Gödel also considered the case where there are a countably infinite collection of formulas. Using the same reductions as above, he was able to consider only those cases where each formula is of degree 1 and contains no uses of equality. For a countable collection of formulas of degree 1, we may define as above; then define to be the closure of . The remainder of the proof then went through as before.
gödel also considered the case where there are a countably infinite collection of formulas using the same reductions as above he was able to consider only those cases where each formula is of degree 1 and contains no uses of equality for a countable collection of formulas of degree 1 we may define as above; then define to be the closure of  the remainder of the proof then went through as before