On Jan 3, 9:07 am, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Jan 2, 12:48 am, Virgil <vir...@ligriv.com> wrote: > > > > > > > > > > > In article > > <de9ee3af-0823-4a99-8216-7b6033235...@po6g2000pbb.googlegroups.com>, > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > On Jan 1, 11:22 pm, Virgil <vir...@ligriv.com> wrote: > > > > In article > > > > <ef09c567-1637-46b8-932a-bcb856e41...@r10g2000pbd.googlegroups.com>, > > > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > > > On Jan 1, 8:59 pm, Virgil <vir...@ligriv.com> wrote: > > > > > > In article > > > > > > <5e016173-aa1b-4834-9d70-0c6b08f19...@jl13g2000pbb.googlegroups. > > > > > > com>, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > > > > > On Jan 1, 7:29 pm, Virgil <vir...@ligriv.com> wrote: > > > > > > > > In article But in that proof Cantor does not require a well > > > > > > > > ordering of the reals, only an arbitrary sequence of reals > > > > > > > > which he shown cannot to be all of them, thus no such > > > > > > > > "counting" or sequence of some reals can be a count or > > > > > > > > sequnce of all of them. -- > > > > > > > > Basically > > > > > > > Nonsense deleted! -- > > > > > > Nonsense deleted, yours? > > > > > Nope! -- > > > > Great: from demurral to denial. > > Seems clear enough: in ZFC, there are uncountably many irrationals, > each of which is an endpoint of a closed interval with zero. And, > they nest. Yet, there aren't uncountably many nested intervals, as > each would contain a rational. > To whit: in ZFC there are and there aren't uncountably many > intervals. > Then, with regards to Cantor's first for the well-ordering of the > reals instead of mapping to a countable ordinal, there are only > countably many nestings in as to where then, the gap is plugged (or > there'd be uncountably many nestings). Then, due properties of a well- > ordering and of sets defined by their elements and not at all by their > order in ZFC, the plug can be thrown to the end of the ordering, the > resulting ordering is a well-ordering. Ah, then the nesting would > still only be countable, until the plug was eventually reached, but, > then that gets into why the plug couldn't be arrived at at a countable > ordinal. Where it could be, then the countable intersection would be > empty, but, that doesn't uphold Cantor's first proper, only as to the > finite, not the countable. So, the plug is always at an uncountable > ordinal, in a well-ordering of the reals. (Because otherwise it would > plug the gap in the countable and Cantor's first wouldn't hold.) > > Then, that's to strike this: > "So, there couldn't be uncountably many nestings of the interval, it > must be countable as there would be rationals between each of those. > Yet, then the gap is plugged in the countable: for any possible value > that it could be. This is where, there aren't uncountably many limits > that could be reached, that each could be tossed to the end of the > well-ordering that the nestings would be uncountable. Then there are > only countably many limit points as converging nested intervals, but, > that doesn't correspond that there would be uncountably many limit > points in the reals. " > Basically that the the gap _isn't_ plugged in the countable. > > Then, there are uncountably many nested intervals bounded by > irrationals, and there aren't. >
Point being there are uncountably many disjoint intervals defined by the irrationals of [0,1]: each non-empty disjoint interval contains a distinct rational. Thus, a function injects the irrationals into a subset of the rationals.