Structure And Randomness In Computability And Set Theory

387 Pages·2020·14.175 MB·English

by  Edited by Douglas Cenzer , Edited by Christopher Porter , Edited by Jindrich Zapletal

Structure and Randomness in
Computability and Set Theory

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Structure and Randomness in
Computability and Set Theory
Edited by
Douglas Cenzer
University of Florida, USA
Christopher Porter
Drake University, USA
Jindrich Zapletal
University of Florida, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Cenzer, Douglas, editor. | Porter, Christopher (Christopher P.), editor. |
Zapletal, Jindrich, editor.
Title: Structure and randomness in computability and set theory / edited by
Douglas Cenzer, University of Florida, Christopher Porter, Drake University,
Jindrich Zapletal, University of Florida.
Description: New Jersey : World Scientific, [2021] | Includes
bibliographical references and index.
Identifiers: LCCN 2020034089 (print) | LCCN 2020034090 (ebook) |
ISBN 9789813228221 (hardcover) | ISBN 9789813228238 (ebook for institutions) |
ISBN 9789813228245 (ebook for individuals)
Subjects: LCSH: Set theory. | Computable functions.
Classification: LCC QA248 .S892 2021 (print) | LCC QA248 (ebook) |
DDC 511.3/22--dc23
LC record available at https://lccn.loc.gov/2020034089
LC ebook record available at https://lccn.loc.gov/2020034090
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A catalogue record for this book is available from the British Library.
Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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(cid:2)c 2021 World Scientific Publishing Company
https://doi.org/10.1142/9789813228238 fmatter
Preface
The goal of this collection of review chapters is to provide an
in-depth overview of work in three areas that have emerged
on the frontier of research in set theory and computability
in recent years: (1) infinitary combinatorics and ultrafilters;
(2) algorithmic randomness and algorithmic information; and
(3) computable structure theory.
The unifying themes of these areas are an emphasis on struc-
ture, randomness, and the interplay between them. Taking the
above-listed three research areas in reverse order, these themes
appear as follows: First, we have an emphasis on the use of
the tools of computability theory to analyze the complexity of
various mathematical structures in computable structure the-
ory. Second, we can study the extent to which effective methods
allow us to classify objects as random or nonrandom, a central
theme in algorithmic randomness, or, in the case of algorithmic
information theory, measuring the amount of algorithmic infor-
mation in an object, placing it on a scale that at the lower levels
corresponds to more structure, while at the higher levels corre-
sponds to more randomness. Third, in infinitary combinatorics,
one can study the extent to which order emerges from disor-
der, particularly in the case of Ramsey theory for infinite sets,
v
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vi Structure and Randomness in Computability and Set Theory
thereby yielding an interesting balance between structure and a
certain kind of randomness.
We now layout the specifics of the book. Part I “Infinitary
Combinatorics and Ultrafilters” deals with a remarkably per-
sistent theme in set theory. Nonprincipal ultrafilters on natural
numbers are well-known to be difficult to analyze and treat in
detail. At the same time, they are exceptionally useful in many
directions, asonecansee fromtheproofofvander Waerdenthe-
orem via the compactification of the semigroup of natural num-
bers or (more recently) Malliaris’s and Shelah’s proof of p = t.
Over the years, set theorists isolated certain critical properties
of ultrafilters which completely determine their combinatorial
features.
As the oldest example of such a critical property, consider
selective ultrafilters: the ultrafilters U which, for every color-
ing c : [ω]2 → 2, contain a set homogeneous for such a color-
ing. It turns out that it is impossible to find further distinc-
tions between selective ultrafilters using formulas from a certain
broad syntactically identified class. In other words, if we know
that a certain ultrafilter is selective, we probably know most
of its other combinatorial properties as well. One way to pre-
cisely formulate this intuition is found in a result of Todorcevic:
grantedsufficientlylargecardinals,everyselective ultrafilterU is
generic over thecanonical model L(R) (thesmallest model ofZF
set theory containing all reals and all ordinals) for the partially
ordered set P of all infinite subsets of ω ordered by inclusion.
Thus, the study of combinatorial properties of U expressible in
the generic extension L(R)[U] reduces to the study of the par-
tial ordering P and is completely independent of U. The model
L(R)[U] attracted plenty of attention over the last 30 years from
authors such as Todorcevic, Di Prisco, Dobrinen, Paul Larson
and Zapletal; it is one of the canonical and best understood
objects in transfinite set theory.
Given the success story of Ramsey ultrafilters, one can ask
whether it is possible to find other properties playing similar
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Preface vii
criticalrole.Theanswer isaffirmativeandthelistofsuch critical
properties is continually growing. Most of the examples found so
far can be restated in terms of partition calculus. For each such
criticalpropertyφ,thereisadefinitionofapartialorderingP such
φ
thatevery ultrafiltersatisfyingφisinfactgenericoverthemodel
L(R)fortheposetP .Thisconvertsthestudyofanultrafilterwith
φ
the property φ to the study of the ordering P . The comparison
φ
of the models L(R)[U] for ultrafilters U of various critical types
thenslowlydissectstheunwieldysetofallultrafiltersintosmaller,
manageable units. This open-ended, bold program has seen con-
