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Topic: Can we count N ?!
Replies: 5   Last Post: Jun 10, 2013 12:06 PM

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Posts: 3,792
From: London
Registered: 2/8/08
Re: Can we count N ?!
Posted: Jun 9, 2013 7:38 AM
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"Julio Di Egidio" <julio@diegidio.name> wrote in message

> By the same token, the sequence of subsets of N shown initially captures
> *all* subsets of N, finite and (potentially) infinite.
> Bottom line, within the potentially infinite, P(N) is countable.

I retract this conclusion as such, which is bogus in light of the definition
of countability: at the moment I see no way out of the fact that counting
the power-set is a super-task, i.e. that we get into the non-standard, and
this is because we need to count terminal nodes of the infinite binary tree
to actually count the infinite sets. But the contention that a theory of
infinite sets cannot have potentially infinite sets rather becomes the
contention that, in a coherent theory of potentially infinite sets, the
power set of a set would only have the set's finite subsets as members. --
Are there set theories with this kind of limited power-set definition?


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