GEB is a marvellous work that is accessible to anyone with reasonably good school grade maths. I chanced upon it by accident in the school library one day and was hooked after a few pages.
Anyway the crux of the matter is that you can very carefully construct a statement about a system that can't be either proven or disproven by that system! I don't have anything like the formal knowledge to really get to grips with it and the discussions on infinities and so on are pretty mind blowing. However, you feel that DH is imparting glimpses into the sheer beauty of the ideas he covers.
Wait 'til you discover what ricercar and quining is all about - bloody lovely. Just read it but take your time. There is something in there for everyone. You often get told by clever people about the links between maths, music and art. Mr H easily gives the best argument I've ever seen that attests to that being true, whilst giving your brain a right good kicking.
There was a Dutchman, a German and an Austrian who walked into a book ...
Close but not quite as that's an inconsistent statement.
"This statement is unprovable." is the approach Godel takes and eliminates the inconsistency. Either that statement is true, in which case it's unprovable, or it's false in which case there exists a proof of a false statement.
It obeys all the "laws" (OK vagaries) of English grammar. There is nothing in the rules of grammar that requires a statement in English to actually be self-consistent. The problem only surfaces once you attach the associated meanings to the various components.
I just picked a classic to start off my prior comment which morphed more into praise for GEB than Goedel's brain breaker.
I have dim memories of a cracking constructive argument starting off with some very basic axioms where one was written as S, and two as SS and so on, then it all went a bit mad but the genius of Hofstadter is to make all that impenetrable palava nigh on accessible to the layman (with a bit of effort from the reader).
The horrendous thing about Goedel is that the final flourish is clearly correct (I assume that responsible adults have filled in the formal bits, I'm sticking to the lies to children version). It is both terrifying and perhaps obvious at the same time. I can imagine the sense of dread when mighty edifices such as Herr Hilbert's suddenly looked a bit shady and then anger, followed by disbelief and finally acceptance as the big hitters really got to grips with it. The world hasn't come to an end because of Goedel but it certainly got a bit more interesting.
I think that it is almost comforting that we have a system that can be complicated enough be to capable of saying things about itself that can't be proven within itself. I think that there is a good chance that our system of mathematics will eventually become complicated enough and no more. Obviously there will always be spherical cows and some absolutely mad numbers and when I say eventually - that will take forever (nearly).
It is still a paradox, a two step one, that drags the whole system into being a paradox. The false statements that have a proof do already abstractly exist.
My take on that statement is that it is not saying anything about the world. It only refers to itself, making it a self-contained mini-universe with no relation to the real world.
So, since it's not saying anything, it's neither true nor false. Only something confusing that feels like it should have some meaning.
There are several levels that you can analyse this. Firstly: Is it English and is it grammatically correct? Seems OK but we can allow English to be a bit slack.
Now, how do we attach meaning to the actual statement. Do the English words actually mean the same thing to me as they do to you? Probably.
Should any statement be applicable to anything - world or otherwise? It is just a statement and not my best man's speech at my brother's wedding 25 odd years ago, which I'm sure you can appreciate that I consider that being rather more important.
So here we have a statement that is unable to be consistent and as you say, it sounds like it should have meaning but doesn't.
So we have a way of saying things with spoken language that are logically inconsistent that are also grammatically correct and that is a sort of flavour of what Goedel proved with his incompleteness theorem.
GEB is a marvellous work that is accessible to anyone with reasonably good school grade maths. I chanced upon it by accident in the school library one day and was hooked after a few pages.
Anyway the crux of the matter is that you can very carefully construct a statement about a system that can't be either proven or disproven by that system! I don't have anything like the formal knowledge to really get to grips with it and the discussions on infinities and so on are pretty mind blowing. However, you feel that DH is imparting glimpses into the sheer beauty of the ideas he covers.
Wait 'til you discover what ricercar and quining is all about - bloody lovely. Just read it but take your time. There is something in there for everyone. You often get told by clever people about the links between maths, music and art. Mr H easily gives the best argument I've ever seen that attests to that being true, whilst giving your brain a right good kicking.
There was a Dutchman, a German and an Austrian who walked into a book ...