3
$\begingroup$

Possible Duplicate:
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)

$\endgroup$
6
  • 1
    $\begingroup$ Re-read the incompleteness theorems ;) $\endgroup$ Oct 3, 2012 at 17:13
  • 1
    $\begingroup$ Long story short: "complete" has at least two meanings. $\endgroup$ Oct 3, 2012 at 17:19
  • $\begingroup$ 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. $\endgroup$ Oct 3, 2012 at 17:21
  • 5
    $\begingroup$ 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! $\endgroup$
    – user642796
    Oct 3, 2012 at 17:27
  • $\begingroup$ @AndréNicolas I slap my forehead and stad corrected. $\endgroup$ Oct 3, 2012 at 17:57

2 Answers 2

9
$\begingroup$

"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.

$\endgroup$
5
$\begingroup$

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).

$\endgroup$

Not the answer you're looking for? Browse other questions tagged .