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

On 9/9/2024 3:13 PM, WM wrote:

On 09.09.2024 21:05, Jim Burns wrote:

On 9/9/2024 6:39 AM, WM wrote:

ℵo unit fractions cannot exist without
1, 2, 3, ... unit fractions before.

>
Each unit.fraction has 1,2,3,... unit.fraction before.
>
It is incorrect (it is a quantifier shift)
to conclude from that correct claim
that
🛇 there are 1,2,3,... unit.fractions before each

>
It is a property of
real points on the positive real line,
that every sequence has a beginning.

That's not a property of these points:
⎛ Each real number > 0 is a rational > 0 or
⎜ between sides of a split of rationals > 0
⎜
⎜ Each rational > 0 is the ratio of
⎜ two integers > 0
⎜
⎜ Each nonempty set S of integers > 0 holds
⎜  minimum.S
⎜ Each integer > 0 has a successor.integer > 0 (+1)
⎜ Each integer > 0 has a predecessor.integer > 0 (-1)
⎝  except the minimum integer > 0 (1).

What alternative configuration can the points have
in your opinion.

The usual configuration is sufficient.
Pause the extra bells and whistles for a moment.
What makes it sufficient is that
( Each integer > 0 has a successor.integer > 0 (+1)
We can define a k.successor k+1 = k∪{k}
The claim that some sequences are infinite
follows from
the claim that, for existing k, k∪{k} exists.
That k∪{k}.claim isn't justified by logic;
instead, it is justified by _being what we mean_
Denying it might not lead to contradiction,
but it definitely leads to irrelevance.

This is the first question:
Do ℵo unit fractions exist smaller than any x > 0,
as Fritsche claims?

You have messed up the quantifier order.

The second question concerns a weaker claim:
Do ℵo unit fractions exist smaller than any 1/n > 0?

ℵ₀.many < ⅟n
k ⟼ ⅟(k+n)
ℵ₀.many < ⅟n′
k ⟼ ⅟(k+n′)
but not the same unit.fractions.
⅟(k+n) ≠ ⅟(k+n′)