On Nov 30, 12:46 pm, Virgil <Vir...@home.esc> wrote: > In article > <8b0d1469-3424-4a9d-8115-7535aadc6...@a10g2000pre.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > > On Nov 29, 11:53 pm, Virgil <Vir...@home.esc> wrote: > > > In article > > > <3bc52c73-77ee-479b-9a28-824fd9a4a...@z10g2000prh.googlegroups.com>, > > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > > > > > > (Binary and ternary (trinary) anti-diagonal cases require > > > > > > > > refinement.) > > > > > > > > But as neither I nor Cantor were not dealing with numbers in any > > > > > > > base, > > > > > > > your objections are, as usual, irrelevant. > > > > > > > No, it was just noted a specific constructive counterexample to that > > > > > > lists of (expansions representing) real numbers don't contain their > > > > > > antidiagonals. > > > > > > It wasn't even that. > > > > > > Marshall > > > > > In binary or ternary an everywhere-non-diagonal isn't not on the list. > > > > What does "an everywhere-non-diagonal" mean? > > > Given a matrix, there is a main diagonal, called "the" diagonal. In > > binary, there's one anti-diagonal. In ternary, base three, an > > everywhere-non-diagonal is different at each place than the diagonal. > > If there are infinitely many infinitely long entries in the list itself, > there are at least as many "diagonals". They can all be reformatted into > entries of base a least 4, in which case the dual representation problem > evaporates as no diagonal has a dual representataion. > > > > > > And does "isn't not on the list" mean the same as "is on the list"? > > > No. It means "isn't necessarily not on the list." > > Then you should have said so. >
I did, but it's OK, I'm glad that you see that there are some differences in the treatments that are radix-specific, there are funny differences between the forms of those results. These can be "quantified", in a sense, in terms of identifying the cost of algorithms in terms of time and space complexity, where as the radix of the list expansions decreases or goes to one, an ever more complex algorithm is necessary to attempt to build an item difference from each element of the list.
> > > > > In any integer base, from 2 on up, and any list, there are constructably > > > as many non-members of the list as members of it. > > > No, that is in integer bases from 4 on up. > > But any base can be converted to its square by taking digits two at a > time instead of one at time, so that every base is effectively treatable > like a base of at least 4. >
I like to take a step back and wonder, and explain, why those things are different.
In binary there's one antidiagonal.
> > > > > > Using AC, in ZFC, given a well-ordering of the reals > > > > Since no one has yet been able to give an explicit well ordering of the > > > reals, we won't give it to you. > > > I don't need it from you, the theory guarantees one exists (if ZFC > > were consistent). > > But does not guarantee that you have access to it.
It doesn't matter its form, its relevant properties are guaranteed by definition.
Its form is the EF or REF, an interesting mathematical primitive with tractable analytical properties.