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

On 14.04.2025 11:03, joes wrote:

Am Sun, 13 Apr 2025 13:30:54 +0200 schrieb WM:

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 set of naturals does not change; every number either is or is not
included in it.

The set of defined natural changes like the set of known prime numbers. If oly defined naturals exist, then we have a potentially infinite collection.

 

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.

Aha! And for the one infinite length we get the undefined expression
inf/inf.

No, we get the limit (here it is appropriate to use it) of the function 2, 2, 2, ... .

 

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.

So there are infinitely large even naturals?

No, they are not natural numbers because all natural numbers are finite, i.e., less than ω. But by counting we can pass ω, according to Hilbert. Why shouldn't we pass it by multiplying?

 

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 actual.

Just no. It would have to be larger than ω (=infinite) itself. Half an
infinity is still infinite.

ω/n is larger than every definable natnumber but not larger than any natnumber.

 

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.

No, I mean in the second set? ω/2 is not even defined.

It is a dark number.

Or do you think
the naturals form consecutive infinities like
1, 2, 3, …, ω/ω, ω/(ω-1), ω/(ω-2), …, ω/3, ω/2, ω ?
 

ω/ω = 1.
We get 1, 2, 3, ..., ω/n, ..., ω-1, ω
Regards, WM