Kurt Gödel: a logical thinker

Having talked about the turing machine and uncomputable problems to some extent in class i became much more interested in the beginnings of the modern computing era as we know it. one of the contributors to this movement towards computing is Kurt Godel. There were no readings assigned for him so i wanted to understand better his contribution to this time period. I found a biography describing briefly his life and his studies on logics. but this on only left me with more questions like: what did he do in the field of Logic mathematics?

What i found was a blog post about his work on the axiom of choice. this one caught my eye especially because in economics we have axioms of choice and i thought for a moment Godel might have been the one to pioneer these theories. but what i found left me even more confused. His mathematical thinking went way over my head. after staring at the strange symbols and explanations for several minutes i was simply more confused.
So i went to the idiot proof source: wikipedia. it was there that i found out that he is best known for his incompleteness theorems. They go something like this:

If the system is consistent, it cannot be complete.

The consistency of the axioms cannot be proven within the system.

"The basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system."



In other words, he helped to father the ideas we discussed in class last time that there are certain problems that cannot be solved. Personally, I think i would rather dwell on those problems that do have a solution, but i am glad that someone thought about these incomplete logics.