Re: Incompleteness of Cantor's enumeration of the rational numbers

WM wrote :

On 21.11.2024 00:01, FromTheRafters wrote:

After serious thinking WM wrote :

On 20.11.2024 21:05, FromTheRafters wrote:

WM wrote on 11/20/2024 :

On 20.11.2024 15:14, FromTheRafters wrote:

WM formulated on Wednesday :

>

>
It does not make it wrong, but it unmasks it at imprecise. That's why I don't like it. We can do better.

>
It works well enough.

>
Really? Then you can answer the following questions:
>
Let every unit interval after a natural number on the real axis be coloured white with exception of the powers of 2 which are coloured black. Is it possible to shift the black intervals so that the whole real axis becomes black?

>
No, of course not.

>
Thank you. You are the first person not bewitched by set theory.

 The real line is infinitely long in both directions. Your shifted intervals are all on the positive real axis.

>
But if Cantor was right, all integers, positive as well as all other could be in bijection with the powers of 2.

Integer powers of two? Yes, both are countable sets.

Depending on the order of the tasks. I think half red or half black.

>
Recidivistic?

 I recognize McDuck in disguise, or your notion that there are fewer primes than there are natural numbers. You keep finding different ways to say the same wrong thing.

>
The analytical limit is wrong in your opinion?

Which limit? Is a half-plane smaller than a whole plane? Is a ray shorter than a line? Is a line segment shorter than a ray? What size symbol could you write on a whole plane that you could not write on a half plane? If McDuck's earnings always outpace his spending how can one consider him ever going bankrupt 'in the limit'? Which limit?