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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite

[the status and nature of infinity as a number]

26 ideas
 18090 Without infinity time has limits, magnitudes are indivisible, and numbers come to an end [Aristotle]
 7555 Zeno achieved the statement of the problems of infinitesimals, infinity and continuity [Russell on Zeno of Citium]
 8738 Postulate 2 says a line can be extended continuously [Euclid, by Shapiro]
 13151 Not all infinites are equal [Newton]
 10856 A truly infinite quantity does not need to be a variable [Bolzano]
 15902 Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine]
 15908 It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine]
 9633 No one shall drive us out of the paradise the Cantor has created for us [Hilbert]
 12460 We extend finite statements with ideal ones, in order to preserve our logic [Hilbert]
 12462 Only the finite can bring certainty to the infinite [Hilbert]
 14420 Infinity and continuity used to be philosophy, but are now mathematics [Russell]
 14119 We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell]
 14133 There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell]
 18708 Infinity is not a number, so doesn't say how many; it is the property of a law [Wittgenstein]
 17809 Gödel showed that the syntactic approach to the infinite is of limited value [Kreisel]
 10491 Infinite natural numbers is as obvious as infinite sentences in English [Boolos]
 9813 Mathematics shows that thinking is not confined to the finite [Badiou]
 13509 We can establish truths about infinite numbers by means of induction [Hart,WD]
 17799 Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
 17797 Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
 18518 Infinite numbers are qualitatively different - they are not just very large numbers [Heil]
 10866 Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg]
 15947 The infinite is extrapolation from the experience of indefinitely large size [Lavine]
 15949 The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
 17730 Combining the concepts of negation and finiteness gives the concept of infinity [Jenkins]
 17923 Intuitionists only accept a few safe infinities [Colyvan]