On Nov 1, 5:25 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 1 Nov., 20:03, William Hughes <wpihug...@gmail.com> wrote: > > > So when we add the complete first column we do no have to add a line > > index. > > So whether or not we have a complete first column the set of line > > indexes > > is identical to the set of column indexes. Stop claiming that if the > > first > > column is complete there must be more lines than columns. > > If the first column is complete and if an actually infinite (complete) > string of digits is to carry any mathematical meaning or can be > detected by mathematical means then there must be more lines than > columns. > > Of course I implicitly made this assumption.
Next time be a bit more explicit. The steps from "an actually infinite string of digits carries mathematical meaning" to "there must be more lines than columns" are a bit obscure.
In any case complete strings of digits (one digit for every natural number) do carry mathematical meaning. Take, e.g. 0.111... This is by definition the limit of the partial sums
S= 0.1 0.11 0.111 ...
This limit is 1/9. We can also consider 0.111... to be the set limit of the sequence of sets, S. In this case 0.111... is the complete sequence of 1's (one 1 for every natural number). Using either of these definitions 0.111... is not a sequence of sequences so how can one ask if the complete sequence of 1's is *in* 0.111... ? One way to look at this is to consider 0.111... to be the union of the sequences in S. Then every sequence from S is in 0.111... However, the complete sequence of 1's *is* the union of the sequences in S. So, using this definition, 0.111... does contain the complete sequence of 1's. There are other less attractive definitions (e.g. the set S) under which 0.111... does not contain the complete sequence of 1's, but these do not change the fact that using the standard definitions, 0.111... carries mathematical meaning.
