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The unlimited; that which goes beyond any fixed bound. Exploration of this notion goes back at least to Zeno of Elea, and extensive mathematical treatment began with Eudoxus of Cnidus (4th c. bc). The mathematical notion was further developed by Cantor and K. T. W. Weierstrass (1815–97) in the 19th century. In the philosophy of space and time problems arise both with the infinitely small (see Bayle's trilemma, Zeno's paradoxes) and with the infinitely large or boundless nature that each seems, intuitively, to possess. Kant argued in the antinomies that consistently regarding space or time as either finite or as infinite, was impossible, and this formed a key element in his idealist theory of them as imposed upon an unknown (noumenal) nature by our own forms of sense. Philosophically one important division is between thinkers such as Cantor who can countenance actual completed infinities, and those such as Aristotle or the mathematician Leopold Krönecker (1823–91) who reject any such notion in favour of merely ‘potential’ infinities.

Subjects: Philosophy.

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