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

On 10/13/2024 5:04 PM, FromTheRafters wrote:

Chris M. Thomasson explained :

On 9/18/2024 7:31 AM, 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.

>
With regard to a very _strict) line of thinking (no mixing and matching), say 100% natural numbers... There is NO "gap" between, say:
>
3 and 4

 No gaps. But there are gaps between these naturals with respect to the rationals. That is, there are rationals which fill the distance between natural numbers (and integers) as embedded in Q.

Wrt breaking the rules, there is a unit fraction half way between any gap, say 1/2 ;^)
Say with 3 and 4:
dif = 4 - 3 = 1
mid = 3 + dif / 2 = 3.5
So, 3.5 is between 3 and 4 wrt a gap and restricting ourselves to unit fractions between a gap... ;^)

 With some algebraic numbers not being in Q, we have reals filling the 'gaps' in Q with respect to the reals but there are no 'gaps' in Q.