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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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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Distinguishability argument x Cantor's arguments?
Posted: Jan 8, 2013 6:15 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 8 Jan., 14:44, mstem...@walkabout.empros.com (Michael Stemper)
>

> > However, ANY real can be distinguished from ANY other by SOME finite
> > initial segment of its decimal, or other base, representation.

>
> The problem is not distinguishing given reals, but how reals can
> uniquely by *given*. That requires a finite definition. But I am
> afraid that you don't even understand what that means.
>
> Regards, WM

A set is determined by any rule for distinguishing whether a test object
is or is not a member. If more than any finite number of things satisfy
such a rule, then the set will have more than any finite number of
members.

If one can show that there is a surjection from N to such a set then
that set is defined to be a countable set.

If one can show that there cannot be any surjection from N to such a set
then that set is defined to be an uncountable set.

That this makes WM unhappy is irrelevant.
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