[FOM] Godel's First Incompleteness Theorem as it possibly relates to Physics

Brian Hart hart.bri at gmail.com
Mon Oct 13 17:24:03 EDT 2008

On Sun, Oct 12, 2008 at 8:54 PM, Paul Budnik <paul at mtnmath.com> wrote:
> A physical theory must be at least potentially infinite for Gödel's
> first Incompleteness Theorem to apply. This means both time and the size
> of structures that can be embedded within it must have no finite bound.
> At the same time Gödel's proof insures that there is always the
> potential for something new and different in such a universe.

Space-time may be emergent from something more fundamental, though
(space-time foam?).  It is unknown whether space-time is infinite, but
if Guth's theory of eternal inflation is correct, then in some sense
the multiversal structure of space-time is infinite in extent, but may
be causally inaccessible to beings within a given branch of it.  An
ontological question regarding the nature of creativity presents
itself: is "newness" just a rehashing of the "old" or is it truly new?
 This question relates to creativity of all kinds: artistic,
mathematical, ontological and is of great metaphysical importance.

> I have long been fascinated by possible connections between Gödel's
> incompleteness results and biological evolution in a potentially
> infinite universe. For example, the one way around the limitations of
> Gödel's proof in a deterministic universe is through ever expanding
> diversity where an ever increasing number of alternative paths are
> pursued without selecting which is correct. With such a process, every
> truth of first order arithmetic can be explored by some path. I suspect
> it is not a coincidence that biological evolution, which created the
> mathematically capable human mind, seems to pursue expanding diversity
> whenever resources make this practical.

Chaitin has pursued investigations into the evolutionary development
of human creativity, particularly the mathematical type, using his
theory of algorithmic information which you may find interesting.

> I created a half hour video "Mathematical Infinity and Human Destiny"
> (http://video.google.com/videoplay?docid=-8677521434864225474) that
> offers philosophical speculation about this and related ideas along with
> scenes of nature as metaphors for the philosophy.

Thank you for sharing this excellent and informative video.

> Paul Budnik
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