Infinity. The ∞ symbol in several typefaces History Ancient cultures had various ideas about the nature of infinity.
The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. Early Greek In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinity; for example, instead of saying that there are an infinity of primes, Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers (Elements, Book IX, Proposition 20).
However, recent readings of the Archimedes Palimpsest have hinted that Archimedes at least had an intuition about actual infinite quantities. Early Indian The Indian mathematical text Surya Prajnapti (c. 3rd–4th century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Infinite Ink: The Continuum Hypothesis by Nancy McGough.
By Nancy McGough (firstname.lastname@example.org) Overview 1.1 What is the Continuum Hypothesis?
People have tried to understand space, time, motion, and the notion of "continuum" for thousands of years. This pursuit lead to the Pythagoreans discovery of irrational numbers, Zeno's paradoxes, infinitesimal calculus, transfinite set theory, relativity theory, quantum physics, and many more intriguing ideas. What do we mean when we say "continuum"? Here's a description Albert Einstein gave on p. 83 of his Relativity: The Special and the General Theory: The surface of a marble table is spread out in front of me.
Hermann Weyl said "let us stick to time as the most fundamental continuum" and gave the following description on p. 92 of his The Continuum: 1.1 What is the Continuum Hypothesis? We still think that the study of the size of the continuum should be our guiding light for further research in set theory. 2.1 Style I link only the first occurrence of a word or phrase. [set, order relation] 2aleph0. Continuum hypothesis. This article is about the hypothesis in set theory.
For the assumption in fluid mechanics, see Fluid mechanics. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose cardinality is strictly between that of the integers and the real numbers. Cardinality of infinite sets Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspondence) between them. Has the same cardinality as With infinite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate. Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers (see Cantor's first uncountability proof and Cantor's diagonal argument). Watch Free Documentary Online.
Documentary examining current ideas about very large numbers and infinity in regards to mathematics and the observable universe.
By our third year, most of us will have learned to count. Once we know how, it seems as if there would be nothing to stop us counting forever. But, while infinity might seem like an perfectly innocent idea, keep counting and you enter a paradoxical world where nothing is as it seems. Mathematicians have discovered there are infinitely many infinities, each one infinitely bigger than the last. And if the universe goes on forever, the consequences are even more bizarre. Older than time, bigger than the universe and stranger than fiction. Georg Cantor. Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔr/ KAN-tor; German: [ˈɡeɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantɔʁ]; March 3 [O.S.
February 19] 1845 – January 6, 1918) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware. The harsh criticism has been matched by later accolades.
Life Youth and studies Cantor, ca. 1870. Teacher and researcher