Date: Jun 9, 2013 9:39 PM
Author: ross.finlayson@gmail.com
Subject: Re: Can we count N ?!

On Sunday, June 9, 2013 4:38:52 AM UTC-7, Julio Di Egidio wrote:
> "Julio Di Egidio" <julio@diegidio.name> wrote in message
>
> news:kp1hkg$8r8$1@dont-email.me...
>
>
>

> > By the same token, the sequence of subsets of N shown initially captures
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> > *all* subsets of N, finite and (potentially) infinite.
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> >
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> > Bottom line, within the potentially infinite, P(N) is countable.
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>
>
> I retract this conclusion as such, which is bogus in light of the definition
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> of countability: at the moment I see no way out of the fact that counting
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> the power-set is a super-task, i.e. that we get into the non-standard, and
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> this is because we need to count terminal nodes of the infinite binary tree
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> to actually count the infinite sets. But the contention that a theory of
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> infinite sets cannot have potentially infinite sets rather becomes the
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> contention that, in a coherent theory of potentially infinite sets, the
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> power set of a set would only have the set's finite subsets as members. --
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> Are there set theories with this kind of limited power-set definition?
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>
>
> Julio


Yes, there are, for example the null axiom set theory with powerset as order type as successor in ubiquitous ordinals.

Regards,

Ross Finlayson