What is the extent of the implications of Kurt Godel's two Incompleteness Theorms?

Question

Or what are some good web sites or books that I could read to find out more about this?

Answer 1

The incompleteness theorems "only" mean that any axiomatic system that is complicated enough to study axiomatic systems itself is not going to be proved entirely, it's as simple as that; the formalism and logicism were proven to be too optimistic. It concerns only formal systems that allow a specific conception of proof.

Don't believe the guys who are going to tell you things about the implications of those theorems in Life, the Universe, and Everything.  

Don't ask Regis Debray, that's for sure.

Answer 2

Simply stated it implies that there will always be true statements which cannot be proven to be true, within an axiomatic system. Godel's theory for the existence of something godlike is also worth examining. Although refuted by Hume on the definition of terms, it is a very interesting logical hypothesis.

Answer 3

Two books you want to read are: 'Godel, Escher, Bach' by Douglas Hofstadter and 'Incompleteness' by Rebecca Goldstein. Douglas Hofstadter's book is a classic, but it's a long read. Rebecca Goldstein's book looks at Godel from a Platonic perspective.