tinualprogressovertheyears,anditisintimatelyconnectedwith
thetoolsofabstractpartitioncalculus.
Part I consists of two chapters. Chapter 1 “Topological
Ramsey Spaces Dense in Forcings”, by Natasha Dobrinen, is
an extensive survey of this area. Dobrinen frames this tradi-
tional field of inquiry using the theory of topological Ramsey
spaces of Stevo Todorcevic, which support infinite-dimensional
RamseytheorysimilarlytotheEllentuckspace.Eachtopological
Ramsey space is endowed with a partial ordering which can be
modified to a σ-closed “almost reduction” relation analogously
to the partial ordering of “mod finite” on [ω]ω. Such forcings
add new ultrafilters satisfying weak partition relations and have
complete combinatorics. In cases where a forcing turned out to
beequivalent toatopologicalRamseyspace,thestrongRamsey-
theoretic techniques have aided in a fine-tuned analysis of the
Rudin–Keisler and Tukey structures associated with the forced
ultrafilter andindiscovering new ultrafilters withcomplete com-
binatorics. This original perspective allows her to organize the
wealth of existing research in a particularly incisive way and to
prove a number of new results as well. Dobrinen’s exposition
provides a long dictionary of critical combinatorial properties
of ultrafilters and pointers for extending this dictionary fur-
ther. Readers interested in using techniques using topological
Ramsey spaces to study ultrafilters with various partition rela-
tions should find Dobrinen’s survey to be instructive.
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viii Structure and Randomness in Computability and Set Theory
Chapter 2 “Infinitary Partition Properties of Sums of
Selective Ultrafilters”, by Andreas Blass, is more specific and
deals with a particular pair of critical combinatorial ultrafilter
properties. It concerns two kinds of ultrafilters on ω2, the first
kind given by the sums of nonisomorphic selective ultrafilters
that are indexed by another selective ultrafilter, and the second
kind given by ultrafilters that are generic with respect to the
forcing the conditions of which are subsets of ω2 that have an
infinite intersection with {n} × ω for infinitely many n ∈ ω.
Although these two kinds of ultrafilters share a number of prop-
erties, such as being Q-points but not P-points and not being at
thetopoftheTukey ordering,theyalsodifferinseveralrespects,
as only ultrafilters of the first kind are basically generated while
only ultrafilters of the second kind are weak P-points. Blass first
shows that the infinitary partition property of ultrafilters of the
first kind is of the same strength with what has been previously
shown about the infinite partition property of ultrafilters of the
second kind. This, in turn, leads to Blass’s second main result,
obtained via an application of complete combinatorics, that the
two kinds of ultrafilters are the same when viewed in different
models of set theory. Lastly, Blass uses this result to account for
how both the similarities and differences between the two kinds
of ultrafilter arise.
Part II “AlgorithmicRandomness and Information” concerns
two different research strands, namely, the study of algorithmi-
cally random sequences under Turing reductions, and the study
of effective notions of Hausdorff dimension defined in terms
of Kolmogorov complexity, the central concept in algorithmic
information theory.
One of the primary aims in the study of algorithmic random-
ness is to study various definitions of algorithmically random
sequences and the properties of such sequences. The most well-
studied definition of algorithmic randomness is Martin-Lo¨f ran-
domness. Martin-L¨of’s original idea behind his definition was
to formalize the notion of an effective statistical test, given
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Preface ix
by a sequence of effectively generated open sets the measures
of which are effectively converging to zero. A sequence that is
not in the intersection of any such test is a Martin-L¨of random
sequence. Alternative definitions of algorithmic randomness can
be obtained, for instance, by modifying the underlying notion of
an effective statistical test, although Martin-L¨of has proven to
be, in certain respects, more well-suited to being studied from
a computability-theoretic point of view compared to alternative
definitions of randomness.
One respect in which Martin-Lo¨f randomness is amenable to
study using tools from computability theory, namely the behav-
ior of random sequences under Turing reductions, is the subject
ofthefirst randomness-theoretic chapter inthisbook,“Limitsof
the Kuˇcera–Ga´cs Coding Method”, by George Barmpalias and
Andrew Lewis-Pye. This chapter discusses an improvement of
methods used independently by Kuˇcera and Ga´cs to prove what
is now considered to be a classical result in the field concerning
the computational power of Martin-L¨of random sequences.
In 1985, Kuˇcera proved that for every sequence A ∈ 2ω, there
is some Martin-L¨of random sequence B such that A ≤ B.
T
Kuˇcera’s proof involved a method of coding according to which
oneencodesanarbitrarysequence A ∈ 2ω byamemberofafixed
Π01 class of positive Lebesgue measure. If we use this method of
coding on a Π0 class consisting of Martin-L¨of random sequences
1
(such as the complement of some level of the universal Martin-
Lo¨f test), we obtain the desired reduction to a random sequence.
Independently, in 1986 Ga´cs proved the same result using an
alternative method of coding, which allowed him to provide a
more fine-grained analysis of the reduction in question, which he
shown can be given by a wtt-functional. That is, Ga´cs proved
that there is a computable function f : ω → ω such that every
sequence A is computable from a Martin-L¨of random sequence
B with the function f bounding the use of this reduction. In
fact, Ga´cs proved that we can take this function to be given by
√
f(n) = n+ nlog(n).