20 research outputs found
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Guessing genericity -- looking at parametrized diamonds from a different perspective
We introduce and study a family of axioms that closely follows the pattern of
parametrized diamonds, studied by Moore, Hru\v{s}\'ak, and D\v{z}amonja in
[13]. However, our approach appeals to model theoretic / forcing theoretic
notions, rather than pure combinatorics. The main goal of the paper is to
exhibit a surprising, close connection between seemingly very distinct
principles. As an application, we show that forcing with a measure algebra
preserves (a variant of) , improving an old result
of M. Hru\v{s}\'ak
The -Ramsey properties may be different
We give examples of -Ramsey spaces that are not -Ramsey under
several assumptions. We improve results from Kubis and Szeptycki by building
such examples from and
. We also introduce a new
splitting-like cardinal invariant and then show that the same holds under
.Comment: 29 page
Completeness for Flat Modal Fixpoint Logics
This paper exhibits a general and uniform method to prove completeness for
certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form
\gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the
language L\sharp (\Gamma) is obtained by adding to the language of polymodal
logic a connective \sharp\_\gamma for each \gamma \epsilon. The term
\sharp\_\gamma (\varphi1, . . ., \varphin) is meant to be interpreted as the
least fixed point of the functional interpretation of the term \gamma(x,
\varphi 1, . . ., \varphi n). We consider the following problem: given \Gamma,
construct an axiom system which is sound and complete with respect to the
concrete interpretation of the language L\sharp (\Gamma) on Kripke frames. We
prove two results that solve this problem. First, let K\sharp (\Gamma) be the
logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint
axiom and a least fixpoint rule, for each fixpoint connective \sharp\_\gamma.
Provided that each indexing formula \gamma satisfies the syntactic criterion of
being untied in x, we prove this axiom system to be complete. Second,
addressing the general case, we prove the soundness and completeness of an
extension K+ (\Gamma) of K\_\sharp (\Gamma). This extension is obtained via an
effective procedure that, given an indexing formula \gamma as input, returns a
finite set of axioms and derivation rules for \sharp\_\gamma, of size bounded
by the length of \gamma. Thus the axiom system K+ (\Gamma) is finite whenever
\Gamma is finite