
Re: Why Wolkenmuekenheim fails to understand set theory
Posted:
Jan 21, 2014 7:13 PM


On 01/21/2014 05:56 PM, Virgil wrote: > In article <9c2d9b8083134d5187707aba48f2c3c1@googlegroups.com>, > mueckenh@rz.fhaugsburg.de wrote: >> On Tuesday, 21 January 2014 05:41:54 UTC+1, Virgil wrote:
>>> WM must be using a highly nonstandard 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.

