Rationals Countability and Cantor's Proof - Gauge-institute.org

Gauge Institute, February 2006
H. Vic Dannon
1 1 1 1 1
1 2 3 4 5
2 2 2 2
1 2 3 4
3 3 3
1 2 3
4 4
1 2
5
1
... ... * ... *
... ... ... ...
... ... ... * ... *
... ... ... ...
... ... ... * ... *
... ... ... ...
... ... .... * ... *
... ... ... ...
... ... ... * ... *
... ... ... ...
* ... * ... *
...
* ... *
...
*
Summing the number of the rationals along the zig-zag, for
n 1,2,3.... ,
1 2 3 .... n
. (1)
0
Thus,
2(12 3 .... n
) .
0
That is,
Tarski ([3], or [4, p.174]) proved that
for any sequence of cardinal numbers,
cardinal m , the partial sums inequalities
1 2
... n
n
2
n
. (2)
0
m ,
1
m ,
2
m m m m, for n 1,2,3....
imply the series inequality
m m ... m ....
m.
1 2
n
m ,…, and a
3
4