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

On 9/16/2024 2:13 PM, Jim Burns wrote:

On 9/15/2024 9:31 PM, Ross Finlayson wrote:

On 09/15/2024 03:07 PM, FromTheRafters wrote:

on 9/15/2024, Ross Finlayson supposed :

On 09/15/2024 11:03 AM, FromTheRafters wrote:

After serious thinking Ross Finlayson wrote :

 

"What, no witty rejoinder?"

>
What you said has no relation to
the 'nextness' of elements in discrete sets.
What is 'next' to Pi+2 in the reals?

>
In the, "hyper-reals", it's its neighbors,
in the line-reals, put's previous and next,
in the field-reals, there's none,
and in the signal-reals, there's nothing.

 

What is the successor function on the reals?
Give me that, and maybe we can find the
'next' number greater than Pi.

>
Ah, good sir, then I'd like you to consider
a representation of real numbers as
with an integer part and a non-integer part,
the integer part of the integers, and
the non-integer part a value in [0,1],
where the values in [0,1], are as of
this model of (a finite segment of a) continuous domain,
these iota-values, line-reals,
as so established as according to the properties of
extent, density, completeness, and measure,
fulfilling implementing the Intermediate Value Theorem,
thus for
if not being the complete-ordered-field the field-reals,
yet being these iota-values a continuous domain [0,1]
these line-reals.

 As n → ∞, (ι=⅟n), ⟨0,ι,2⋅ι,...,n⋅ι⟩ → ℚ∩[0,1]
 ℚ∩[0,1] is not complete.
ℚ∩[0,1] has one connected component,

Sorry. I meant the opposite of that.
ℚ∩[0,1] is NOT connected, NOT "continuous".
For each irrational x ∈ (ℝ\ℚ)∩[0,1]
ℚ∩[0,x) and ℚ∩(x,1] are open in ℚ∩[0,1]

being what you (RF) call "continuous".
ℚ∩[0,1] has no points next to each other.
 

I wonder what you think of something like Hilbert's
"postulate of continuity" for geometry, as with
regards to that in the course-of-passage of
the growth of a continuous quantity, it encounters,
in order, each of the points in the line.

 That sounds like the Intermediate Value Theorem,
where "encounters each" == 'no skips".
 The Intermediate Value Theorem
is equivalent to
Dedekind completeness.