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 mater 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 where, as we know modern mathematics is incomplete, there are to be discovered true things about its domain, not its theorems.
And no, there's no proof that ZF (as a general foundation for modern mathematics) is consistent and complete, and there aren't that it's consistent, either, and Goedel shows, in modern mathematics, it's not both. And, measure theory uses countable additivity (of the non- finite non-zero infinitesimal differential patches) for real analysis, and concrete mathematics uses asymptotics and sometimes, regular infinite ordinals: with no applied results solely due transfinite cardinals, and indeed transfinite set theory is somewhat a raw, disposed shoehorn of real analysis. And half of the integers are even.
Let's work more on posts, and progress, than replies. Quit worrying so much about covering your ass, as getting your head out of it.
And as described above, a breadth-first traversal, sees different results for the tree's "anti-diagonal", as that it's simply at the end of the traversal, end-to-end, point-to-point.
Then the question arises, diagonal of what? Where's the middle? Where's the square.
Draw a line: without putting pencil to paper. That's mathematics.