On Jan 3, 6:26 am, Ralf Bader <ba...@nefkom.net> wrote: > WM wrote: > > On 1 Jan., 19:19, Zuhair <zaljo...@gmail.com> wrote: > >> The distinguishability argument is a deep intuitive argument about the > >> question of Countability of the reals. It is an argument of mine, it > >> claims that the truth is that the reals are countable. However it > >> doesn't claim that this truth can be put in a formal proof. > > > The distinguishability argument is neither deep nor intuitive. > > It is not even an argument, just question-begging. >
Call it what may you, what is there is:
(1) ALL reals are distinguishable on finite basis
(2) Distinguishability on finite basis is COUNTABLE.
So we conclude that:
"The number of all reals distinguishable on finite basis must be countable".
Since ALL reals are distinguishable on finite basis, then:
"The number of all reals is countable".
Because generally speaking no set contain more elements than what it CAN have. So you cannot distinguish more reals than what you CAN distinguish. Since all reals are distinguished by finite initial segments of them, and since we only have COUNTABLY many such finite initial segments, then for the first glance it seems that there ought to be COUNTABLY many reals so distinguished. This is what our intuition would expect!
Nobody can say that this simple and even trivial line of thought have no intuitive appeal. Definitely there is some argument there, at least at intuitive level.
However Cantor's arguments all of which are demonstrated by explicit and rigorous formal proofs have refuted the above-mentioned intuitive gesture, however that doesn't make out of Cantor's argument an intuitive one, no, Cantor's argument remains COUNTER-INTUITIVE, it had demonstrated a result that came to the opposite of our preliminary intuitive expectation.
> > And is > > not an argument of yours since you do not even understand its > > implications. It is simply the basis of the axiom of extensionality. > > How should we distinguish elements if they could not be distinguished? > > > we arrive finally at: > > >> FINAL CONCLUSION: > > >> The number of all reals is COUNTABLE. > > > Of course this would be the result if "countable" was a sensible > > notion. > > You even know what the result would be if non-sensible notions involved > were sensible. Mückenheim, you are either the Greatest Genius Of All Times > or one of the greatest idiots.