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Re: [xml-dev] Infinity
• From: Dimitre Novatchev <dnovatchev@gmail.com>
• To: Norman Gray <norman@astro.gla.ac.uk>
• Date: Sun, 4 Mar 2018 13:59:06 -0800

> 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).

Yes Norman,

"One of Cantor's most important results was that the cardinality of the continuum ({\displaystyle {\mathfrak {c}}}) is greater than that of the natural numbers ({\displaystyle \aleph _{0}}); that is, there are more real numbers R than whole numbers N. (see Cantor's diagonal argument or Cantor's first uncountability proof)."

I hope that no mathematician is reading this forum ...

Cheers,
Dimitre

On Sun, Mar 4, 2018 at 1:27 PM, Norman Gray <norman@astro.gla.ac.uk> wrote:
>
> 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,
>
> 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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--
Cheers,
Dimitre Novatchev
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