Gödel's Incompleteness Phenomenon -- Computationally
It is argued that Gödel's completeness theorem is equivalent to completability of consistent theories, and Gödel's incompleteness theorem expresses that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any decidable theory can be extended to a complete and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively enumerable theories. Tarski's theorem is deduced by relativizing the incompleteness theorem to definable oracles.