Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Distinguishability argument x Cantor's arguments?
Posted:
Jan 8, 2013 6:15 PM
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In article
<99114492-6fad-4ac1-a30e-ff70224adbbe@c14g2000vbd.googlegroups.com>,
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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