I am currently learning about Godel's Incompleteness Theorems and would like to check that my understanding is correct, and discuss the real world implications that follow. Does anyone here have enough understanding of his work or of formal maths or logic to carry on a productive discussion? I am especially interested in gaining a deeper understanding of what I think are the implications for science, religion, and AI programming, not just for maths.
Ultimately, I believe that the Incompleteness Theorem must be viewed as a corollary to Computability Theory, for while the Incompleteness Theorem came first, it was only a specific instance within Turing and Church's Computability Theory which can apply the principles of the Incompleteness beyond Number Theory to all other fields of Mathematics and Computer Science. Essentially the Incompleteness theorem can logically be derived from the halting problem. I dont know what your experience with Computation and Complexity theory is, but this site offers a good introduction:
http://plato.stanford.edu/entries/computability/As far as discussing the theory, I can give it a go, I will confess that my Computability theory is a bit rusty as I haven't used it in a few years, but it would do me good to brush up on it anyway (but I must warn you, while I always prefered computability theory, I was always much better at complexity theory). However, I dont know how popular this type of conversation would be, I tried to introduce a few class mates at Holy Cross to the Subject of Computability, but the usual conclusion to this introduction was a mocking of my textbook for it's claim to be entirely self-contained, requiring no prior knowledge -- which was technically true, but extensive mathematical experience is generally required to realize this.
If you simply want to discuss the incompleteness theorem in laymans terms and apply them to non-mathematical situatoins, that's pretty straightforward, the logical conclusion is relativism, but if you want to actually make progress I would really suggest the conversation centre around Computability.
As for a discussion of science, religion, etc. appart from mathematics, this quite confuses me. Did not Pythagoras teach us that 'Number is the ruler of forms and ideas and the cause of gods and demons;' then again, he also didn't believe in irrational numbers.
In any case, if you want to discuss something, shoot; no need to ask permission.
