In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 11 Jan., 23:30, Virgil <vir...@ligriv.com> wrote: > > > > > But my question aimed at the application of undefined reals in > > > mathematics. > > > > Without them one cannot have a complete infinite Archimedean ordered > > field such as the real number field. > > And with them one cannot have it either.
Perhaps not in WMytheology, but one can most other places.
> Try to discern a real that is not in the Binary Tree constructed from > a countable set.
I do not find any reals IN binary trees. > > > > > No, you misunderstand again. Cantor's opinion was (and did not change > > > until he died) that undefined items are nonsense. And ofcourse he was > > > absolutely right. > > > > But he still showed that the set of real numbers, i.e., the objects > > forming the unique Archimedean complete totally ordered field was not a > > countable set in the sense that no surjection from |N to that set is > > possible. > > He assumed that a set of all naturals exist, which is an assumption as > wrong as the assumption that a set of naturals can have cardinality 10 > and sum 10.
It is only "wrong" in dim dark places like WMYTHEOLOGY. > > > > One may know that a real is between 0.1 and 0.2 but still not finitely > > definable. In fact one may know a real accurate to any finite number of > > decimals places but still have it undefineable any further. > > which is tantamount to *not* having any real number but only a > rational interval.
So, according to WM, one can have an interval without having any of its members? Typical! > > > > > So if you only know its first n digits, that number is one of those > > undefineables that WM claims do nt exist.. > > No, there is no number known. Of course there are definable rationals > and reals in the interval.
Then one has at least one number. > > > > > There are two cases: > > > 1) If a Cantor list is finitely defined, then you know the entry in > > > every line and you know every digit of the diagonal. > > > > So that every list of finitely defined basal numerals, with base >=4, > > is incomplete since its antidiagonal is not listed. > > Not a list which contains all (terminating) representations of > rationals.
Such a list is trivially incomplete without ever worrying about antidiagonals, since there is no base in which every rational has a terminating represention, much less very real. > > > > > 2) If a Cantor list is undefined and has only, as usual, the first > > > three lines and then an "and so on", then you do neither know the > > > following entries nor the digits of the diagonal. Nothing is > > > "discernible" then except the theorem that two decimals which differ > > > at some place are not identical. But that is not a deep recognition. > > > > But any assertion that a list of basal numerals is COMPLETE is > > falsified by the existence of anti-diagonals which are provably not in > > the original list. > > It is already falsified by the non-existence of the sets of naturals > mentioned above.