Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 10/14/24 6:35 AM, WM wrote:

On 12.10.2024 22:47, Richard Damon wrote:

On 10/12/24 2:39 PM, WM wrote:

On 11.10.2024 20:02, Jim Burns wrote:
>

"Potential ℕ" vs. "actual ℕ" leaves unchanged
which claims we reason to.

>
Potential infinity: All natural numbers when double yield natural numbers but larger numbers are among the result.
In actual infinity no new naturals can be created but since doubling doubles the value of each number, numbers of the second number class are created.

 

No, in actual infinity no new natural can be created,

 True.
 

but every natural number can find the one that is twice itself in the set,

 False. n*2 > n for all numbers, even for infinite numbers.
 Regards, WM

Nope, that is a property of finite numbers, and infinite ordinals only. It doesn't apply for infinite cardinal numbers. (Omega * 2 is bigger than Omega, but Aleph_0 * 2 is just Aleph_0)
For finite numbers n*2 is also finite, and thus less than Omega, and in the actually infinite set, that doesn't require the creation of new numbers, as the full infinite set was there to begin with, even if you didn't notice them when you looked. (And that doesn't make them "dark")