[XMLDEV Mailing List Archive Home] [By Thread] [By Date] [Recent Entries] [Reply To This Message] Re: Infinity
Peter, hello. On 3 Mar 2018, at 22:05, Peter Hunsberger wrote: I'm fairly sure the set of real numbers has a larger cardinality than the integers (I say this with some diffidence, though, since I've never covered this formally, so I'm basing this on a mixture of incidental reading and Wikipedia).On Sat, Mar 3, 2018 at 7:33 AM Norman Gray <norman@astro.gla.ac.uk> (By the way, I take it that we are both taking 'real number' to mean the mathematical reals rather than floating point numbers  Liam touches on this). The Wikipedia page I quoted [1] mentions that \aleph_1 is the cardinality of the ordinal numbers, and explicitly states that 'The cardinality of the set of real numbers [...] is 2^{\aleph_0}' (and goes on to imply that this is indeed larger than \aleph_0 given certain hypotheses). Also, Cantor's diagonal argument [2] explicitly shows (if I recall and understand it correctly) that there is no onetoone correspondence between the integers and the reals. That is, although the integers can indeed be mapped to a set of a points on a real line, they can be mapped only to a _subset_ of those points, and in any such mapping there will be points on the real line which do not correspond to an integer. There's a onetoone correspondence from integers to rationals, and to the set of algebraic numbers (the set of solutions to polynomials), so both of those sets are of cardinality \aleph_0. The latter set of course excludes the transcendental numbers, but I don't _think_ the main point depends directly on the existence or not of transcendental numbers. There are a number of subtleties here which I would be reluctant to speak confidently about, but I think the main statement ('more reals than integers') stands. Best wishes, Norman [1] https://en.wikipedia.org/wiki/Aleph_number [2] https://en.wikipedia.org/wiki/Cantor's_diagonal_argument  Norman Gray : https://nxg.me.uk SUPA School of Physics and Astronomy, University of Glasgow, UK
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