Goedel's incompleteness theorem
To paraphrase Boolos paraphrasing Goedel paraphrasing himself:
It is true that 2 plus 2 is 4. I hope you can at least believe that much. Of course, it can be proved that 2 plus 2 is 4. It can also be proved that it can be proved that 2 plus 2 is 4. Moreover, it can be proved that it can be proved that it can be proved that 2 plus 2 is 4. We could continue in this seemingly trivial manner, but what we are hoping for is that 2 plus 2 is not 5. Fortunately, it can be proved that 2 plus 2 is not 5, and that too can be proved. Now just to be on the safe side what we really should ask is if it can be proved that it can’t be proved that 2 plus 2 is 5. The catch is that it can’t! If it could, then it can be be proved that 5 is not 5, which is absurd. In fact, no claim of the form “claim X can’t be proved” can be proved. So thanks to Goedel and his insanity it can be proved that if it can be proved that it can’t be proved that 2 plus 2 is 5 , then it can be proved that 2 plus 2 is 5. I would rather take an incomplete theory over an inconsistent one any day. Wouldn’t you?
I’ll leave whoever remains interested with their choice of some pretty tanpura drones in the pitches of C# and A, respectively (which I suggest listening to with eyes closed and nice headphones).
C#
A
