Everybody knows that I don't miss an opportunity to make a fool of myself.
That is why I am exposing this problem:
This property (countable) is normally accepted by proving that the subset of R (real number) , D= [0,1] is not countable.
Nobody would dare to contradict it (Cantor proof!).
But I am on the verge to loose what is left of myself-confidence because I don't find what is wrong in this proof:
First, we know that the set N²= {(n,m), n and m belong to N } is countable . (SCHAUM The algebra of sets Chap 2 p 46).
Next we can map any number that belong to D with N² (it must be the weak part):
0,001035 =1035 10E-6 that can be mapped with (1035, 6) of N².
So how is it that D isn't countable?
What is wrong in my reasoning?
I am not completely dumb so I should soon or later find the answer!
haqueoui!
We can continue the foolishness with this :
We can construct any element of R by picking an element in N and in D:
if x= 250,001035 the we need 250 in N and 0,001035 in D.
So we can map any element of R with NxD.
Because D is countable, we can map it again with (NXN).
And N² being countable so is R.
