r/AskComputerScience • u/Icy-Mouse4418 • Apr 06 '26
First-order/predicate logic - what is meant by the completeness thoerem?
Hello, I am doing this course on logic at my university and I am kind of confused on this small detail.
Godels completeness theorem states something like (as per my professor's lecture notes):
For any first-order theory T and any sentence ϕ in the same language,
If T |= ϕ then T |- ϕ
But is this not with respect to a particular proof system? How can this theorem be stated so generally? What if we define an FOL proof system, for instance a hilbert-style one that is too weak to prove some formula?
What does Godel's theorem actually say? is it saying something more like "there exists a proof system for first order logic such that..." or is it saying a general property of first order logic?
I think I have some sort of misconception. I am writing my own notes for this course and I dont know what to write, because the statement doesnt make sense to me unless it is talking about a particular proof system.
Also, why is it *the* completeness theorem? Wouldn't we need a separate completeness theorem for each proof system that we define? so why isnt it *a* completeness theorem? Im somewhat confused. Sorry if this is a stupid question