On Dec 30 2012, 1:44 pm, "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 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. >
Rather casually: rays through ordinal points.
Draw the binary tree with the endpoints of the paths of the finite tree to points 0, ..., 2^n-1.
Consider rays from the origin, to the ordinal points 0, ..., 2^n-1. They are countable.
Then with rays from the origin through 0, ..., 2^w-1, they are countable.