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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,

Here is what Wilipedia says about this at: https://en.wikipedia.org/wiki/Cardinality

"*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)."

> 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,

Here is what Wilipedia says about this at: https://en.wikipedia.org/wiki/Cardinality

"

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/

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--

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.

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@lists.xml.org

> subscribe: xml-dev-subscribe@lists.xml.org

> 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.

**References**:**Infinity***From:*Michael Kay <mike@saxonica.com>

**Re: [xml-dev] Infinity***From:*Elliotte Rusty Harold <elharo@ibiblio.org>

**Re: [xml-dev] Infinity***From:*"Norman Gray" <norman@astro.gla.ac.uk>

**Re: [xml-dev] Infinity***From:*Peter Hunsberger <peter.hunsberger@gmail.com>

**Re: [xml-dev] Infinity***From:*"Norman Gray" <norman@astro.gla.ac.uk>

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