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. Try to discern a real that is not in the Binary Tree constructed from a countable set.
> > 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.
> 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 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.
> > 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. > > > 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.