I realized recently that I did not understand well the completeness theorem of Godel, and how it interacts with the incompleteness theorems.
What I understand now (and you will see my understanding is not consistent !)
- Incompleteness means that, as long as I have some kind of arithmetic power (let's say multiplication) in my axioms, my theory is incomplete : there are (even FOL) formulae that can't be proved nor disproved.
- Completeness means that for any First Order Logic (FOL) formula, if this formula is true in all the models of my theory, there is a proof of that formula in my theory.
So I take a theory (with potent arithmetic) where integers are the same in every model of the theory (axiom of infinity ?), and I look at the FOL formulas on those integers. I feel that there is a contradiction between 1. and 2., because my formulae "live" only in one model of $\mathbb N$ but there are some of them without proof.
As I feel confident with the incompleteness theorem, I think I misunderstand the completeness theorem.
I found that question but I don't understand the answer that says the theory is not complete but the logic used in the language is. Can someone elaborate ?
Thank you very much !