Re: Cantor Diagonal Proof

On 2025-04-17 23:45:46 +0000, Lawrence D'Oliveiro said:

On Tue, 15 Apr 2025 19:44:11 +0100, Mike Terry wrote:
 

It is not a proof by induction as it makes no use of an induction
hypothesis PHI(n), and does not have any essential induction step PHI(n)
--> PHI(n+1).

 It does. It shows that, if the first N digits match, then so does the
(N+1)th digit. Given that it matches the first digit, those are your two
requirements for proof by induction.
 

Nonsense!  RH's example using your very list constructs the
anti-diagonal 0.111111... which is NOT IN YOUR LIST.

 But at every stage, by induction, it matches an element in the list. You
only get a number that’s not in the list when you get to the end of the
construction. But it’s an infinite construction!

Likewise, at stage N the first N digits differ form the first N digits of
each of the first N elemenst of the list so it s none of the first N
elements of the list. Therefore it is different from every number in the
list.
A list that both contains a number that is different from every number in
the list does not exist.
--
Mikko