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

On 9/19/2024 6:38 AM, WM wrote:

On 18.09.2024 16:31, Jim Burns wrote:

On 9/18/2024 8:39 AM, WM wrote:

On 16.09.2024 19:30, Jim Burns wrote:

On 9/15/2024 3:47 PM, WM wrote:

I don't believe in gaps on the real line.

>
There aren't gaps and there aren't next.numbers
in numbers.situating.splits of rationals with
countable.to.numerators.and.denominators

>
So what is next instead?

>
What is between one and the next?
A gap.
>
There is no gap in the real line.
>
There is no next in the real line.
If there were, there'd be a gap.

>
Try to use logic.
Either there is a point next to zero
or there is no point next to zero,

Consider ABCDEFGHIJKLMNOPQRSTUVWXYZ
Q is between A and Z
A and Z are not next to each other.
Nothing is between P and Q
P and Q are next to each other.
⎛ There is a gap between P and Q
⎜ A gap isn't some _presence_
⎜ A gap is an _absence_ of whatever.
⎜
⎜ Either there is a point with
⎜ an _absence_ between it and zero,
⎜ or there _isn't_ a point with
⎜ an _absence_ between it and zero.
⎜
⎜ It is the latter which holds for
⎝ the gapless real line.
Consider the real line.
For any two points x and y,
(x+y)/2 is between x and y.
x and y are not next to each other.
There are no two points such that
(x+y)/2 is NOT between x and y.
There are no two points next to each other.

or there is no point next to zero,
that means there is nothing, i.e., a gap.

If there is nothing (no points) between 0 and x
then there IS a point next to 0:  x
Consider again ABCDEFGHIJKLMNOPQRSTUVWXYZ
and being.next.to and having.a.gap.