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

Am 02.10.2024 um 14:56 schrieb joes:

Am Wed, 02 Oct 2024 13:10:09 +0200 schrieb WM:

On 01.10.2024 22:05, Jim Burns wrote:

∀n ∈ ℕ: 1/n - 1/(n+1) > 0 shows ∀n ∈ ℕ: 1/n > 1/(n+1) [...]
which shows each unit fraction 1/n is not first.

Right.

No.

Yes.

∀n ∈ ℕ: 1/n - 1/(n+1) > 0 does not prove that n+1 is a natural
number.

@Mückenheim:
Right. But the following PEANO-AXIOM
       An e IN: S(n) e IN
zusammen mit der Definition
       n+1 := s(n)
beweist es, Du Depp.
Since from them we get the theorem:
       An e IN: n+1 e IN    (*).
Hint: Jim DIDN'T claim that your trivial statement proves that n+1 is a natural number, but that it proves that for each and every unit fraction there's a SMALLER one [which it does together with (*)].
[Well, actually, for showing this we would also have to refer to a definition of "unit fraction": u is a /unit fraction/ iff there's an n e IN such that u = 1/n.]

Note the infinite sequence 1, 2, 3, ..., ω-2, ω-1, ω. It consists of infinitely many finite numbers.

@Mückenheim:
Red' doch keine solche Scheiße daher, Mückenheim!
1. Ist ω GANZ GEWISS keine "finite number", sondern /the smallest infinite ordinal number/.
2. Sind die Ausdrücke "ω-2", "ω-1" nicht definiert. Du redest also wieder mal saudummen Scheißdreck daher.
3. Ist 1, 2, 3, ... ω "technisch gesehen" keine "unendliche Folge" (jedenfalls keine mit Indexmenge IN).
Kurz und gut: Nichts als purer Sachwachsinn, der auch nicht das Geringste mit dem in Rede stehenden Sachverhalt zu tun hat.