811 research outputs found

    The Galvin property under the Ultrapower Axiom

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    We continue the study of the Galvin property. In particular, we deepen the connection between certain diamond-like principles and non-Galvin ultrafilters. We also show that any Dodd sound ultrafilter that is not a pp-point is non-Galvin. We use these ideas to formulate an essentially optimal large cardinal hypothesis that ensures the existence of a non-Galvin ultrafilter, improving on results of Benhamou and Dobrinen. Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a κ\kappa-complete ultrafilter on κ\kappa has the Galvin property if and only if it is an iterated sum of pp-points

    Transferring Compactness

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    We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a cardinal κ\kappa that is nn-dd-stationary for all n∈ωn\in \omega but not weakly compact. This is in sharp contrast to the situation in the constructible universe LL, where κ\kappa being (n+1)(n+1)-dd-stationary is equivalent to κ\kappa being Πn1\mathbf{\Pi}^1_n-indescribable. We also show that it is consistent that there is a cardinal κ≤2ω\kappa\leq 2^\omega such that Pκ(λ)P_\kappa(\lambda) is nn-stationary for all λ≥κ\lambda\geq \kappa and n∈ωn\in \omega, answering a question of Sakai.Comment: Corrected some typo

    Hilbert Spaces Without Countable AC

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    This article examines Hilbert spaces constructed from sets whose existence is incompatible with the Countable Axiom of Choice (CC). Our point of view is twofold: (1) We examine what can and cannot be said about Hilbert spaces and operators on them in ZF set theory without any assumptions of Choice axioms, even the CC. (2) We view Hilbert spaces as ``quantized'' sets and obtain some set-theoretic results from associated Hilbert spaces.Comment: 51 page

    Views from a peak:Generalisations and descriptive set theory

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    This dissertation has two major threads, one is mathematical, namely descriptive set theory, the other is philosophical, namely generalisation in mathematics. Descriptive set theory is the study of the behaviour of definable subsets of a given structure such as the real numbers. In the core mathematical chapters, we provide mathematical results connecting descriptive set theory and generalised descriptive set theory. Using these, we give a philosophical account of the motivations for, and the nature of, generalisation in mathematics.In Chapter 3, we stratify set theories based on this descriptive complexity. The axiom of countable choice for reals is one of the most basic fragments of the axiom of choice needed in many parts of mathematics. Descriptive choice principles are a further stratification of this fragment by the descriptive complexity of the sets. We provide a separation technique for descriptive choice principles based on Jensen forcing. Our results generalise a theorem by Kanovei.Chapter 4 gives the essentials of a generalised real analysis, that is a real analysis on generalisations of the real numbers to higher infinities. This builds on work by Galeotti and his coauthors. We generalise classical theorems of real analysis to certain sets of functions, strengthening continuity, and disprove other classical theorems. We also show that a certain cardinal property, the tree property, is equivalent to the Extreme Value Theorem for a set of functions which generalize the continuous functions.The question of Chapter 5 is whether a robust notion of infinite sums can be developed on generalisations of the real numbers to higher infinities. We state some incompatibility results, which suggest not. We analyse several candidate notions of infinite sum, both from the literature and more novel, and show which of the expected properties of a notion of sum they fail.In Chapter 6, we study the descriptive set theory arising from a generalization of topology, κ-topology, which is used in the previous two chapters. We show that the theory is quite different from that of the standard (full) topology. Differences include a collapsing Borel hierarchy, a lack of universal or complete sets, Lebesgue’s ‘great mistake’ holds (projections do not increase complexity), a strict hierarchy of notions of analyticity, and a failure of Suslin’s theorem.Lastly, in Chapter 7, we give a philosophical account of the nature of generalisation in mathematics, and describe the methodological reasons that mathematicians generalise. In so doing, we distinguish generalisation from other processes of change in mathematics, such as abstraction and domain expansion. We suggest a semantic account of generalisation, where two pieces of mathematics constitute a generalisation if they have a certain relation of content, along with an increased level of generality

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    The six blinds and the elephant or an interdisciplinary selection of measurement features

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    We propose here selected actual features of measurement problems based on our concerns in our respective fields of research. Their technical similarity in apparently disconnected fields motivate this common communication. Problems of coherence and consistency, correlation, randomness and uncertainty are exposed in various fields including physics, decision theory and game theory, while the underlying mathematical structures are very similar.Comment: 28 pages, 11 figures, 1 table. Proceedings of the XL Workshop on Geometric Methods in Physics, Bialowieza, 202

    More Ramsey theory for highly connected monochromatic subgraphs

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    An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey Theorem on uncountable cardinals asserting that if we color edges of the complete graph we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hru\v{s}\'ak and Shelah are answered by showing that assuming the consistency of suitable large cardinals the following are relatively consistent with ZFC\mathsf{ZFC}: κ→hc(κ)ω2\kappa\to_{hc} (\kappa)^2_\omega for every regular cardinal κ≥ℵ2\kappa\geq \aleph_2 and ¬CH+ℵ2→hc(ℵ1)ω2\neg\mathsf{CH}+ \aleph_2 \to_{hc} (\aleph_1)^2_\omega. Building on a work of Lambie-Hanson, we also show that ℵ2→hc[ℵ2]ω,22\aleph_2 \to_{hc} [\aleph_2]^2_{\omega,2} is consistent with ¬CH\neg\mathsf{CH}. To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.Comment: Number 1242 on Shelah's publication list. 18 page

    L1\mathrm{L}^1 full groups of flows

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    We introduce the concept of an L1\mathrm{L}^{1} full group associated with a measure-preserving action of a Polish normed group on a standard probability space. Such groups are shown to carry a natural separable complete metric, and are thus Polish. Our construction generalizes L1\mathrm{L}^{1} full groups of actions of discrete groups, which have been studied recently by the first author. We show that under minor assumptions on the actions, topological derived subgroups of L1\mathrm{L}^{1} full groups are topologically simple and -- when the acting group is locally compact and amenable -- are whirly amenable and generically two-generated. L1\mathrm{L}^{1} full groups of actions of compactly generated locally compact Polish groups are shown to remember the L1\mathrm{L}^{1} orbit equivalence class of the action. For measure-preserving actions of the real line (also often called measure-preserving flows), the topological derived subgroup of an L1\mathrm{L}^{1} full groups is shown to coincide with the kernel of the index map, which implies that L1\mathrm{L}^{1} full groups of free measure-preserving flows are topologically finitely generated if and only if the flow admits finitely many ergodic components. The latter is in a striking contrast to the case of Z\mathbb{Z} -actions, where the number of topological generators is controlled by the entropy of the action. We also study the coarse geometry of the L1\mathrm{L}^{1} full groups. The L1\mathrm{L}^{1} norm on the derived subgroup of the L1\mathrm{L}^{1} full group of an aperiodic action of a locally compact amenable group is proved to be maximal in the sense of Rosendal. For measure-preserving flows, this holds for the L1\mathrm{L}^{1} norm on all of the L1\mathrm{L}^{1} full group.Comment: Expanded and reworked monograph version of the paper. We added that L1 OE implies flip Kakutani equivalence for flows, pointed out that most L1 full groups contain a copy of L1 for their natural L1 metric, and showed that the latter is maximal in the sense of Rosendal on the derived L1 full group. We also added new appendice

    Maximal Hardy Fields

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    We show that all maximal Hardy fields are elementarily equivalent as differential fields, and give various applications of this result and its proof. We also answer some questions on Hardy fields posed by Boshernitzan.Comment: 470 pp. This document is not intended for publication in its current for
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