Can we enumerate finite sequences which have no halting continuation?
Not an answer, but too long to be a comment:


  Specifically, as I understand it the Completeness Theorem guarantees a finite proof for any first-order logical statement which holds in all countable models of a first-order theory. (This is my restatement of Wikipedia's "Every syntactically consistent, countable first-order theory has a finite or countable model.")


I'm not a logician so I don't know if the latter statement is true, but it's definitely not a restatement of the first statement.

The statement you quoted from Wikipedia says if you have countably many axioms and they're consistent then there exists at least one model of the theory with countably many elements. 

What Godel's completeness theorem says that if a statement is true then it has a proof.

What you're missing to bridge the two is "a statement true in every countable model is true in every model."  I don't know if this is true or not (logicians would probably know off the tops of their heads).