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Re: [xml-dev] Infinity
• From: "Norman Gray" <norman@astro.gla.ac.uk>
• To: "Peter Hunsberger" <peter.hunsberger@gmail.com>
• Date: Sun, 04 Mar 2018 21:27:13 +0000

Peter, hello.

On 3 Mar 2018, at 22:05, Peter Hunsberger wrote:

On Sat, Mar 3, 2018 at 7:33 AM Norman Gray <norman@astro.gla.ac.uk>

It will be, but since there are as many elements in that set as there
are positive integers (they can be put into a one-to-one
correspondence), it is no bigger or smaller an infinity than the number
of integers. In contrast, the number of real numbers is a 'larger
infinity' than the number of integers. If you wish to further explore
this rabbit hole, see <https://en.wikipedia.org/wiki/Aleph_number> and
work outwards...

and reals are both of cardinality Aleph naught. The easiest way to
conceptualize this equivalence is to think of them both as being mappable
to a set of points on a line.
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).

(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 one-to-one 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 one-to-one 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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