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

On 9/9/24 11:08 AM, WM wrote:

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. Note that real points are in question. Real points fixed on the real line.

The number of unit fractions smaller than any positive x is Aleph_0. PERIOD. They are all diffferent.
That does NOT mean there must be a first, THAT is irrational finite thinking, which doesn't work on unbounded sets, and your using it has exploded your brain into smithereens by its contradictions.

  > 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 it contains almost all of them, an Aleph_0 of them while missing only a finite number of them.
A set of size Aleph_0 can (repeatedly) loose finite number of elements and maintain its size. This fact seems to be beyond your ability to understand.

 Regards, WM

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