TITLE: Some questions about mathematical content of Gödel's Completeness Theorem
QUESTION [5 upvotes]: I'm trying to refresh my knowledge about mathematical logic and I'm still unsatisfied with my insight of Gödel's Completeness Theorem.
In my only "raid" into MathOverflow, I posed a similar question about Gödel's Completeness Theorem. 
I received a good answer from Joel David Hamkins about this issue, but I was not sufficiently prepared to prosecute that discussion.
Gödel's proof is equivalent to Konig's Lemma [Gödel's paper says "it follows by familiar arguments" and a footnote by the editor comments : "Apparently by Konig's infinity lemma"].
R.Smullyan, into his First-Order Logic (1968), in his exposition of Compactness Theorem for propositional logic, refers to Konig's Lemma (pag.31); but he also points out (pag.34) "that although we used both analytic tableaux and Konig's lemma in our proof of the compactness theorem, neither is really essential". This is not clear to me ...

Question. Is it possible to "avoid" $\mathsf {ZF}$ and formalize Gödel's original proof only into f-o $\mathsf {PA}$ ?

REPLY [4 votes]: The completeness theorem is stated for arbitrary (even possibly uncountable) theories. If you restrict the theorem to only countable theories, then it is possible to prove the completeness theorem in weak subsystems of second-order arithmetic, such as $\mathsf{WKL}_0$, which has a consistency strength lower than PA. 
The challenge with proving the completeness theorem in PA itself is stating the completeness theorem in PA itself. There is no good way to quantify over models in the language of first-order arithmetic, nor to quantify over arbitrary infinite theories (even countable ones). 
I have no idea about Kleene's comment.