Re: Replacement of Cardinality

On 7/30/24 1:37 PM, WM wrote:

Le 30/07/2024 à 03:18, Richard Damon a écrit :

On 7/29/24 9:11 AM, WM wrote:

 

But what number became ω when doubled?

 ω/2

And where is that in {1, 2, 3, ... w} ?

>
No, that is w double, what number in the first set became w?
>

>
Every natural number when doubled is a Natural Number.

>
No.

>
Why not? WHich ones don't?

 ω/2 and larger.

Which is what number?
The input set was the Natural Numbers and w, so you are just proving you are lying,

>

>

Note, ω-1 doesn't exist in the base transfinite numbers, just as -1 doesn't exist in the Natural Numbers, you can't go below the first element.

>
If all natural numbers exist, then ω-1 exists.

>
Why?

 Because otherwise there was a gap below ω.

But you combined two different sets, so why can't there be a gap?
That is just the nature of unbounded sets.

 

That is unavoidable. You believe in the magical appearance of infinitely many unit fractions. That breaks logic and mathematics.

>
Nope,

 ∀n ∈ ℕ: 1/n - 1/(n+1) > 0. Note the u niversal quantifier.

Right, so we can say that ∀n ∈ ℕ: 1/n > 1/(n+1), so that for every unit fraction 1/n, there exists another unit fraction smaller than itself. The only way to have a smallest unit fraction is to have a largest natural number, at which point they aren't even "potentially infinite" as you have established a finite limit to them.
Remember, one property of Natural numbers that ∀n ∈ ℕ: n+1 exists.

 Regards, WM