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LudovicoVan
Posts:
4,034
From:
London
Registered:
2/8/08


Re: Can we count N ?!
Posted:
Jun 9, 2013 7:38 AM


"Julio Di Egidio" <julio@diegidio.name> wrote in message news:kp1hkg$8r8$1@dontemail.me...
> 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 powerset is a supertask, i.e. that we get into the nonstandard, 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 powerset definition?
Julio



