Godel's Theorem

[Imparcticle]

I was just reading about Godel's Theorem. I was unable to grasp the exact meaning of this sentence:

"These propositions, termed Gödel propositions, can be shown to exist by giving a procedure for returning a Gödel proposition from a set of axioms. This procedure forms the basis for proving Gödel's theorem"


from the paragraph:


Using the axioms we can derive propositions about the axioms. Gödel's theorem states that for any given axiomatic system there exists propositions that are either undecidable, or that the axiomatic system is incomplete. These propositions, termed Gödel propositions, can be shown to exist by giving a procedure for returning a Gödel proposition from a set of axioms. This procedure forms the basis for proving Gödel's theorem.


I don't understand what is meant by "..retruning a Godel proposition.."

[matt grime]

computer speak: just means given the set of axioms there is a method of examining them and then writing down a proposition which is consistent with the axioms, and whose negation is also consistent with the axioms. Meaning if we assume it true there are no contradictions, and if we assume its negation is true there are still no contradictions. Example, the continuum hypothesis and the axioms of ZFC (godel propositions are a little unrealistic; formally they are correct, but practically they aren't propositions you might come across 'naturally')

[Gza]

You've probably already heard about or even read it, but I recommend Godel Escher Bach by Douglas H. It was the only layman book I read that broke down Godels argument for me in an understandable fashion. I read it when I was like ten and it still made sense to me, despite my age. That is the sign of a good author, since the idea itself isn't at all intuitive or obvious, especially to a kid.