The Incompleteness Theor....

I hadn't actually read the section of Logicomix which contained the picture I posted as the heading for yesterday's entry and so did not realise that it was the moment that Russell discovered the self-containing set paradox - Does the set of sets which do not contain themselves contain itself? The answer being the simultaneously rather glib and profound - If it does it doesn't and if it doesn't is does! A lovely bit of self-reference. Actually I was going to write "deeply, profoundly" but that would be what I hope is a tautology.

My hypnagogic dreams for last night involved quite a lot about the Incompleteness Theorem. Principia Mathematica is Russell and Whitehead's attempt to define logic completely; it contains 362 pages that prove that 1+1 =2 and yet Kurt Godel proved that proof would always require something outside the proof meaning that mathematics will always be reliant on something else - see God Over Djinn etc. Of course this applies to the whole universe. Not that any of us will ever be able to define all knowledge but I have the nagging feeling that no matter what we try to do to extend our knowledge, something will always be out of our reach - an analogy to the "something outside the system" that Godel realised. This suggests that there is an infinity of extras outside any defined system. Is the whole Universe like this - is there always going to be an infinity of knowledge outside what we already know? If we assume there is a finite number of particles in the Universe, then there must be a finite amount of information in it - realising of course that a large finite number used as the basis for combinations to describe information results in a much greater number; however it is still finite. Then of course extending the system might apply to whatever there is outside the Universe that we have access to, though this seems to knock over whatever is considered the current First Cause. All this has given me a headache which seems well within the system and therefore probably on page 2 or 3 of PM so I think it is time to go off and think of something slightly simpler.