In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> Am Donnerstag, 5. Dezember 2013 19:36:16 UTC+1 schrieb Michael F. Stemper: > > > > > And the Set of All Definitions, when properly And Logically implemented, > > > cannot be put into Bijection with N. > > > > > > > > It can't? Why not? A definition must be a finite string. As far as > > > > I can tell, the set of all finite strings is countable, and may be > > > > bijected with N. Since the set of definitions is a subset of the set > > > > of finite strings, it must be bijectable with N or a subset of N. > > > > The official trick is this: Since definition cannot be defined, it is not > clear which finite words are definitions. So there is no list. But it is > clearly sufficient to estimate that the set is not empty, since there are > defined numbers, and that it cannot have cardinality larger aleph_0.
There are definitions of things other than definition that do exist, including a definition of "countable" and there are valid proofs that some sets do not qualify as being countable by such a definition.
One such widely accepted definition of a set, A, have a cardinality no larger than aleph_0, or, equivalently, being "countable" requires that one be able to show the existence of a surjection from |N to set S before claiming it to have a cardinality no larger than aleph_0, or, equivalently, being "countable".
Thus by claiming that no set can have a cardinality greater than aleph_0, WM is also claiming that for every set he can PROVE the existence of a surjection from |N to that set.
Let WM start by proving such a surjection from |N to the set of reals! --