''The Higher Infinite: Large Cardinals in Set Theory from their Beginnings'' is a
monograph
A monograph is a specialist work of writing (in contrast to reference works) or exhibition on a single subject or an aspect of a subject, often by a single author or artist, and usually on a scholarly subject.
In library cataloging, ''monograph ...
in
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
by
Akihiro Kanamori
is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinal property, large cardinals, ''The Higher Infinite''. He has written several essays on the history of mathematics, especia ...
, concerning the history and theory of
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least Î ...
s, infinite sets characterized by such strong properties that their existence cannot be proven in
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
(ZFC). This book was published in 1994 by
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
in their series Perspectives in Mathematical Logic, with a second edition in 2003 in their Springer Monographs in Mathematics series, and a paperback reprint of the second edition in 2009 ().
Topics
Not counting introductory material and appendices, there are six chapters in ''The Higher Infinite'', arranged roughly in chronological order by the history of the development of the subject. The author writes that he chose this ordering "both because it provides the most coherent exposition of the mathematics and because it holds the key to any epistemological concerns".
In the first chapter, "Beginnings", the material includes
inaccessible cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of ...
measurable cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivisio ...
indescribable cardinal
In mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language ''Q''. There are many different types of indescribable cardinals corresponding to different choices of languages ...
s. The chapter covers the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
and
inner model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let L = \langle \in \rangle be ...
s,
elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one often ...
s and
ultrapower
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...
s, and a result of
Dana Scott
Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, Ca ...
that measurable cardinals are
inconsistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
with the
axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible universe, constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann unive ...
.
The second chapter, "Partition properties", includes the
partition calculus In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets.
Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom.
R ...
of
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
and
Richard Rado
Richard Rado FRS (28 April 1906 – 23 December 1989) was a German-born British mathematician whose research concerned combinatorics and graph theory. He was Jewish and left Germany to escape Nazi persecution. He earned two PhDs: in 1933 from th ...
,
trees
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
and
Aronszajn tree In set theory, an Aronszajn tree is a tree (set theory), tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a Cardinal number, cardinal ''κ'', ...
s, the
model-theoretic
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the st ...
study of large cardinals, and the existence of the set 0# of true formulae about indiscernibles. It also includes
Jónsson cardinal In set theory, a Jónsson cardinal (named after Bjarni Jónsson) is a certain kind of large cardinal number.
An uncountable cardinal number κ is said to be ''Jónsson'' if for every function ''f'': ºsup><ω → κ there is a set ''H'' of or ...
s and
Rowbottom cardinal In set theory, a Rowbottom cardinal, introduced by , is a certain kind of large cardinal number.
An uncountable cardinal number \kappa is said to be ''\lambda- Rowbottom'' if for every function ''f'': kappa;sup><ω → λ (whe ...
s.
Next are two chapters on "Forcing and sets of reals" and "Aspects of measurability". The main topic of the first of these chapters is forcing, a technique introduced by
Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
for proving consistency and inconsistency results in set theory; it also includes material in
descriptive set theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ot ...
. The second of these chapters covers the application of forcing by
Robert M. Solovay
Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.
Biography
Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on '' ...
to prove the consistency of measurable cardinals, and related results using stronger notions of forcing.
Chapter five is "Strong hypotheses". It includes material on
supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties.
Formal definition
If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...
huge cardinal In mathematics, a cardinal number κ is called huge if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point (set theory), critical point κ and
:^M \subset M.\!
Here, ''&al ...
extendible cardinal In mathematics, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look simila ...
s, on
strong cardinal In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal.
Formal definition
If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and there ...
s, and on
Woodin cardinal In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number \lambda such that for all functions
:f : \lambda \to \lambda
there exists a cardinal \kappa < \lambda with
:
and an s.
The book concludes with the chapter "Determinacy", involving the
axiom of determinacy
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of ...
and the theory of infinite games. Reviewer Frank R. Drake views this chapter, and the proof in it by
Donald A. Martin
Donald Anthony Martin (born December 24, 1940), also known as Tony Martin, is an American set theorist and philosopher of mathematics at UCLA, where he is an emeritus professor of mathematics and philosophy.
Education and career
Martin rece ...
of the
Borel determinacy theorem In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is Determinacy, determined, meaning that one of the two players will have a winning strategy for the game. A Gale-Stewart ga ...
, as central for Kanamori, "a triumph for the theory he presents".
Although quotations expressing the philosophical positions of researchers in this area appear throughout the book, more detailed coverage of
issues in the
philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
regarding the
foundations of mathematics
Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...
Philip Welch
Philip David Welch (born 6 January 1954) is a British mathematician known for his contributions to logic and set theory. He is Professor of Pure Mathematics at the School of Mathematics, University of Bristol. He is currently President of the B ...
Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...