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The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. [1]
Actual infinity. In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, [1] involves infinite entities as given, actual and completed objects. Since Greek antiquity, the concept of actual infinity has been a subject of debate among philosophers. Also, the question of whether the Universe is ...
Absolute infinite. The absolute infinite (symbol: Ω), in context often called " absolute ", is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite. Cantor linked the absolute infinite with ...
Aleph number. Aleph-nought, aleph-zero, or aleph-null, the smallest infinite cardinal number. In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor [1] and are named ...
Infinity symbol. The infinity symbol (∞) is a mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate, [1] after the lemniscate curves of a similar shape studied in algebraic geometry, [2] or "lazy eight", in the terminology of livestock branding. [3]
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. [1] It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers.
Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world; for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence. [16] "Actual infinity does not exist.
Infinity (philosophy) In philosophy and theology, infinity is explored in articles under headings such as the Absolute, God, and Zeno's paradoxes. In Greek philosophy, for example in Anaximander, 'the Boundless' is the origin of all that is. He took the beginning or first principle to be an endless, unlimited primordial mass (ἄπειρον ...