Thursday, September 18, 2008

daydreams on a strange loop...

I came up with these fanciful meanderings after reading Godel, Escher, Bach by Hofstadter a few years ago... I was thinking about it again today so I dug it out of some old linear algebra class notes.  It's fairly crazy in the details, but the "dance" of it is fun... don't take it too seriously, unless you take it as a warning of the dangers of sleep deprevation mixed with studies in linear algebra...  


The fundemental problem that Godel's Incompleteness Theorem describes is that any untrivial formal system is either incomplete or inconsistent.  However, a super-system may be built around the first system that does completely and consistently describe it!  The trick is, you have to go "outside" the system you want to describe in order to fully describe it (Hofstatder calls this a "strange loop").

The reason this isn't a terribly useful solution is that the super-system itself can be shown to be incomplete or inconsistent.  In fact, no matter how many systems you layer on top of each other to fix things, there is always one outermost layer that suffers from this problem.

So, combining two other things I read about (Godel numbering and Hilbert spaces) :
What if it were possible to somehow compose an infinite number of formal systems such that they converge on a finite system?  Could the outer "error" (required to describe inner systems completely) diminish wrt the total in a manner analogous to a limit?  Could the limit then approach a system that is both complete and consistent in spite of Godel's incompleteness theorem?
(ED: this suddenly reminded me of a recent article I read about Negative Databases for some reason...)

Godel numbering allows any theory or formal system to be represented in mathematical form. So, it follows that there must exist a set of Godel numbers that correspond to math in Hilbert spaces.  

Hilbert spaces are very interesting, because even though points in such spaces have an infinite number of coordinates, angles and distances measured between any such points are finite!  Weird huh?  (common applications are quantum physics, signal processing, fourier transforms, etc.)

(ED: heh, imagining extra dimensions reminds me of that great short story "Dreams in the Witch House" by H.P. Lovecraft... creepy!)

Since points in these spaces are vectors with an infinite number of coordinates, it seems reasonable that each point vector in this space could be interpreted as a Godel numbering representing a potential formal system.

Since there are an infinite number of points, there must be an infinite number of formal systems being represented... yet, it's possible to define a line of finite measure from the origin of this space to any coordinate point given an angle and a distance (using polar coordinates).  

A line represents an infinite set of points... does this imply that a finite Godel numbering can somehow represent an infinite set of formal systems?  

(ED: woah woah woah Tex, this is quite a leap...)

This raises even more questions:
What's the the representation of the differential term used in the limit?  
Does the differential actually converge towards a finite value?

(ED: it just kind of ends here... I probably wised up by this point.  Sorry, no breakthroughs!)

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