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

On 9/9/24 6:27 AM, WM wrote:

On 08.09.2024 22:11, Richard Damon wrote:

On 9/8/24 3:48 PM, WM wrote:

 

Select any gap between one of the first ℵo unit fractions and its neighbour. Call its size x. Then ℵo unit fractions cannot fit into the interval (0, x), independent of the actual size.
>

But that is changing the value of x in the middle of the problem which isn't allowed.

 No.

>
Given that new x, we can choose a new set of Aleph_0 unit fractions below that x.

 ℵo unit fractions are claimed to be smaller than every x > 0. If that is true then I can choose as the x one of the ℵo intervals between two of them.
 Regards, WM
 

No, because the claim is GIVEN an x, we can do "Y", that is make the Aleph_0 unit fractions below it. Until you have chosen your x, we don't need to provide those unit fractions, so, you can't use them to create your x.
It seems your ignorance extends to ordering.
To follow YOUR idea, then *YOU* get stuck in the infinite loop of every time you change your x, the unit fractions change so you need to change to another x, and the unit fractions change again.
Its sort of like the problem with the request for the biggest number you can think of, because as soon as you think of it, you can imagine something bigger, but that doesn't show that there actually is a biggest number that actually exists, because our thinking of them doesn't make them come into existance, it was the original definition of them.