About First Order Logic an Godel incompleness theory.

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There's something I'm failing to grasp: Are Gödel's theorems of incompleteness a first order logic result or a second order logic result? What confuses me is the fact that in Gödel's original work, he talks about predicate variables which I believe are second order entities, however I think that by using "Type Theory" it is possible to talk about predicate variables with out using second order logic. Is this correct?