Why, and how?
Can this be explained in a kind of dumbed-down way? (I am very not a mathematician.)
I am not a mathematician either - so anyone who wants, feel free to improve on this!
Gödel was interested in the question of what could be proved with a given set of axioms. An axiom is just something that you assume from the outset to be true. Mathematicians has believed that a small set of axioms were sufficient to create the whole of mathematics by just using them to create truths - which you could think of as extra, but redundant axioms. Thus, you only need a tiny number of axioms for integer arithmetic to re-create an awful lot of number theory maths.
However, Gödel showed that once you had enough axioms to describe integer arithmetic, something strange happened - there were facts that could not be proved or disproved. Of course, you could just add the new fact to the set of axioms (or you could add the negation of that fact if you preferred). That way you had an augmented set of axioms that encompassed that dodgy fact F, but his theorem shows that there is another fact F' waiting to mess up the new axiom set. You can carry this process on indefinitely, with the original nice clean axiom set getting grubbier and grubbier as you add in ad-hoc extra stuff!
Penrose is interested in whether consciousness could arise purely from a physical process. He observes that a physical process is analogous to an axiom set, in that it can generate new facts (or theorems if you prefer) in a mechanical sort of way - there are just way more axioms than a mathematician would care to use. This means that any ordinary physical process - particularly a computer program - will be subject to the restrictions of Gödel's theorem. Since mathematicians seem to transcend Gödel's theorem as if they don't work from a set of axioms, but see through to the truth of things in some other way (inevitably this is disputed...) Penrose concludes that mathematicians (and by extension all of us) are not conscious by virtue of an axiom set (again disputed...), however his conclusion - at least in public - is that there must be some new physics which is 'non-computable' so it can't be described as a set of axioms.
What I find fascinating about all this, is that if you look at an artificial intelligence program, they usually respond to simple textual inputs (say) with an appropriate response - so for example, they may have been given the fact that birds fly, so someone types in:
Q: "Can a sparrow fly?"
C: "Yes!"
So the computer has the right answer. but then you try:
Q: "Can a penguin fly?"
Now the computer will get the wrong answer until it is given a list of birds that can't fly. This might seem reasonable enough, but then it goes on:
Q: "Can a dead bird fly?"
To answer this, the computer needs to have yet another 'axiom' that dead birds of any species don't fly! But even this soon breaks down.
Q: "Can a dead bird fly in an airplane?"
This seems awfully analogous to the cascade of extra axioms that I described above.
The problem is that the axioms never capture the essence of what real people think about - which somehow transcends a mere mechanical process.
David