> (1) ALL reals are distinguishable on finite basis > (2) Distinguishability on finite basis is COUNTABLE.
What does this mean? If you have two _different_ reals r1 and r2, then you can establish this fact in finite time. The set of reals that are describable by finite strings over a finite character set is countable. However, not all reals have that property.
> 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:
You seem to use "distinguishable" in two different ways.
Seeing your argument reminds me of the old chestnut about cats: A cat has three tails. Proof: No cat has two tails. A cat has one tail more than no cat. QED.
> "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. > > Zuhair