Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"]

On 13.04.2025 10:20, joes wrote:

Am Thu, 03 Apr 2025 22:17:33 +0200 schrieb WM:

On 03.04.2025 16:40, Alan Mackenzie wrote:

Ifff the natural numbers are an actually infinite set, then its
elements are invariable and fixed.

How else could it be.

The alternative is potential infinity.

The density of the natural numbers on the real axis is greater than the
density of the even natural numbers.

In which sense? There are infinitely many of either.

For all infinitely many finite lengths the densities are 2 to 1.

 

Therefore the doubled natural numbers cover twice as many space than
before.

Again, how? How much?

They cover the ordinal axis between 0 and 2ω evenly.

Natural numbers not available before
have been created. This is possible only based on potential infinity. Or
all natural numbers have been doubled, then the result stretches
farther, namely beyond all natural numbers.

No, there is no natural number whose double is larger than ω.

Wrong if infinity is actaul.

It is more suggestive to double the set ℕ U {ω} = {1, 2, 3, ..., ω}
with the result
{1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2} .
What elements fall between ω and ω*2?

Where is ω?

On the ordinal axis immediately after all natural numbers.
Regards, WM