Set Theory (Stanford Encyclopedia of Philosophy)Set theory is the mathematical theory of well-determined collections, called sets , of objects that are called members , or elements , of the set. Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets. The theory of the hereditarily-finite sets, namely those finite sets whose elements are also finite sets, the elements of which are also finite, and so on, is formally equivalent to arithmetic. So, the essence of set theory is the study of infinite sets, and therefore it can be defined as the mathematical theory of the actual—as opposed to potential—infinite. The notion of set is so simple that it is usually introduced informally, and regarded as self-evident.
Naive Set Theory: Introduction
Discrete Mathematics/Naive set theory
When we talk of set theory , we generally talk about collections of certain mathematical objects. In this sense, a set can be likened to a bag, holding a finite or conceivably infinite amount of things. Sets can be sets of sets as well bags with bags in them. However, a set cannot contain duplicates -- a set can contain only one copy of a particular item. When we look at sets of certain types of numbers, for example, the natural numbers, or the rational numbers, for instance, we may want to speak only of these sets.
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Set theory , branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. Between the years and , the German mathematician and logician Georg Cantor created a theory of abstract sets of entities and made it into a mathematical discipline. This theory grew out of his investigations of some concrete problems regarding certain types of infinite sets of real numbers. A set, wrote Cantor, is a collection of definite, distinguishable objects of perception or thought conceived as a whole.