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

On 06.10.2024 18:07, joes wrote:

Am Sun, 06 Oct 2024 16:15:27 +0200 schrieb WM:

On 06.10.2024 14:01, joes wrote:

Am Sun, 06 Oct 2024 12:32:16 +0200 schrieb WM:

On 06.10.2024 11:49, Moebius wrote:

Am 06.10.2024 um 10:40 schrieb WM:

On 06.10.2024 05:35, Moebius wrote:

Am 05.10.2024 um 22:38 schrieb WM:

On 05.10.2024 22:13, Moebius wrote:

Am 05.10.2024 um 22:01 schrieb WM:
>
ω/2 * 2 e IN,

Nein.

Doch, weil IN gegenüber der Multiplikation ABGESCHLOSSEN ist. (Das
kannst Du sogar in deinem Bestseller nachlesen, falls Du es
inzwischen vergessen haben solltest.) Mit ω/2 e IN wäre auch ω/2 *
2 e IN (weil 2 e IN ist).

Das gilt für

die Menge IN (und deren Elemente), wie schon gesagt.
In Zeichen: An,m e IN: n * m e IN.

Komisch auch, dass ω/2*2 (ist die Reihenfolge wichtig?)

Nein.

Interessant. ω*2 /2 ist ja eindeutig unendlich, aber ω/2 *2 nicht?

ω*2 /2 = ω =  ω/2 *2

Not all infinite sets can be compared by size, but we can establish some
useful rules.

Schwach.
 

 The rule of subset proves that every proper subset has fewer elements
than its superset. So there are more natural numbers than prime numbers,
|N| > |P|, and more complex numbers than real numbers, |C| > |R|.  Even
finitely many exceptions from the subset-relation are admitted for
infinite subsets. Therefore there are more odd numbers than prime
numbers |O| > |P|.
>
 The rule of construction yields the numbers of integers |Z| = 2|N| +

1

and the number of fractions |Q| = 2|N|^2 + 1 (there are fewer rational
numbers Q# ). Since all products of rational numbers with an irrational
number are irrational, there are many more irrational numbers than
rational numbers |X| > |Q#|.
>
 The rule of symmetry yields precisely the same number of real
geometric points  in every interval (n, n+1] and with at most a small
error same number of odd numbers and of even numbers in every finite
interval and in the whole real line.

How small an error? Surely there is one more even number.

[3, 5] has one more odd number.

 

This theory makes the number of natural numbers (and of course of other
sets too) depending on the numerical representation. The set {1, 11,
111, ...} of natural numbers has only comparatively few elements.
Therefore the set of natural numbers in unary or binary notation has
fewer, in hexadecimal notation more than |N| elements. The set {10, 20,
30, ...} has |N|/10 elements, but if the zeros are only applied as
decoration, this set, like the set {1', 2', 3', ...}, has |N| elements.
It will be a matter of future research to investigate the effect of
different numerical systems in detail.

I don’t understand this argument. What is the cardinality of the set
{1, 11, 111, …} ? What of {0, 1, 10, 11, 100, …} in binary and decimal?
What is a decoration different from?

It is a symbol added to a number but not belonging to the numeral.
Regards, WM