Re: Infinity

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
>
>
> _______________________________________________________________________
>
> XML-DEV is a publicly archived, unmoderated list hosted by OASIS
> to support XML implementation and development. To minimize
> spam in the archives, you must subscribe before posting.
>
> [Un]Subscribe/change address: http://www.oasis-open.org/mlmanage/
> Or unsubscribe: xml-dev-unsubscribe@l...
> subscribe: xml-dev-subscribe@l...
> List archive: http://lists.xml.org/archives/xml-dev/
> List Guidelines: http://www.oasis-open.org/maillists/guidelines.php



--
Cheers,
Dimitre Novatchev
---------------------------------------
Truly great madness cannot be achieved without significant intelligence.
---------------------------------------
To invent, you need a good imagination and a pile of junk
-------------------------------------
Never fight an inanimate object
-------------------------------------
To avoid situations in which you might make mistakes may be the
biggest mistake of all
------------------------------------
Quality means doing it right when no one is looking.
-------------------------------------
You've achieved success in your field when you don't know whether what you're doing is work or play
-------------------------------------
To achieve the impossible dream, try going to sleep.
-------------------------------------
Facts do not cease to exist because they are ignored.
-------------------------------------
Typing monkeys will write all Shakespeare's works in 200yrs.Will they write all patents, too? :)
-------------------------------------
Sanity is madness put to good use.
-------------------------------------
I finally figured out the only reason to be alive is to enjoy it.