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 seem so convinced of it after all.
Languages belong to real life. Nothing in real life is infinite. > > > > > > 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? > > 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.
Further uncountably many languages would not help in mathematical discourse. There only all languages spoken by the participants can be used. >
> > > > 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. > > > 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.
Is it really so difficult to understand that? Is it because you have never mistrusted set theory for many years?