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

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Posts: 4,165
From: London
Registered: 2/8/08
Re: Can we count N ?!
Posted: Jun 10, 2013 12:06 PM
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"Ross A. Finlayson" <> wrote in message
> On Sunday, June 9, 2013 4:38:52 AM UTC-7, Julio Di Egidio wrote:
>> "Julio Di Egidio" <> wrote in message
>> news:kp1hkg$8r8$

>> > 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
>> embers. --
>> Are there set theories with this kind of limited power-set definition?

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

Thanks for the kick-start. Browsing browsing, I've got to KPU, Goedel's
constructible universe, the axiom of constructibility, etc. Deemed as
"unnecessarily restrictive" by a standard that promotes incompleteness and
paradoxes. I'll also investigate further the surreal number system.


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