Benjamin Rin

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5

A semantics for quantified modal logic is presented that is based on Kleene's notion of realizability. This semantics generalizes Flagg's 1985 construction of a model of a modal version of Church's Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church's Thesis and a variant of a modal set theory due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra's generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott's graph model (for the untyped lambda calculus) which witnesses the failure of the stability of non-identity.

Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the iterative conception of set has been widely recognized as insufficient to establish replacement and recursion, its supplementation by considerations pertaining to algorithms suggests a new and philosophically well-motivated reason to believe in such principles.

In [7], open questions are raised regarding the computational strengths of so-called ∞-α -Turing machines, a family of models of computation resembling the infinite-time Turing machine model of [2], except with α -length tape . Let TαTα denote the machine model of tape length α . Define that TαTα is computationally stronger than TβTβ precisely when TαTα can compute all TβTβ-computable functions ƒ: min2→min2 plus more. The following results are found: Tω1≻TωTω1≻Tω. There are countable ordinals α such that Tα≻TωTα≻Tω, the smallest of which is precisely γ , the supremum of ordinals clockable by TωTω. In fact, there is a hierarchy of countable TαTαs of increasing strength corresponding to the transfinite Turing-jump operator ▼. There is a countable ordinal μ such that neither Tμ⪰Tω1Tμ⪰Tω1 nor Tμ⪯Tω1Tμ⪯Tω1—that is, the models TμTμ and Tω1Tω1 are computation-strength incommensurable . A similar fact holds for any larger uncountable device replacing Tω1Tω1. Further observations are made about countable TαTα