Continuous first order logic for unbounded metric structures

We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than the unit ball approach (which has been the common approach so far to Banach space structures in continuous logic), as well as of applying in situations where the unit ball approach does not apply (i.e., when the unit ball is not a definable set). We also introduce the process of single point \emph{emboundment} (closely related to the topological single point compactification), allowing to bring unbounded structures back into the setting of bounded continuous first order logic. Together with results from \cite{BenYaacov:Perturbations} regarding perturbations of bounded metric structures, we prove a Ryll-Nardzewski style characterisation of theories of Banach spaces which are separably categorical up to small perturbation of the norm. This last result is motivated by an unpublished result of Henson

A Generalization of Martin's Axiom

We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $\omega_1$ that don't seem to have been considered in the literature before.Comment: 36 page

A Search for various Double Beta Decay Modes of Cd, Te and Zn Isotopes

Various double beta decay modes of Cd, Zn and Te isotopes are explored with the help of CdTe and CdZnTe semiconductor detectors. The data set is splitted in an energy range below 1 MeV having a statistics of 134.5 g$\cdot$d and one above 1 MeV resulting in 532 g$\cdot$d. No signals were observed in all channels under investigation. New improved limits for the neutrinoless double beta decay of Zn70 of $T_{1/2} > 1.3 \cdot 10^{16} yrs$ (90% CL), the longest standing limit of all double beta isotopes, and 0$\nu\beta^+$EC of Te120 of $T_{1/2} > 2.2 \cdot 10^{16} yrs$ (90% CL) are given. For the first time a limit on the half-life of the 2$\nu$ECEC of $^{120}$Te of $T_{1/2} > 9.4 \cdot 10^{15} yrs$ (90% CL) is obtained. In addition, limits on 2$\nu$ECEC for ground state transitions of Cd106, Cd108 and Zn64 are improved. The obtained results even under rough background conditions show the reliability of CdTe semiconductor detectors for rare nuclear decay searches.Comment: Extended introduction and summar

A new foundational crisis in mathematics, is it really happening?

The article reconsiders the position of the foundations of mathematics after the discovery of HoTT. Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one. We reiterate a pluralist vision of the foundations of mathematics. The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.Comment: Final versio

Combinatorics of Open Covers VI: Selectors for Sequences of Dense Sets

We consider the following two selection principles for topological spaces: [Principle 1:] { For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space; [Principle 2:]{ For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, such that the union of these finite sets is dense in the space. We show that for separable metric space X one of these principles holds for the space C_p(X) of realvalued continuous functions equipped with the pointwise convergence topology if, and only if, a corresponding principle holds for a special family of open covers of X. An example is given to show that these equivalences do not hold in general for Tychonoff spaces. It is further shown that these two principles give characterizations for two popular cardinal numbers, and that these two principles are intimately related to an infinite game that was studied by Berner and Juhasz

The combinatorics of the Baer-Specker group

Denote the integers by Z and the positive integers by N. The groups Z^k (k a natural number) are discrete, and the classification up to isomorphism of their (topological) subgroups is trivial. But already for the countably infinite power Z^N of Z, the situation is different. Here the product topology is nontrivial, and the subgroups of Z^N make a rich source of examples of non-isomorphic topological groups. Z^N is the Baer-Specker group. We study subgroups of the Baer-Specker group which possess group theoretic properties analogous to properties introduced by Menger (1924), Hurewicz (1925), Rothberger (1938), and Scheepers (1996). The studied properties were introduced independently by Ko\v{c}inac and Okunev. We obtain purely combinatorial characterizations of these properties, and combine them with other techniques to solve several questions of Babinkostova, Ko\v{c}inac, and Scheepers.Comment: To appear in IJ