Re: how

Le 01/05/2024 à 00:49, Richard Damon a écrit :

On 4/30/24 8:56 AM, WM wrote:

Le 27/04/2024 à 19:27, Richard Damon a écrit :

On 4/27/24 10:36 AM, WM wrote:

 

But since it isn't in the set, you can't use it as the value to "step back" to.

>
The question was whether it is an upper bound. Bounds need not belong to a set.

>
They do if you want to use them to step into the set, like you did.

 I use the upper bond only as upper bound.

 But then you can't talk about using it to "step back" from omega.

I can't because almost all natural numbers are dark.

 Yes, ALL Natural Numbers are less than omega.

Yes.

 That doesn't mean there is a largest Natural Number.
 In fact, the fact that the smallest Upper Bound we can find is Omega, which is out of the set of Natural Numbers means that there is no "Highest" Natural number.

The findable natural numbers constitute only a small initial segment of ℕ. Every findable number has ℵo successors. But these successors are natural numbers too which can be subtracted from ℕ, such than no natural numbers remain, only collectively.

 But there isn't a specific number that is the end.

There is no such number findable.

 That is the problem with your logic.

That is not a problem with logic. It is only a problem for people who cannot think farther than the definable or findable numbers reach. Intellectual inertia.
It is so easy:
Every findable number has ℵo successors. ==> Not all can be found.
All natural numbers have no successors. ==> There are more.
Regards, WM