Peter, hello.
On 3 Mar 2018, at 22:05, Peter Hunsberger wrote:
> On Sat, Mar 3, 2018 at 7:33 AM Norman Gray <>
>
>>
>> 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
