2

Could someone please help me out. In the book example, the arrows go in one way and the answer that i came up the arrows goes in a different way. Does the path you have to go through cantors zigzag really matter ?

enter image description here

user602774
  • 1,093
  • 7
  • 19
  • 31
  • 1
    Any method you give for enumerating/counting the rational numbers, which eventually lists any given rational number, constitutes a valid proof that the rationals are countable. Cantor gave 1/1, 1/2, 2/2, 1/3, etc. Equally valid would be 1/1, 1/2, 2/2, 1/3, 2/3, 3/3, ... and 1/1, 2/2, 1/2, 2/3, 1/3, 3/3, ... Makes no difference, as long as your enumeration doesn't take infinitely long to get to any fixed (finite) rational number. – Patrick87 Feb 08 '12 at 15:10

1 Answers1

2

in the proof "infinite triangular matrix of rationals", the matrix will be constructed like this: 

1/1  1/2  1/3  1/4  ... 
2/1  2/2  2/3  2/4  ... 
3/1  3/2  3/3  3/4  ... 
..

you have to find a function (or a bijective relation between set and natural numbers) which counts the rationals like f(x) = y it means (i.e.) 

f(1) = 1/1 
f(2) = 1/2 
f(3) = 2/2

and so on. In cantor's system, he counts the rationals with a certain function which can be found in his proof, see below on page 7:

http://www.gauge-institute.org/zigzag/cantorzigzagP.pdf

if you can find a function which counts these in this way as you painted, why not ;-) maybe you get the next nobel prize in math.

Erhan Bagdemir
  • 5,231
  • 6
  • 34
  • 40
  • 9 years late to the party...but H. Vic Dannon is a crank. He's really not a good resource for this sort of thing. Claiming that Cantor is wrong, when just about every mathematician out there accepts his work, takes a lot more evidence than an 8-page treatise with giant fonts and margin. – Adam Smith Nov 20 '21 at 08:11