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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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Posts: 18,076
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 <> wrote:
> In article
> <>,
>  WM <> wrote:

> > On 2 Jan., 22:19, Zuhair <> 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
and so on.

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

Regards, WM

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