Re: Replacement of Cardinality

Le 28/07/2024 à 19:07, Richard Damon a écrit :

On 7/28/24 7:55 AM, WM wrote:

Mistake! ℕ U ω = {1, 2, 3, ..., ω}

 And who was using that set?

I.

 

 

yields the set
{2, 4, 6, ..., ω, ω+2, ω+4, ..., ω*2}.
>

>
Why?

 See the correction.

 But what number became ω when doubled?

ω*2

 Every natural number when doubled is a Natural Number.

No.

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.

 But, just as we can expand the Natural Numbers to the Integers, and get negative numbers, we also might be able to define an extention to the transfinite numbers that can have a ω-1 element.

ω-1 is not transfinite but cisfinite.

Using the unit fractions itelligent readers understand that there must be a first one after zero. Others must believe in the magical appearance of infinitely many unit fractions.
 

Nope, since that implies there is a highest Natural Number, which breaks their definition,

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