In article <email@example.com>, firstname.lastname@example.org wrote:
> On Friday, 2 May 2014 21:04:55 UTC+2, Virgil wrote:
> > > But we know that every expression that can appear in eternity in > > > the whole, possibly infinite, universe belongs to a countable > > > set.
> > But the set of all subsets, or power set, is of provably greater > > cardinality then any set itself, so the existence of even one countably > > infinite set requires the existene of uncountably infinite sets. > > Then the only conclusion is that countability is nonsense.
Then one runs into the some problem with a finite set of definitions, being unable to define every one of its own subsets, and thus requiring undefinable elements in the set space.
> > > Therefore also every > > > expression that can appear in the mathematical discourse belongs to a > > > countable set.
> > That may be true in WMytheological discourse,
> It is true everywhere in set theories with the notion of countability
If there is set of definitions, whereever does one find enough definitions to define all its subsets? : > > a > b > aa > ab > ba > bb > aaa > ... > > > > > Nothing that can appear in this discourse, which *is* mathematics, can be > > > uncountable.
If there are more that a finite number of definitions allowed, there are automatically uncountaly many SETS of definitions, though not all those sets of definitions will be individually defineable > > > > > So that WM requires existence of sets for which there are no sets of all > > subsets? > > All that exists in mathematics is (at most) countable.
But if countable sets then necessarily uncountable power sets, at least outside of WM's wild weird world of WMytheology.
And in any case more sets than definitions for them