On Dec 30, 3:21 pm, Virgil <vir...@ligriv.com> wrote: > In article > <fb7fa7ab-bc25-4cb9-9b7c-a073c2fe7...@i2g2000pbi.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > > > > > > > > On Dec 30, 1:15 pm, Virgil <vir...@ligriv.com> wrote: > > > In article > > > <8a425f72-80f2-4aee-9bb9-01f1c6f12...@vb8g2000pbb.googlegroups.com>, > > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > > > requires that they be listable, but one can prove that they are not > > > > > listable by showing that no list of them can be complete. > > > > > And no matter how vociferously WM tries to argue otherwise, in standard > > > > > mathematics that can all be done. > > > > > -- > > > > > Well, not when "standard" was "pre-Cantorian" > > > > I used only the present tense which eliminates pre-Cantorianism. > > > -- > > > Then you shouldn't discount the future > > I don't, but neither do I pretend to predict it, the way you do. > > And, as thing stand in the present, standard mathematics supports the > Cantor diagonal argument and that every Complete Infinite Binary Tree > which really is a Complete Infinite Binary Tree instead of one of WM's > corrupted versions of one, must have uncountably many paths. > --
Until you find applications for transfinite cardinals, it's all "pretend": pure, abstract mathematics.
Unless you're a Platonist now.
I'm a Platonist: those things are real. Show an application solely due transfinite cardinals.
And the universe is infinite, and it's real. And: as mathematical object: it's its own powerset.
And, there may always be more than our "standard" mathematics, but, existence is that, for what we know it.