On 4 Feb., 20:31, fom <fomJ...@nyms.net> wrote: > On 2/4/2013 2:15 AM, WM wrote: > >
> > 2) An uncountable set has (infinitely many) more elements than a > > countable set. > > By "more," you mean that the construction of a new name > may be accomplished and by "infinitely many" you mean that > consecutive constructions can always be performed sequentially > without end from any initial finite configuration of names.
More means more than all rational points of the universe. Gödel and Cohen doubted the continuum hypothesis. They estimate the cardinality of the continuum as being muchg larger. > > > 3) Every real number has at least one unique representation as an > > infinite binary string (some rationals have even two representations > > but that's peanuts). > > By "uniqueness", you mean there is a strategy for > constructing names that always allows you to differentiate > a single object from a plurality on the basis of "naming"
Yes. If you cannot select a particulat number like 3/4 or pi, you cannot work with it. > > > 4) In many cases the string cannot be defined by a finite word. > > What would be the limitation here? Is it the negative logic > of "since there are more numbers than names..."?
Of course. > > > 5) Without loss of information the first bits of two strings, if > > equal, need not be written twice. > > This starts to become a little problematic. Now, your numbers > are turning into classes of numbers. And, your names are > turning into the names for canonical representatives of those > classes if the partition is viewed as an equivalence partition.
Don't see problems where no problems are. Whether I write 3.14000... and 3.14159... or write 000... 3,14 159... with connecting edges as a guides for the eye does not make any difference.