On Saturday, May 3, 2014 1:57:37 PM UTC-7, muec...@rz.fh-augsburg.de wrote: > On Saturday, 3 May 2014 20:42:10 UTC+2, Zeit Geist wrote:
> > While your at it, why don't you prove there are NOT Uncountably many Countable Languages. > > A language need be spoken or written, that means it must be learned. For that sake you need a dictionary, at least one. But the number of dictionaries in the infinite and eternal universe is at most countable.
You continuously confuse things that Do Exist ( the Physical Universe ) with things that Can Exist ( the Logical/Mathematical Universe ).
> > You seem so convinced of it after all. > > Languages belong to real life. Nothing in real life is infinite.
As I said above, continuously.
Or, is that supposed to be your proof.
> > > > 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. > > Let everybody consisting of at least one hydrogen atom in the universe present countably many dictionaries. How man dictionaries would be existing?
Again the continuous confusion.
As you said above, nothing Infinite Exists in "Reality". Hence, we are obviously not talking about what "Really" Exists.
And anyway, what each Being present Uncountably many Dictionaries?
> > You are the one who needs to prove your claim. > > Languages belong to real life. real life is never infinite. Only fools can believe and swindlers can assert that uncountably many languages are possible.
Take Set of Aleph_0 Symbols, call it S. For Each Infinite Subset of S, say T, let the elements of T be the language L_T. There are Uncountably many Languages of the form L_T.
Furthermore, if we were to take the Subsets themselves as the Language, we have an Uncountable Language.
> Further uncountably many languages would not help in mathematical discourse. There only all languages spoken by the participants can be used.
That why I said pick one. If its Well Constructed that all you need.
This discussion of Language has nothing to do with you Inability to produce a Definable Listing of All the Definble Reals.
> > > > 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. > > Why should I? Nothing of that kind can be proven because there is no definition of definition. Your claim has absolutely no relation to the fact that every countable set has at most countable subsets, whether or not they are defined.
You CLAIM that the Set of All Definable Reals is Countable. Since you are so hung-up on "Definabliity", this Set MUST be Definable.
So, give me that Definition, or Withdraw your claim.
And, any Definable Listing of the Subsets of an Infinte Countable Definable Set is Incomplete!
> > 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. > > That would be necessary if listing the elements was the only possibility to prove countability of a set. But it is not. And just this schism makes the contradcition: The set has no bijection with |N, but its superset has, proving the set countable.
First, in your argument is that Assumption that you have a Definition of All Definable objects. Also, as you say, the set has no Bijection with N; hence, it is Uncountable,
> Is it really so difficult to understand that? Is it because you have never mistrusted set theory for many years?
No, it's because you are mistaken in very fundamental ways.