On 14 Feb 2011, at 11:17, Stephen Paul King wrote:
From: Bruno Marchal
Sent: Monday, February 14, 2011 4:49 AM
To: [email protected]
Subject: Re: Plato's Heaven
On 14 Feb 2011, at 09:40, Stephen Paul King wrote:
snip
{SPK]
Allow me to add a comment to this brilliant argument. Following
Jason’s description of Platonia, Plato’s heaven, it would seem to
include all possible descriptions of itself and thus is in a way
like a set that contains all subsets of itself including at least
one subset that is identical to itself. Is this not a paradox?
>Not at all. UD* contains many UD* which contains many UD* ad
infinitum. There is no paradox because UD* is infinite. It is no
more paradoxical than the Mandelbrot set, which is "made of"
Mandelbrot sets.
[SPK]
OK, how does the UD* relate to Numbers?
By the UD, which can be seen as a number, which, in relation with
other number, does emulate the computations.
Is the UD* like an inch worm that travels Platonia measuring them?
UD* is the same as the truth an d falsity of the arithmetical
sentences having the simple existential form ExP(x, y) P decidable
(the Sigma_1 sentences). UD* is the tiny effective part of the
arithmetical truth.
Seriously, how is the difference between one number and another
knowable to the UD* if the UD* is almost every where in Platonia.
This UD* seems remarkably similar to an eternal, omniscient and
omnipotent being!
It is as eternal as the number 7.
It is not omniscient, nor omnipotent. It is less "scient" and less
"potent" than any Löbian machine, and of course a fortiori much less
so than the divine hypostases (the G* part of the G/G* variant).
UD* is just the collection of all computations. I suspect it to be
coded by the rational Mandelbrot set, which is a good illustration for
it. But some Penrose pavage will, do, or you can write it in Lisp, as
I did. I mean it is a very concrete and specific object, which,
assuming comp, can play the role of the effective (computably
generable) part of arithmetical Platonia (which is non computable).
Bruno
http://iridia.ulb.ac.be/~marchal/
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