On 01/21/2014 05:56 PM, Virgil wrote: > In article <firstname.lastname@example.org>, > email@example.com wrote: >> On Tuesday, 21 January 2014 05:41:54 UTC+1, Virgil wrote:
>>> WM must be using a highly non-standard definition of countability if he >>> claims that sets which cannot be listed can be counted. >>> In standard mathematics "countable" and "listable" are equivalent. > >> A set that is not uncountable but cannot be listed is not a set in standard >> set theory. > > In standard mathematics and standard set theory, countability and > listability are equivalent properties, and any set that has one has > both, and any set that lacks either lacks both. > An the presence or absence of either of these in no way limits what > constitutes a set, > >> For instance the set of all real numbers that can only be defined >> as limits. > > Unless one can unambiguously determine of every real whether it an be > defined OTHER than by a limit, that property is far too ambiguous to > define a set at all.
Of course, if you define the reals as "equivalence classes of Cauchy sequences of rationals", then all reals are defined by limits. By definition.
-- Michael F. Stemper No animals were harmed in the composition of this message.