811 research outputs found
The Galvin property under the Ultrapower Axiom
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 -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 -complete ultrafilter on has the Galvin property if and only
if it is an iterated sum of -points
Transferring Compactness
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 that is
--stationary for all but not weakly compact. This is in
sharp contrast to the situation in the constructible universe , where
being --stationary is equivalent to being
-indescribable. We also show that it is consistent that there
is a cardinal such that is
-stationary for all and , answering a
question of Sakai.Comment: Corrected some typo
Hilbert Spaces Without Countable AC
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
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
LIPIcs, Volume 261, ICALP 2023, Complete Volum
The six blinds and the elephant or an interdisciplinary selection of measurement features
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
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 : for every regular
cardinal and . Building on a work of Lambie-Hanson, we also show that
is consistent with
. 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
full groups of flows
We introduce the concept of an 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 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 full groups are topologically simple and -- when
the acting group is locally compact and amenable -- are whirly amenable and
generically two-generated. full groups of actions of compactly
generated locally compact Polish groups are shown to remember the
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
full groups is shown to coincide with the kernel of the index
map, which implies that 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 -actions, where the number of topological generators is
controlled by the entropy of the action.
We also study the coarse geometry of the full groups. The
norm on the derived subgroup of the 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 norm on all of the 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
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
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