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

Am Mon, 09 Sep 2024 17:08:49 +0200 schrieb WM:

On 09.09.2024 13:49, joes wrote:

Am Mon, 09 Sep 2024 12:27:47 +0200 schrieb WM:

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.
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.

More precisely: every positive x has infinitely many smaller unit
fractions (mind the quantifier order).

The quantifier order related to the problem is this: NUF(x) = ℵo means:
There exist ℵo unit fractions smaller than any x > 0. If this is not
true, then there are fewer. How many unit fractions are smaller than any
x > 0. THAT is the question. None. But all are differente. Hence there
must be a first one smaller than all other unit fractions.

The wrong quantifier order is: „There is a fixed infinite set of unit
fractions, which are all less than any positive x.” The right one is:
Any positive x has AN infinite set of unit fractions less than it.
(Those sets are different; the one <0.4 is missing 1/2 compared the
set of UFs <0.75 but of course still infinite.)
How do you get to the „hence”?

A number is not an interval.

An interval has a length that can be expressed by a real number:
1/n - 1/(n+1) = x .
Then the interval (0, x) contains not all unit fractions, for instance
not 1/n.

But still infinitely many, since only finitely many are missing.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.