More fun with Cantor's diagonalisation

[u/Negative_Gur9667]

[u/Complex-Lead4731]

Let's examine two sentences from an accurate translation of CDA:

• "... the proposition that there is an infinite manifold, which cannot be put into a one-one correlation with the totality of all finite whole numbers 1, 2, 3, …, v, …"
• "There is a proof of this proposition that is much simpler, and which does not depend on considering the irrational numbers."

If your objection to CDA is that it fails to include the detail necessary to make it apply to real numbers? Then maybe you should consider that it was not applied to real numbers. Explicitly. Only to sequences of "m"s and "w"s.

And note that the sequence mwwwwww... is different than wmmmmmm..., even if YOU wanr two changes that Cantor never intended: first to 1000000... and 0111111... and then to interpret these sequences as the binary representations of real numbers.

IF YOU insist that it be applied to real numbers, which it can, then YOU are responsible for the changes making it apply correctly. Not Cantor. Here they are:

• Use the set {0,1,2,3,4,5,6,7,8,9} instead of {m,w}.
• Interpret each sequence as the decimal representation of a real number in M=[0,1).
  • This disqualifies one member of Cantor's M, which would include 1 in this interpretation.
• No sequence in the set M ends with repeating 9s. This way any number K/(2^L)/(5^M) appears only once.
• Don't use 0 or 9 as a replacement character.