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Cantor said: The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extra worldly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical ...
In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, [1] involves infinite entities as given, actual and completed objects. The concept of actual infinity has been introduced in mathematics near the end of the 19th century by Georg Cantor , with his theory of infinite sets , later formalized into ...
The argument's central thesis is the metaphysical impossibility of a temporally past-infinite universe and of actual infinities existing in the real world, traced by Craig to 11th-century Persian Muslim scholastic philosopher Al-Ghazali.
Any subset of the real numbers is either finite, or countably infinite, or has the cardinality of the real numbers. In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers : 2 ℵ 0 = ℵ 1 {\displaystyle 2^{\aleph _{0}}=\aleph _{1}} , or even shorter with beth numbers : ℶ ...
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.
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.
The Economist's review says The Beginning of Infinity is "equally bold" as Deutsch's previous book The Fabric of Reality, and "its conclusions are just as profound. Mr Deutsch argues that decent explanations inform moral philosophy, political philosophy and even aesthetics. He is provocative and persuasive."
The term goes back to Leibniz's theory of possible worlds, [2] used to analyse necessity, possibility, and similar modal notions.In short, the actual world is regarded as merely one among an infinite set of logically possible worlds, some "nearer" to the actual world and some more remote.