On Saturday, May 3, 2014 2:39:05 AM UTC-7, muec...@rz.fh-augsburg.de wrote: > On Saturday, 3 May 2014 10:25:00 UTC+2, Zeit Geist wrote: > > On Saturday, May 3, 2014 12:48:09 AM UTC-7, muec...@rz.fh-augsburg.de wrote: > > > On Friday, 2 May 2014 23:20:55 UTC+2, Zeit Geist wrote: > > > > > Can you please provide a Countable List > > > > Every list is countable. > > > > > of ALL Finitely Definable Real Numbers? > > > > Remember, this List must have a Finite Definition. > > > > I use set theory. All finite expressions in all possible (non-formal and formal) languages form a countable set. This is undisputed except by extreme-matheologians who wish to create uncountably many languages or uncountable alphabets. I do not argue with them. > > > But there are Uncountable many Languages. > > As I said, I do not discuss with fools.
While your at it, why don't you prove there are NOT Uncountably many Countable Languages. You seem so convinced of it after all.
> > Just pick one. > > Impossible without a dictionary. A language has to be known by the user. For that sake there has to be a dictionary. You can costruct it, if you like, but there can never more than a finite number of languages exist.
That is up to you to present your Dictionary. You are the one who needs to prove your claim.
> > > A real number is defined by (at least) one or (frequently) more of those expressions. Therefore all real numbers form a subset of the set of all finite expressions. According to ZF set theory a subset of a countable set is countable. > > > But how do you know that that countable subset is Finitely Definable? > I do not know it. I know that every subset of a countable set is countable in ZF.
So you say, but YOU need to prove that Countable Set is Finitely Definable.
> > Give an Explicit Definition of that List. > > Why? I show that the assumption of infinite lists, i.e., of bijections with |N, lead to contradictions with the rule that subsets cannot have larger cardinality than sets. >
No, you have Not!
If claim that your Collection of Definitions of Real is Numbers is Countable, then to prove that, you must provide a Finite Definable List of all of those Definitions.
You are, after all, the one insisting on Finite Definitions.