transfinite number
In mathematics, transfinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to q ...
s, by Polish mathematician
Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and t ...
. It was published in 1958 by
Państwowe Wydawnictwo Naukowe
Wydawnictwo Naukowe PWN (''Polish Scientific Publishers PWN''; until 1991 ''Państwowe Wydawnictwo Naukowe'' - ''National Scientific Publishers PWN'', PWN) is a Polish book publisher, founded in 1951, when it split from the Wydawnictwa Szkolne i P ...
, as volume 34 of the series Monografie Matematyczne of the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, the book has four chapters on
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
s, their arithmetic, and series and products of cardinal numbers, comprising approximately 50 pages. Following this, four longer chapters (totalling roughly 180 pages) cover orderings of sets,
order type
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...
s,
well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-or ...
Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Ces ...
according to which the collection of all ordinal numbers cannot be a set. Three final chapters concern
aleph number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named a ...
s and the
continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
, statements equivalent to the axiom of choice, and consequences of the axiom of choice.
The second edition makes only minor changes to the first except for adding footnotes concerning two later developments in the area: the proof 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 ...
of the independence of the continuum hypothesis, and the construction by Robert M. Solovay of the
Solovay model In the mathematical field of set theory, the Solovay model is a model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measu ...
in which all sets of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s are
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
.
Audience and reception
Sierpiński was known for his significant contributions to the theory of transfinite numbers;, reviewer
Reuben Goodstein
Reuben Louis Goodstein (15 December 1912 – 8 March 1985) was an English mathematician with a strong interest in the philosophy and teaching of mathematics.
Education
Goodstein was educated at St Paul's School in London. He received his Master ...
calls his book "a goldmine of results", and similarly
Leonard Gillman
Leonard E. Gillman (January 8, 1917 – April 7, 2009) was an American mathematician, emeritus professor at the University of Texas at Austin. He was also an accomplished classical pianist.
Biography
Early life and education
Gillman was born i ...
writes that it is highly valuable "as a compendium of interesting mathematical information, presented with care and clarity". Both Gillman and John C. Oxtoby call the writing style "leisurely" and "unhurried", and although Gillman criticizes the translation from an earlier Polish-language manuscript into English as unpolished, and points to several errors in the bibliography, he does not find the writing in the text of the book to be problematic.
In the 1970 text ''General Topology'' by Stephen Willard, Willard lists this book as one of five "standard references" on
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 ...