The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all facts about the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem shows that if such a system is also capable of proving certain basic facts about the natural numbers, then one particular arithmetic truth the system cannot prove is the consistency of the system itself.
Gödel's theorem shows that, in theories that include a small portion of number theory, a complete and consistent finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program. Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent.
There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.
He attempts to discover The Ultimate Question by extracting it from his brainwave patterns, as abusively suggested by Marvin the Paranoid Android, when a Scrabble-playing caveman spells out forty two. Arthur pulls random letters from a bag, but only gets the sentence "What do you get if you multiply six by nine"?
“ "Six by nine. Forty two."
"That's it. That's all there is."
"I always thought something was fundamentally wrong with the universe"
Six times nine is, of course, fifty-four. The program on the "Earth computer" should have run correctly, but the unexpected arrival of the Golgafrinchans on prehistoric Earth caused input errors into the system—computing (because of the garbage in, garbage out rule) the wrong question—the question in Arthur's subconscious being invalid all along.
The fact that truth transcends theoremhood , in any given formal system, is called "incompleteness" of that system .
Douglas R. Hofstadter: Gödel, Escher, Bach (1999) p. 86
What would there be that is not bizarre?
It seems we both agree that reductionist science only goes so far. How about a few more steps of progress using holistic approaches as advocated by Monica Anderson and others. Are there others? Who?