical biology. It is the one I have use
to build amoeba (self-reproducing programs), planaria (self
regenerating programs) and dreaming machine (programs
capable of conceiving themselves in alternate anticipations).
Solovay's arithmetical completeness theorems of G and G* are
themselves the most pretty application of 2-REC. George Boolos
use the DL instead. They are quite related.
If, however, self referential systems are limited to a finite number of
reflections, such as the human mind is capable of, then these paradoxes
may go away. With one reflection a knave would says: I am a knight.
With two reflections he would say I am a knave. With three reflections,
I am a knight. With four I am a knave. and so on. With an infinite
number of reflections he would remain Forever Undecided.
I am not sure if Physics is derived from an ideal infinite
self-referential systems or from a more human and messy finite system and
I cannot think of an obvious and clear-cut justification for either
approach. What do you think?
Physics will come from the *machine* messy
finite system, not necessarily *human*.
Actually the machine's psychology G and G*
is also the psychology of some infinite machine
and there exists for each sort of alpha-machine
(alpha ordinal) a related physics. So by testing those
physics we will be able to evaluate our own degree
of comp/non-comp. My thesis shows that quantum sort of logic
appears right at the starting point alpha = omega point.
And now the solution of the problems:
(I see most people have find the solutions (and even new problem and
solution, thanks to John and Russell) ... modulo some details.
The native of that Island are all either knight or knaves
and knight always tell the truth, and knaves always lie.
You go there.
Problem 1. A native tell you I am a knight. Is it possible to deduce
the native's type?
Problem 2. You meet someone on the island, and he tells you
I am a knave. What can you deduce?
Problem 1. The answer is NO. The assertion I am a knight
can be made by a knight (who then tells the truth as it should),
but can be made by a knave (who then tells a lie as it should).
as it should giving the rule of the island. OK?
Problem 2. Someone says I am a knave. Suppose it is a
knave: well then he says the truth so it cannot be a knave.
Suppose it a knight: well then it is a lie so it cannot be a knight, either.
So it is neither a knight, nor a knave, but we have been said
that all the native of the island are either knight or knave.
So we can deduce that it cannot be a native of the
island. That someone (the trap!) is most probably
a (lying) tourist, or a mad explorator ... OK?
;-)
Bruno
http://iridia.ulb.ac.be/~marchal/
Re: All possible worlds in a single world cosmology?
In reply to posts by Hal Finney and Bruno Marchal-- Hal: I found the paper you referred to, and it certainly has some very interesting ideas, for example the idea that the arrow of time is actually an anthropic artefact. I admit that I have much reading to do if I am to understand the paper properly, but I am not sure what I was proposing - that all possible worlds will at some stage exist - is the same thing as the Poincare recurrences discussed in this paper. It possible that only a subset of possible events (everything that has occurred so far, and then some) will cycle endlessly, and if so, as Nietzche commented, that'll suck (or words to that effect). I probably gave the wrong example when I proposed as my unlikely event the formation of an exact copy of our solar system and its inhabitants in far future interstellar space; much more interesting would be a rather different copy, where you would be resurrected with intact personality and memories of your past life, with enhanced intelligence and physical abilities, and a whole new civilization with scientific wonders, intelligent aliens, and things so strange that no-one today has even imagined them, all to explore. Of course, you will also experience burning in hellfire as the flipside to this happy state, but who was it that said it was better to burn than to disappear? Bruno: I agree that my four assumptions are dubious, but I chose them, for the sake of argument, as being (a) most inimical towards Many Worlds theories, (b) closest to what most people would think of as common sense, and (c) least controversial/ most conservative in the scientific community. I do think they are internally consistent, even if they are completely wrong. I do not understand your comment that by saying the universe is unique, finite, expanding and cooling forever, it is contradictory to allow that my example of an unlikely event will occur as time approaches infinity. The increase in entropy and cooling which go with the model I suggested are average trends over time. It is possible within this long term decline to have pockets of order/ decreasing entropy, both in classical statistical mechanics and quantum mechanics. It is a mathematical fact, independent of the actual physics, that given enough time (and eternity is certainly enough time), any event that is possible, however close to zero its probability per unit time, will occur with probability arbitrarily close to 1. What rather surprised me, however, is the fact that the last statement is only true in general if the probability per unit time stays constant or increases with increasing time; if it decreases, limiting towards zero as time approaches infinity, then it is possible that this event, which still always has non-zero probability per unit time, may never actually occur. For example, if Pr(P)=1/(t^2), as t goes from 2 to infinity, the cumulative probability that P will occur at some point is 1/2. Stathis Papaioannou _ Protect your inbox from harmful viruses with new ninemsn Premium. Go to http://ninemsn.com.au/premium/landing.asp?banner=emailtagreferrer=hotmail
Re: ... cosmology? KNIGHT KNAVE
- Original Message - From: Russell Standish [EMAIL PROTECTED] To: John M [EMAIL PROTECTED] Cc: [EMAIL PROTECTED]; Bruno Marchal [EMAIL PROTECTED] Sent: Tuesday, July 20, 2004 7:29 PM Subject: Re: ... cosmology? KNIGHT KNAVE Russell, your solution (in your attachment) is the right one: the 'double reverse': eliminting the negative by double negation, while the 'double positive' still stays positive. ^^ What would your other brother say is the road to Baghdad? Then take the other direction! Cheers ^ How would that be altered in George Levy's nth 'reflection'? (just kidding). John M
