This statement is not provable.
Let 's call the above statement S. So what S says is thatS is not provable
If this statement is false (meaning S is false) then the opposite statement must be true. The opposite statement is
S is provable
and it must be true. But wait a minute. If S is provable, meaning that there exists a proof of its truth, then S is true. But wait a minute again. We started by assuming that S is false. A contradiction... This means that our initial assumption was wrong and therefore S is true. But what does this mean ? This means that what it says is true, S is actually not provable. So S is true but non-provable. This is non-completeness. Our inability to reach every existing truth. Simple, elegant and yet so deep.
5 comments:
Excellent explanation! But what triggered you to write about it?
I was certain that especially you would like it. I 'm glad I was right. Actually, this post is not an explanation. It is an (over)simplification of the actual proof that Goedel constructed. I 'm reading a book about Goedel 's theorems these days and I liked so much the part about the proof to the first theorem that I felt I had to share.
If it is Nagel's "Goedel's proof" then you should also read Goedel, Nagel, minds and machines.
I should also try and at last finish "Goedel, Escher, Bach"
A good book is also "Infinity and the mind" by Rudy Rucker
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