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

On 09.10.2024 19:08, joes wrote:

Am Wed, 09 Oct 2024 14:48:17 +0200 schrieb WM:

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

That is a weakness of your notion of cardinality.

The weaker weakness of Cantors Cardinality is that it is complete nonsense.

How do you compare finite sets?

By their numbers of elements.

Nonsense. Only potential infinity is used. Never the main body is
applied.

What "main body"?

The actually infinite numbers of dark elements.

 

For Cantor's enumeration of all fractions I have given a simple
disproof.

Your "proofs" tend to be nonsense.

Theorem: If every endsegment has infinitely many numbers, then
infinitely many numbers are in all endsegments.
Proof: If not, then there would be at least one endsegment with less
numbers.

I struggle to follow this illogic. Why should one segment have less
numbers?

All have the same numbers, namely ℕ. Some of the first numbers are transformed from contents to indices and than lost. But almost all numbers, namely ℵo, remain (because after every definable natnumber n ℵo numbers follow). If the intersection is less than ℵo, at least one endsegment must have fewer than ℵo numbers.

 

Note: The shrinking endsegments cannot acquire new numbers.

Not necessary, they already contain as many as needed.

It would be nessessary if all are infinite but their intersection is empty. Then the infinitely many numbers cannot be the same in all endsegments. Consider this finite example:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
{2, 3, 4, 5, 6, 7, 8, 9, 10}
{3, 4, 5, 6, 7, 8, 9, 10}
{4, 5, 6, 7, 8, 9, 10}
{5, 6, 7, 8, 9, 10}
{6, 7, 8, 9, 10}
{7, 8, 9, 10}
{8, 9, 10}
{9, 10}
{10}
{ } .
Satz: Jede Menge, die mindestens drei Elemente enthält (nennen wir diese Mengen DE), enthält drei Elemente gemeinsam mit allen DE-Mengen. Es gibt also drei Elemente, die in allen DE-Mengen enthalten sind. Das ist ein erlaubter Quantorentausch.
Regards, WM