Boolos-style proofs of limitative theorems

In his famous paper announcing the incompleteness theorem, Gödel remarked that, though his argument is analogous to the Richard and the Liar paradoxes, “Any epistemological antinomy could be used for a similar proof of the existence of undecidable propositions.” ([7] Note 14). It is interesting that, despite the fact that the soundness of arguments like Gödel’s one built on self–reference (or diagonalization) was often questioned (of course, from a philosophical not a mathematical point of view), the first attempt to support Gödel’s claim and prove the theorem using another paradox (and hence without recourse to diagonalization) came only recently. In 1989, formalizing the Berry paradox consisting in the fact that the least integer not nameable in fewer than nineteen syllables has just now been named in eighteen syllables, G.Boolos proved the semantic version of the incompleteness theorem to the effect that there are arithmetical sentences that are true but unprovable in Peano arithmetic (see [3]). The proof, as Boolos notes at the end of his paper, “unlike the usual one, does not involve diagonalization”. Not much later, in a letter, he adds “What strikes the author as of interest in the proof via Berry’s paradox is [. . . ] that it provides a different sort of reason for the incompleteness [. . . ] ” (cf. [4]). Perhaps Boolos’s proof was one of the factors that have inspired a wave of “proving old results in a new way” (see e.g. [1] and the references given there). Nevertheless, unlike the proof theoretical methods used in both Gödel’s original proof and Boolos’s one, most of these new proofs apply sophisticated model theoretical methods that can hardly be considered “finitistic”. On the other hand, Boolos’s proof can straightforwardly be extended to yield simple proofs of some fundamental theorems that are related very closely to the incompleteness theorem and to each other. The two versions of Gödel’s first incompleteness theorem (the semantical and syntactical one describing respectively the relation between truth and provability and that of provability and refutability) together with their strengthening (the Gödel–Rosser theorem), Church’s theorem on the undecidability of provability, and Tarski’s theorem on the undefinability of truth, in a sense, constitute a complete circle of mutually related statements answering some basic questions on provability and truth. The close connection between these fundamental results is also witnessed by the fact that their standard proofs have essentially the same structure : they all can be derived from a general formal version of the Liar paradox, that is, they can be considered as different formal resolutions of this paradox (cf. [11]). Now, as we shall show below, almost the same can be said if we replace the Liar paradox by Berry’s one. Actually, without any essential modification, the idea underlying Boolos’s proof of incompleteness can be used to provide “diagonalization–free” proofs of all the basic limitative theorems mentioned above. After fixing notation and giving the definition of basic notions, we first mimic Boolos’s proof in a slightly more detailed form than that in which the original proof was given so that we can continue the proof in different directions, which is just what we shall do. Let us first fix any one of the standard first order languages of arithmetic. By a formula (resp. sentence, term etc.) we mean a formula (resp. sentence, term etc.) of this language. Theories are arbitrary sets of sentences. Robinson arithmetic (cf. [8] I.1.1) will be denoted by Q. We shall denote the standard model of Q (as well as its universe) by ω, and say that a sentence is true (resp. a set is definable, defined etc.) if the sentence considered is true (resp. the set is