Re: Cantor Diagonal Proof

On 4/14/25 6:46 PM, Lawrence D'Oliveiro wrote:

On Sun, 13 Apr 2025 19:57:14 -0400, Richard Damon wrote:
 

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?

 See my construction of the flipped-integer list.
 

Which assumes the computablility of the list that wasn't computable.

But he didn't use "induction" to make his proof.

 Precisely.

Because there is no need to.

 

He used basic inspection.

 How do you conduct an infinity of “basic inspections”?

By doing it. It is a simple nature of the Universal Conditional.
If for All x, we can show a difference, then there are No x without a difference.
We don't need "induction" to prove the full set, as the proof of each element is direct.
We can disprove that the diagonal is the Nth number, as it differs from that at the Nth digit, and thus CAN'T be the same number. We don't need to perform an induction to get to the Nth number, we can prove it directly from that number.

 

That is merely an application of simple logic.

 “We hold these truths to be self-evident”, is it?

Sure. If all are different, None are the same.

 

But we don't need an axiom to show it works.

 Yes we do. Because otherwise we can use induction to show it doesn’t.

No you can't. Your "induction" assumed something that was incorrect, and thus flawed.