On proofs of the incompleteness theorems based on Berry's paradox by Vopenka, Chaitin, and Boolos

By formalizing Berry's paradox, Vopenka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We shall show also that Vopenka's proof can be reformulated in arithmetic by using the arithmetized completeness theorem.