ZFC set theory,first order theory

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What is the difference between Gödel's Completeness and Incompleteness Theorems?
what is the relationship between ZFC and first-order logic?

I am a bit confused by a few things that I have read recently.

I have read that ZFC is a first order theory and that any part of mathematics can be expressed in ZFC. Now I know that first order logic is complete, however this would seem to contradict the incompleteness theorems (with I have a basic understanding of). I was wondering where I have gone wrong?

Thanks very much for any help (sorry for the silly question)

First order logic is a complete theory (I think). But ZFC consists of first order logic plus the relation $\epsilon$ and the axioms of set theory and is incomplete. — Michael Greinecker, Oct 03 '12 at 17:21
Proof once again the Gödel was the greatest logician ever: not only did he prove an amazing theorem in his dissertation, but about two years later he proved its negation! — user642796, Oct 03 '12 at 17:27
http://math.stackexchange.com/questions/14709/what-is-the-difference-between-godels-completeness-and-incompleteness-theorems/14710#14710 — Andrés E. Caicedo, Oct 03 '12 at 19:54

score 10 · Answer 1 · answered Oct 03 '12 at 17:30

"Complete" means two different things for a logic (such as first-order-logic) versus for a theory in that logic.

A logic is complete iff: Every sentence that has no counterexample-model can be proved.

A theory is complete iff: Every sentence that has no proof-of-its-negation can be proved.

First-order logic is complete in the first sense. ZFC is (assuming it is consistent) incomplete in the second sense -- that is, there are sentences that ZFC neither proves nor disproves. That's completely compatible with the logic being complete; it just means that for each such sentence there are models of ZFC where it is true, and other models where it is false.

score 5 · Answer 2 · answered Oct 03 '12 at 17:32

As in the comments is said, the word 'complete' has 2 different meanings.

That the first order logic is complete, is meant that it is complete w.r.t the corresponding first order models, that is: a formula is valid in all models iff it has a proof (a deduction consisting of finitely many formulas, using some specific deduction rules, like modus ponens..)

That ZFC is incomplete, is meant it is incomplete as an axiom system: there is a formula $\phi$ such that neither $\phi$ nor $\lnot\phi$ is not provable from ZFC. (And, in fact, it will be still incomplete if adding any more axioms).

ZFC set theory,first order theory

2 Answers2