Infinity represents something that is boundless or endless or something that is greater than any real or natural number. It is often denoted by the symbol of infinity presented here.

Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of much discussion among philosophers. In the seventeenth century, with the introduction of the symbol of infinity and infinitesimal calculus, mathematicians began working with infinite series and what some mathematicians (including l’Hôpital and Bernoulli) considered to be infinitely small quantities, but infinity is still associated with endless processes. . While mathematicians struggled with the basics of calculus, it remained unclear whether infinity could be considered a number or a quantity and, if so, how it could be achieved. At the end of the 19th century, Georg Cantor extended the mathematical study of infinity by studying infinite sets and numbers, showing that they could be of different sizes. For example, if a line is considered as a set of all its points, their infinite number (i.e., the power of the line) is greater than the number of integers. In this use, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

The mathematical concept of infinity refines and expands the old philosophical concept, especially by introducing infinity of many different dimensions of infinite sets. Among the axioms of Zermelo-Fraenkel set theory, on which the most modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics, which may seem to have nothing to do with them. For example, Wiles’ proof of Fermat’s Last Theorem is implicitly based on the existence of very large infinite sets to solve a long-term problem that is formulated in terms of elementary arithmetic.

Ancient cultures had different ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in a precise formalism, as modern mathematics does, but approached infinity as a philosophical concept.

Early Greek

The oldest recorded idea of ​​infinity may be the idea of ​​Anaximander (c. 610 – c. 546 BC), a Greek philosopher of doctrinaire. He uses the word apeiron, which means “unlimited,” “indefinite,” and can be translated as “infinite.”

Aristotle (350 BC) distinguished potential infinity from real infinity, which he considers impossible due to the various paradoxes it seems to produce. It has been argued that, according to this view, the Hellenistic Greeks had a “horror of infinity,” which, for example, would explain why Euclid (c. 300 BC) did not say that there are an infinity of prime numbers, but rather “Prime numbers are more than any set of assigned prime numbers. “It is also argued that proving this theorem, Euclid” was the first to overcome the horror of infinity. “There is a similar controversy over Euclid’s parallel postulate, sometimes translated.

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