I think some clarification is in order.
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There is nothing wrong with Gödel’s Incompleteness Theorem.
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There is nothing wrong with Mises’ Action Theorem.
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There is nothing wrong with Praxeology (The Action Theorem applied)
There is a reason the Action Theorem is not analytical a priori, but synthetic a priori.
Mises, Rothbard, Hoppe would all agree that the validity of the Action Theorem does not rest on pure formal logic. Once validated, however, it becomes possible to deduce praxeological laws logically.
Perhaps we should add Alonzo Church just for good measure? Entscheidungsproblem and all that.