Absoluteness via Resurrection

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Velickovi\'c. We introduce a stronger form of resurrection axioms (the \emph{iterated} resurrection axioms $\textrm{RA}_\alpha(\Gamma)$ for a class of forcings $\Gamma$ and a given ordinal $\alpha$), and show that $\textrm{RA}_\omega(\Gamma)$ implies generic absoluteness for the first-order theory of $H_{\gamma^+}$ with respect to forcings in $\Gamma$ preserving the axiom, where $\gamma=\gamma_\Gamma$ is a cardinal which depends on $\Gamma$ ($\gamma_\Gamma=\omega_1$ if $\Gamma$ is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover we outline that simultaneous generic absoluteness for $H_{\gamma_0^+}$ with respect to $\Gamma_0$ and for $H_{\gamma_1^+}$ with respect to $\Gamma_1$ with $\gamma_0=\gamma_{\Gamma_0}\neq\gamma_{\Gamma_1}=\gamma_1$ is in principle possible, and we present several natural models of the Morse Kelley set theory where this phenomenon occurs (even for all $H_\gamma$ simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.Comment: 34 page

Algebraic Properties of Qualitative Spatio-Temporal Calculi

Qualitative spatial and temporal reasoning is based on so-called qualitative calculi. Algebraic properties of these calculi have several implications on reasoning algorithms. But what exactly is a qualitative calculus? And to which extent do the qualitative calculi proposed meet these demands? The literature provides various answers to the first question but only few facts about the second. In this paper we identify the minimal requirements to binary spatio-temporal calculi and we discuss the relevance of the according axioms for representation and reasoning. We also analyze existing qualitative calculi and provide a classification involving different notions of a relation algebra.Comment: COSIT 2013 paper including supplementary materia

Quantum field theory with a fundamental length: A general mathematical framework

We review and develop a mathematical framework for nonlocal quantum field theory (QFT) with a fundamental length. As an instructive example, we reexamine the normal ordered Gaussian function of a free field and find the primitive analyticity domain of its n-point vacuum expectation values. This domain is smaller than the usual future tube of local QFT, but we prove that in difference variables, it has the same structure of a tube whose base is the (n-1)-fold product of a Lorentz invariant region. It follows that this model satisfies Wightman-type axioms with an exponential high-energy bound which does not depend on n, contrary to the claims in the literature. In our setting, the Wightman generalized functions are defined on test functions analytic in the complex l-neighborhood of the real space, where l is an n-independent constant playing the role of a fundamental length, and the causality condition is formulated with the use of an analogous function space associated with the light cone. In contrast to the scheme proposed by Bruning and Nagamachi [J. Math. Phys. 45 (2004) 2199] in terms of ultra-hyperfunctions, the presented theory obviously becomes local as l tends to zero.Comment: 25 pages, v2: updated to match J. Math. Phys. versio