On Sunday, June 9, 2013 4:38:52 AM UTC-7, Julio Di Egidio wrote: > "Julio Di Egidio" <firstname.lastname@example.org> wrote in message > > news:email@example.com... > > > > > 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? > > > > Julio
Yes, there are, for example the null axiom set theory with powerset as order type as successor in ubiquitous ordinals.