Abstract In his seminal 1931 paper, Gödel had shown the existence of finitary statements that required infinitary resources to prove them. This led him to postulate that the unlimited transfinite iteration of the power-set operation is necessary to account for finitary mathematics (Gödel’s Doctrine; GD ). GD garnered support over the course of the twentieth century because of the production of other finitary independence results (Paris–Harrington theorem, etc.). In order to counter the platonistic assumptions taken to be implicit in GD , Solomon Feferman challenged the significance of these independence results on the following grounds: (i) they are “artificial”; (ii) their proofs do not ultimately require infinitary set-existence principles but only the weaker 1-consistency of such principles; and (iii) the technique of “proof-theoretic reduction” shows that these finitary results can be proved using weaker resources than was initially thought. Although his technical results are unimpeachable, I argue that Feferman’s case against GD is overstated. First, via a detailed examination of particular independence results, I show that these should be construed as reasonably “mathematical” given their fruitfulness in mathematical practice. Second, I argue that, when we weigh Feferman’s anti-platonist program against its epistemic costs, his objections (ii) and (iii) are much less compelling. In particular, Feferman’s proposals leave us with (a) less than satisfactory proofs of individual results and (b) a less than satisfactory account of the interplay between metamathematics and mathematics, viz., a loss of global explanatory structure. That is, the higher infinite plays both a significant verificational role and a significant explanatory role in proving results about the finite. Thus, I provide a reassessment and partial vindication of an epistemic version of GD . I conclude by sketching a proposal for how this epistemic version of GD might be used to provide a novel argument for a form of mathematical platonism.
Abstract We study Kripke completeness for predicate modal logics with propositional axioms of bounded alternativity corresponding to the condition on Kripke frames that every world sees at most n worlds. We prove that minimal predicate modal logics with axioms of bounded alternativity, as well as their extensions with the axiom of reflexivity, are strongly Kripke complete, whereas the addition of the axiom of transitivity results in Kripke incomplete logics, both with and without reflexivity. We show that these incomplete logics are members of an uncountable family of Kripke incomplete logics and give explicit axiomatizations of completions for some of them.
Abstract The choice of Gödel numberings is a source of indeterminacy in metamathematics. For instance, Gödel’s second incompleteness theorem fails under certain Gödel numberings. Nevertheless, it is widely believed that our standard metamathematical theorems hold for all reasonable Gödel numberings. This article investigates what makes a Gödel numbering reasonable and develops a general framework for addressing philosophical questions about Gödel numberings and, more generally, the arithmetisation of syntax.
Astract We construct an uncountable sequence of minimal varieties of modal algebras which do not contain any modal algebra whose Boolean reduct forms a powerset algebra. By algebraizabilty and duality between modal algebras and neighborhood frames, this yields a result stating that there exists an uncountable set of Post complete and neighborhood unsound (strongly incomplete) congruential modal logics. Our result resolves an open problem posed by Peter Fritz in his paper from 2016. Furthermore, it shows a sharp contrast with the lattice of normal modal logics (NMLs), since every NML is known to be Kripke (and therefore neighborhood) sound.
