What is the proof that First-order logic is complete?

Question

According to Wikipedia, first order logic is complete. What is the proof of this?

(Also, in the same paragraph, it says that its undecidable. Couldn't you just enumerate all possible proofs and disproofs to decide it though?)

Answer 1

I won't cover why first order logic is complete - an adequate textbook on mathematical logic will provide the proof. I recommend Mathematical Logic by Kunen, but it may be a bit more extensive than your needs.

As for decidability: Every necessarily true statement is provable, and every necessarily false statement has a proof of its negation (that's what it means to be "complete") but there are statements which are neither. For example, $(\forall x)P(x)$ may be either true or false depending on the universe of discourse and the definition of $P$. On such a statement, your proposed algorithm would never halt, because it would never find a proof or a disproof; but at no point could we be sure that no proof or disproof exists.

Answer 2

I would recommend Lou Van den dries' lecture notes in first order logic. He does it in three chapters. Here's a link https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www.math.uiuc.edu/~henson/Math570/Fall2009/Math570notes.pdf&ved=0ahUKEwiR--aI5KnPAhVS0GMKHdYADEAQFggbMAA&usg=AFQjCNFRgSAJkPtYyywzaa5YWygqpRZAhQ