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LudovicoVan
Posts:
4,025
From:
London
Registered:
2/8/08


Re: Can we count N ?!
Posted:
Jun 10, 2013 12:06 PM


"Ross A. Finlayson" <ross.finlayson@gmail.com> wrote in message news:16674288b8f748a9a4a16320bf18fe47@googlegroups.com... > 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 >> embers.  >> Are there set theories with this kind of limited powerset definition? > > Yes, there are, for example the null axiom set theory with powerset as > order > type as successor in ubiquitous ordinals.
Thanks for the kickstart. 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.
Julio



