Definition 1: |?A?| = |?B?| Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B. Such sets are said to be equipotent, ...
Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Characteristic Property o ...
In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. However, as first pointed out by Abraham Fraenkel in a 1921 letter to Zermelo, this theory was incapable of proving ...
Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspondence) between them. Intuitively, for two sets S and T to have the same cardinality ...
A logical system or, for short, logic, is a formal system together with a form of semantics, usually in the form of model-theoretic interpretation, which assigns truth values to sentences of the forma ...
Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one.This is to say, that if you interpret the strings in su ...
The primary concept is the Theory of Forms. The only true being is founded upon the forms, the eternal, unchangeable, perfect types, of which particular objects of sense are imperfect copies. The mult ...
An attractive fixed point of a function f is a fixed point x0 of f such that for any value of x in the domain that is close enough to x0, the iterated function sequencex,\ f(x),\ f(f(x)),\ f(f(f(x))), ...
First example: the integers One of the most familiar groups is the set of integers Z which consists of the numbers..., ?4, ?3, ?2, ?1, 0, 1, 2, 3, 4, ..., together with addition.The following prop ...
A set is a well defined collection of distinct objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other ...
The abstract concept of negative numbers was recognized as early as 100 BC – 50 BC. The Chinese Nine Chapters on the Mathematical Art (Chinese: Jiu-zhang Suanshu) contains methods for finding the are ...
N. Bourbaki suggested an explication of the concept "mathematical structure" in their book "Theory of Sets" (Chapter 4. Structures) and then defined on that base, in particular, a very general concept ...
Albany is one of the oldest surviving European settlements from the original thirteen colonies and the longest continuously chartered city in the United States. The area was originally inhabited by Al ...
Moore was born in South London on 4 November 1873. His eldest brother was Thomas Sturge Moore, a poet, writer and engraver.In 1892, he was educated at Dulwich College and then attended Trinity College ...
According to a family tree prepared in Jerusalem after World War II, Wittgenstein's paternal great-grandfather was Moses Meier, a Jewish land agent who lived with his wife, Brendel Simon, in Bad Laasp ...
