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Topic: Distinguishability argument x Cantor's arguments?
Replies: 15   Last Post: Jan 9, 2013 4:32 PM

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mueckenh@rz.fh-augsburg.de

Posts: 15,238
Registered: 1/29/05
Re: Distinguishability argument x Cantor's arguments?
Posted: Jan 4, 2013 6:47 AM
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On 3 Jan., 22:31, Virgil <vir...@ligriv.com> wrote:
> In article
> <3d65ff59-bf7e-445b-aad6-77d4ece64...@p17g2000vbn.googlegroups.com>,
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 2 Jan., 22:19, Zuhair <zaljo...@gmail.com> wrote:
>
> > > On the other hand Cantor have presented many arguments all of which
> > > are rigorously formalized in second order logic under full semantics,
> > > and those arguments PROVED that there are uncountably many reals

>
> > that can be distinguished by their finite initial segments.
>
> A decimal, or other base, expression is not a actually a number but
> merely a numeral, a representation of or name for a number.
>
> Actually, no real can be distinguished from ALL others by ANY finite
> initial segment of its decimal, or other base, representation.


That means that every real that shall be used in mathematics needs a
finite representation over a countable alphabet (even a finite
alphabet - but that does not matter here). Mathematics of real numbers
cannot be done in another way. Examples:

0,25 and only zeros following
0,111... and only 1's following
pi
e
crt(3)
and so on.

All distinguishable reals (and all reals that can appear in Cantor's
mathematics) belong to this set (as Cantor himself clearly
recognized).

Regards, WM



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