Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker , now seen as a founder of mathematical constructivism , did not.
Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" " Cantor's paradise " resulting from the power set operation. This utility of set theory led to the article "Mengenlehre" contributed in by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around , when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes.
Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox : consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In Cantor had himself posed the question "What is the cardinal number of the set of all sets? Russell used his paradox as a theme in his review of continental mathematics in his The Principles of Mathematics. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment.
The work of Zermelo in and the work of Abraham Fraenkel and Thoralf Skolem in resulted in the set of axioms ZFC , which became the most commonly used set of axioms for set theory. The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology— category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A.
Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation, also called set inclusion. As insinuated from this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined.
Just as arithmetic features binary operations on numbers , set theory features binary operations on sets. Some basic sets of central importance are the empty set the unique set containing no elements; occasionally called the null set though this name is ambiguous , the set of natural numbers , and the set of real numbers. A set is pure if all of its members are sets, all members of its members are sets, and so on.
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In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets.
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Sets in the von Neumann universe are organized into a cumulative hierarchy , based on how deeply their members, members of members, etc. The rank of a pure set X is defined to be the least upper bound of all successors of ranks of members of X. Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams.
The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes. The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:.
The above systems can be modified to allow urelements , objects that can be members of sets but that are not themselves sets and do not have any members. NF and NFU include a "set of everything, " relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Yet other systems accept classical logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory , in which the value of an atomic formula embodying the membership relation is not simply True or False.
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The Boolean-valued models of ZFC are a related subject. Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs , manifolds , rings , and vector spaces can all be defined as sets satisfying various axiomatic properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica , it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic.
For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set. Set theory as a foundation for mathematical analysis , topology , abstract algebra , and discrete mathematics is likewise uncontroversial; mathematicians accept that in principle theorems in these areas can be derived from the relevant definitions and the axioms of set theory.
Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath , includes human-written, computer-verified derivations of more than 12, theorems starting from ZFC set theory, first-order logic and propositional logic. Combinatorial set theory concerns extensions of finite combinatorics to infinite sets.
Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces.
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It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.
The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses , and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending "relativizing" it to make it more broadly applicable. A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations.
This has important applications to the study of invariants in many fields of mathematics. In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A.
Professor Suppes in Axiomatic Set Theory provides a very clear and well-developed approach. For those with more than a classroom interest in set theory, the historical references and the coverage of the rationale behind the axioms will provide a strong background to the major developments in the field. Book Reg. Product Description Product Details One of the most pressing problems of mathematics over the last hundred years has been the question: What is a number? Reprint of the original, edition. Set Theory and the Continuum Hypothesis. An Outline of Set Theory.
Set Theory and the Continuum Problem. Boolean Algebra and Its Applications. Schimmerling's new book, A Course on Set Theory , looks like a nice and compact introduction. Henle, An Outline of Set Theory is a problem-oriented text. It has a section on Goodstein's theorem. A very rich collection of results on ordinal and cardinal arithmetic. Heinz Bachmann, Transfinite Zahlen. Unfortunately, I don't know any German. As far as I understand, this book contains some results not found in Sierpinski's book.
Cohen, Set Theory and the Continuum Hypothesis. I guess the last chapter on forcing is quite dated.
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But the previous chapters are insightful. This is a more specialized book with a quick review of basics. It looks like this book is still published: Amazon link. There are some online texts available as well. I remember seeing notes by Steve Jackson, J. Donald Monk and Sy Friedman among others. I used the book Set Theory by Andras Hajnal and Peter Hamburger and got the impression since I was taking the class during a program in Hungary that it was a common book there.
It has a good introduction to naive set theory and a lot of more advanced topics in combinatorial set theory as well. An excellent "Outline of the elements of naive set theory" as the author himself describes the book. The purpose of the book is to equip the beginning student of advanced mathematics with the necessary minimum of set theory "with minimum of philosophic discourse and logical formalism". This means the following: the axioms of set theory are stated and discussed in the most gentle way, followed by an account of all classical topics treated under the label of set theory in the same manner.
This is a great place to start for anyone.
Axiomatic Set Theory
If after reading Halmos, you develop an appetite for a more formal treatment of ZF theory , Suppes is a great companion to continue this journey with as recommended in Halmos Be prepared for some logical formalism and great historical references. Both books are available from Dover, and it is not a bad idea to get the original printing by van Nostrand of either.
Have fun! Frank Drake's book, "Set Theory: An Introduction to Large Cardinals" already mentioned by Ali Kare deserves your attention even if you're not particularly interested in large cardinals. Despite its subtitle, it contains very nice presentations of a lot of general set-theoretic background before getting to the large cardinals. I had a solid book for my studies. This might be considered "Baby Kunin," although it is described as a first year graduate course level.
It has set theory and logic:. Home Questions Tags Users Unanswered.