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Re: Can we count N ?!
Posted:
Jun 9, 2013 9:39 PM


On Sunday, June 9, 2013 4:38:52 AM UTC7, Julio Di Egidio wrote: > "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
Yes, there are, for example the null axiom set theory with powerset as order type as successor in ubiquitous ordinals.
Regards,
Ross Finlayson



