Set Theory: An Introduction to Independence Proofs
Encyclopedia
Set Theory: An Introduction to Independence Proofs is an important textbook and reference work in set theory
by Kenneth Kunen
. It starts from basic notions, including the ZFC axioms, and quickly develops combinatorial notions such as trees
, Suslin's problem
, ◊, and Martin's axiom
. It develops some basic model theory
(rather specifically aimed at models of set theory) and the theory of Gödel's constructible universe L.
The book then proceeds to exposit the method of forcing
. Through exercises, the reader learns to apply the method to prove logical independence results in set theory.
This book is not really for beginners, but graduate students with some minimal experience in set theory and formal logic will find it a valuable self-teaching tool, particularly in regards to forcing. Some find it easier to read than a true reference work such as Thomas Jech
's Set Theory. It is the standard textbook from which to learn forcing, though it has the disadvantage that the exposition of forcing relies somewhat on the earlier presentation of Martin's axiom, and the style is perhaps overly concise. John L. Bell's Set Theory: Boolean-Valued Models and Independence Proofs is an alternative, though it presents the topic from the standpoint of the more conceptually elegant though less easily used technique of Boolean-valued model
s of set theory. Jech's presentation is a hybrid of the former two styles.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
by Kenneth Kunen
Kenneth Kunen
Herbert Kenneth Kunen is an emeritus professor of mathematics at the University of Wisconsin–Madison who works in set theory and its applications to various areas of mathematics, such as set-theoretic topology and measure theory...
. It starts from basic notions, including the ZFC axioms, and quickly develops combinatorial notions such as trees
Tree (set theory)
In set theory, a tree is a partially ordered set In set theory, a tree is a partially ordered set (poset) In set theory, a tree is a partially ordered set (poset) (T, In set theory, a tree is a partially ordered set (poset) (T, ...
, Suslin's problem
Suslin's problem
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin in a work published posthumously in 1920....
, ◊, and Martin's axiom
Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
. It develops some basic model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
(rather specifically aimed at models of set theory) and the theory of Gödel's constructible universe L.
The book then proceeds to exposit the method of forcing
Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
. Through exercises, the reader learns to apply the method to prove logical independence results in set theory.
This book is not really for beginners, but graduate students with some minimal experience in set theory and formal logic will find it a valuable self-teaching tool, particularly in regards to forcing. Some find it easier to read than a true reference work such as Thomas Jech
Thomas Jech
Thomas J. Jech is a mathematician specializing in set theory who was at Penn State for more than 25 years. He was educated at Charles University and is now at the of the Academy of Sciences of the Czech Republic....
's Set Theory. It is the standard textbook from which to learn forcing, though it has the disadvantage that the exposition of forcing relies somewhat on the earlier presentation of Martin's axiom, and the style is perhaps overly concise. John L. Bell's Set Theory: Boolean-Valued Models and Independence Proofs is an alternative, though it presents the topic from the standpoint of the more conceptually elegant though less easily used technique of Boolean-valued model
Boolean-valued model
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean...
s of set theory. Jech's presentation is a hybrid of the former two styles.