Re: Cantor Diagonal Proof

On 4/13/25 5:30 PM, Lawrence D'Oliveiro wrote:

Here’s another way to look at the Cantor construction.
 It is possible to construct a list of numbers, ostensibly from ℝ, where it
is provable, by induction, that the Cantor construction cannot produce a
number not in the list. This shows that you cannot prove, a priori, the
validity of the Cantor construction.

Why not? What step do you object to? GIVEN the list of numbers, we can show by simple pairing that the number created by the method of the diagonal proof can not be equal to any of the numbers.

 But then, you cannot prove, a priori, the validity of proof by induction,
either. Instead, it has to be added as an explicit axiom when constructing
the integers ℤ.

But he didn't use "induction" to make his proof. He used basic inspection. If the diagonal number, by its definition, will differ from any number on the list by at least one of its digits, it can not be equal to any of them.
That is merely an application of simple logic.

 In the same way, you can add “the Cantor construction works” as an axiom
when constructing the set of reals ℝ. Otherwise, ℝ is just the set of
computable numbers.

But we don't need an axiom to show it works. Cantor's proof OF THE UNCOUNTABILITY OF THE REALS just follows from simple logic, by showing that by its defintion it differs from all the numbers on the list.