§1. Introduction.
Among the most remarkable discoveries in set
theory in the last quarter century is the rich
structure of the arithmetic of singular cardinals,
and its deep relationship to large cardinals. The
problem of finding a complete set of rules
describing the behavior of the continuum function
2ℵα for
singular ℵα's, known as the
Singular Cardinals
Problem, has been attacked by many
different techniques, involving forcing, large
cardinals, inner models, and various combinatorial
methods. The work on the singular cardinals
problem has led to many often surprising results,
culminating in a beautiful theory of Saharon
Shelah called the pcf theory (“pcf” stands for
“possible cofinalities”). The most striking result
to date states that if
2ℵn
< ℵω for every
n = 0, 1, 2, …, then
2ℵω <
ℵω4.
In this paper we present a brief history
of the singular cardinals problem, the present
knowledge, and an introduction into Shelah's pcf
theory. In Sections 2, 3 and 4 we introduce the
reader to cardinal arithmetic and to the singular
cardinals problems. Sections 5, 6, 7 and 8
describe the main results and methods of the last
25 years and explain the role of large cardinals
in the singular cardinals problem. In Section 9 we
present an outline of the pcf theory.
§2. The arithmetic of cardinal
numbers. Cardinal numbers were
introduced by Cantor in the late 19th century and
problems arising from investigations of rules of
arithmetic of cardinal numbers led to the birth of
set theory. The operations of addition,
multiplication and exponentiation of infinite
cardinal numbers are a natural generalization of
such operations on integers. Addition and
multiplication of infinite cardinals turns out to
be simple: when at least one of the numbers
κ,
λ is infinite then both
κ +
λ and
κ·λ
are equal to max {κ,
λ}. In contrast with +
and ·, exponentiation presents fundamental
problems. In the simplest nontrivial case,
2κ
represents the cardinal number of the power set
P(κ),
the set of all subsets of
κ. (Here we adopt the
usual convention of set theory that the number
κ is identified with a
set of cardinality κ,
namely the set of all ordinal numbers smaller than
κ.
Borel's conjecture in topological groups