Some questions about mathematical content of Gödel's Completeness Theorem

Solution 1:

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).

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