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Topic: § 422 What is a definition of a number?
Replies: 53   Last Post: Jan 23, 2014 9:13 AM

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 mueckenh@rz.fh-augsburg.de Posts: 13,458 Registered: 1/29/05
Re: § 422 What is a definition of a number?
Posted: Jan 22, 2014 7:26 AM

On Wednesday, 22 January 2014 12:13:38 UTC+1, Wizard-Of-Oz wrote:

>
> Perhaps you are simply talking about transcendental numbers .. ones that
>
> can't be expressed algebraically?

Transcendental numbers cannot be expressed as zero's of poloynomials with rational coefficients. Nevertheless they can be expressed algebraically.
>
>
>
> If so, how does that mean the end of mathemtaics as we know it

Please read the § 422 again and again anbd again until you will have understood that there are two kinds of definitions, namely the finite ones like "Liouvilles number" or "SUM 1/10^n!" and the infinite ones, namely listing all digits digit by digit. § 422 shows first, that the latter definition does not exist and second, that, if it existed, it would not allow for uncountably many numbers yet.

Further we have already arrived, in another thread, at the conclusion that all possible definitions of the set of real numbers form a sub-countable set, such that an uncountable set of transcendentals can be excluded anyhow.

Regards, WM