Set theory

The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s.

Its foundational appeal, together with its paradoxes, and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for logicians and philosophers of mathematics.

Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

In his work, he (among other things) expanded on Galileo's paradox, and introduced one-to-one correspondence of infinite sets, for example between the intervals

However, he resisted saying these sets were equinumerous, and his work is generally considered to have been uninfluential in mathematics of his time.

With the development of calculus in the late 17th century, philosophers began to generally distinguish between actual and potential infinity, wherein mathematics was only considered in the latter.

[3] Carl Friedrich Gauss famously stated: "Infinity is nothing more than a figure of speech which helps us talk about limits.

Bernhard Riemann's lecture On the Hypotheses which lie at the Foundations of Geometry (1854) proposed new ideas about topology, and about basing mathematics (especially geometry) in terms of sets or manifolds in the sense of a class (which he called Mannigfaltigkeit) now called point-set topology.

Set theory, as understood by modern mathematicians, is generally considered to be founded by a single paper in 1874 by Georg Cantor titled On a Property of the Collection of All Real Algebraic Numbers.

This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections (see: Controversy over Cantor's theory).

[a] Dedekind's algebraic style only began to find followers in the 1890s Despite the controversy, Cantor's set theory gained remarkable ground around the turn of the 20th century with the work of several notable mathematicians and philosophers.

For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined.

There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets.

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.

The New Foundations systems of NFU (allowing urelements) and NF (lacking them), associate with Willard Van Orman Quine, are not based on a cumulative hierarchy.

In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold.

An enrichment of ZFC called internal set theory was proposed by Edward Nelson in 1977.

For example, mathematical structures as diverse as graphs, manifolds, rings, vector spaces, and relational algebras can all be defined as sets satisfying various (axiomatic) properties.

[16] Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory.

For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.

Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy.

Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.

[20] Ludwig Wittgenstein condemned set theory philosophically for its connotations of mathematical platonism.

As reviewers Kreisel, Bernays, Dummett, and Goodstein all pointed out, many of his critiques did not apply to the paper in full.

[27][28] Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone spaces.

In the US in the 1960s, the New Math experiment aimed to teach basic set theory, among other abstract concepts, to primary school students, but was met with much criticism.

The math syllabus in European schools followed this trend, and currently includes the subject at different levels in all grades.

Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition[32]) of sets (e.g. "months starting with the letter A"), which may be useful when learning computer programming, since Boolean logic is used in various programming languages.

Likewise, sets and other collection-like objects, such as multisets and lists, are common datatypes in computer science and programming.