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Topic: Matheology § 400
Replies: 79   Last Post: Dec 1, 2013 4:22 PM

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 LudovicoVan Posts: 3,201 From: London Registered: 2/8/08
Re: Matheology § 400
Posted: Nov 27, 2013 4:39 AM

"Virgil" <virgil@ligriv.com> wrote in message
news:virgil-EA4B8D.02104527112013@BIGNEWS.USENETMONSTER.COM...
> In article <l749ei\$h3i\$1@dont-email.me>,
> "Julio Di Egidio" <julio@diegidio.name> wrote:

>> "Virgil" <virgil@ligriv.com> wrote in message
>> news:virgil-DBC840.12472425112013@BIGNEWS.USENETMONSTER.COM...

>> > WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:
>> >

>> >> A number is definable by its digits or by its construction rule. The
>> >> set
>> >> of
>> >> definable numbers is closed under all mathematical operations, i.e.,
>> >> every
>> >> definable number leads always to other definable numbers. The set of
>> >> all
>> >> definable numbers is a subset of the countable set of all finite
>> >> definitions.

>> >
>> > But the set of ALL real numbers is not!
>> >
>> > There are domains in mathematics which require the existence of more
>> > numbers than can be given finite definitions.
>> >
>> > The standard definition of the real number field is such a domain.
>> >
>> > Individual sets of such numbers can be found but not individual members
>> > of such sets.

>>
>> Although, if you will pardon a beginner's question, it sounds true that
>> from
>> definable numbers and definable operations we just get other definable
>> numbers, so it would be nice to hear where/how exactly we get to
>> undefinable
>> numbers, and, indeed, what "undefinable number" means at all.

>
> The properties required by the standardly defined real number field can
> be shown to require the existence of "more" numbers that there are
> definitions, uncountably many real numbers but only countably many
> definitions.
>
> WM chooses to refer to those extra reals as undefineable numbers.
>
> But they are collectively defineable, and only individually undefineable.

What escapes my understanding is that, if we start from any definable list
of definable reals, the anti-diagonal of the list is itself definable. So,
while I am not contending that the reals are countable, I still do not get
how undefinable reals enter the picture.

Julio

Date Subject Author
11/25/13 wolfgang.mueckenheim@hs-augsburg.de
11/25/13 Virgil
11/26/13 wolfgang.mueckenheim@hs-augsburg.de
11/26/13 Michael Klemm
11/26/13 William Elliot
11/26/13 wolfgang.mueckenheim@hs-augsburg.de
11/26/13 Virgil
11/27/13 wolfgang.mueckenheim@hs-augsburg.de
11/27/13 Virgil
11/26/13 Virgil
11/27/13 wolfgang.mueckenheim@hs-augsburg.de
11/27/13 Virgil
11/27/13 wolfgang.mueckenheim@hs-augsburg.de
11/27/13 Virgil
11/27/13 LudovicoVan
11/27/13 Virgil
11/27/13 LudovicoVan
11/27/13 wolfgang.mueckenheim@hs-augsburg.de
11/27/13 fom
11/27/13 wolfgang.mueckenheim@hs-augsburg.de
11/27/13 Virgil
11/27/13 wolfgang.mueckenheim@hs-augsburg.de
11/27/13 Virgil
12/1/13 albrecht
11/27/13 fom
11/28/13 wolfgang.mueckenheim@hs-augsburg.de
11/28/13 Virgil
11/28/13 wolfgang.mueckenheim@hs-augsburg.de
11/28/13 Virgil
11/29/13 wolfgang.mueckenheim@hs-augsburg.de
11/29/13 Virgil
11/29/13 wolfgang.mueckenheim@hs-augsburg.de
11/29/13 Virgil
11/28/13 fom
11/28/13 wolfgang.mueckenheim@hs-augsburg.de
11/28/13 Virgil
11/29/13 wolfgang.mueckenheim@hs-augsburg.de
11/29/13 Tucsondrew@me.com
11/29/13 wolfgang.mueckenheim@hs-augsburg.de
11/29/13 Virgil
11/30/13 wolfgang.mueckenheim@hs-augsburg.de
11/30/13 Virgil
11/30/13 Tucsondrew@me.com
11/30/13 wolfgang.mueckenheim@hs-augsburg.de
11/30/13 Virgil
11/30/13 Virgil
11/30/13 Virgil
11/30/13 Virgil
11/29/13 Virgil
11/29/13 wolfgang.mueckenheim@hs-augsburg.de
11/29/13 Virgil
11/30/13 wolfgang.mueckenheim@hs-augsburg.de
11/30/13 Virgil
11/30/13 Tucsondrew@me.com
11/30/13 Virgil
11/30/13 Virgil
11/30/13 Virgil
12/1/13 Virgil
11/29/13 Virgil
11/29/13 wolfgang.mueckenheim@hs-augsburg.de
11/29/13 Virgil
11/30/13 wolfgang.mueckenheim@hs-augsburg.de
11/30/13 Virgil
11/27/13 fom
11/27/13 fom
11/28/13 wolfgang.mueckenheim@hs-augsburg.de
12/1/13 Virgil
11/27/13 Virgil
11/27/13 wolfgang.mueckenheim@hs-augsburg.de
11/27/13 Virgil
11/27/13 Virgil
11/27/13 wolfgang.mueckenheim@hs-augsburg.de
11/27/13 Virgil
11/27/13 wolfgang.mueckenheim@hs-augsburg.de
11/27/13 Virgil
12/1/13 Virgil