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Re: [xml-dev] Infinity

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...

Actually no, and thankfully the Wikipedia page gets this right. Integers
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,


[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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