Sujet : Re: Replacement of Cardinality
De : invalid (at) *nospam* example.invalid (Moebius)
Groupes : sci.logic sci.mathDate : 05. Aug 2024, 01:24:13
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <v8p63d$a0fn$2@dont-email.me>
References : 1 2 3 4 5 6 7 8 9 10 11 12
User-Agent : Mozilla Thunderbird
Am 05.08.2024 um 01:07 schrieb Chris M. Thomasson:
On 8/4/2024 8:35 AM, WM wrote:
Le 04/08/2024 à 02:15, Moebius a écrit :
Am 03.08.2024 um 21:54 schrieb Jim Burns:
On 8/3/2024 10:23 AM, WM wrote:
>
NUF(x) = ℵ₀ for all x > 0 is wrong.
>
Nonsense.
>
Actually, Ax > 0: NUF(x) = ℵ₀.
>
You mean that there are ℵ₀ unit fractions smaller than all positive x?
Obviously not.
What I mean is that for all positive x there are ℵ₀ unit fractions smaller than x.
Impossible. [...] Not even one unit fraction can be smaller than all positive x.
No one (except WM) claimed that there's a unit fraction which is smaller than all positive x.
Huh?
WM is constantly mixing up
∀x > 0: ∃^ℵ₀ u ∈ ⅟ℕ: u < x (true)
with
∃^ℵ₀ u ∈ ⅟ℕ: ∀x > 0: u < x (false) .
(Here ⅟ℕ = {1/n : n e IN} is the set of all unit fractions.)
Say x = 1/2, there are infinite smaller unit fractions, say, 1/4, 1/5, 1/6, ect... However there is only one larger one, 1/1. See? No smallest one for 1/0 is not a unit fraction! There is a largest one, 1/1...
They tend to zero, but there is no smallest one...
Yeah.
Proof: If s is a unit fraction then 1/(1/s + 1) is a unit fraction which is smaller than s (for each and every s).
See?
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