Virgil
Posts:
9,012
Registered:
1/6/11


Re: Finitely definable reals.
Posted:
Jan 12, 2013 4:37 PM


In article <42e944227b70405888d70035491b19c1@f4g2000yqh.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> 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. > Try to discern a real that is not in the Binary Tree constructed from > a countable set.
I am not aware of any real numbers that need appear in a Complete Infinite Binary Tree. There are various ways of associating sets ofreal numbers with such trees, but none that are inevitable. > > > > > 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.
The set of allnaturals existing may be wrong in WM's WMYTHEOLOGY, but WM has shown no reason why it must be wrong elsewhere, such as in ZFC.
And a lot of perfectly good math is based on its being right. > > > > 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. Any real interval of length greater than 0 must have more than one member, so is not the same as a single number. > > > > > 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.
Its not being known is the point!
> Of course there are definable rationals > and reals in the interval.
And all sorts of reals whose exact values are not known. > > > > > 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.
For any place value base, the nonterminating rationals are at least as numerous as the terminating ones, so all lists of only the terminating representations is necessarily incomplete in the raionals, much less in the reals. > > > > > 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 antidiagonals which are provably not in > > the original list. > > It is already falsified by the nonexistence of the sets of naturals > mentioned above.
Which "sets of naturals" mentioned above does WM claim do not exist?
As far as I am aware, WM's claim is both false and unprovable by any math outside WMytheology 

