Re: [xml-dev] Infinity

["Norman Gray" <norman@astro.gla.ac.uk>]

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