Talk:Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum

Disjoint family of sets

How does the statement

"Thus the sets in $\family {F_i}$ are disjoint."

follow from the lines above? The reasoning seems to be that for an arbitrary $\ds S \in \cup_{k \mathop \in \N} F_k$, there is some first $k$ such that $S \in F_k$. And thus, the sets are disjoint? It might follow, but I don't! ––St.nerol (talk) 20:34, 4 September 2023 (UTC)

It is wrong because $F_2=X$, isn't it? --Usagiop (talk) 21:14, 4 September 2023 (UTC)

Oh, I didn't even think of that! I was more confused about the reasoning itself. Even if there is some first set in a sequence containing a given element, it says nothing about its containment in later sets. –St.nerol (talk) 10:55, 5 September 2023 (UTC)

Two proofs?

Now there is a "Proof 2" which actually works, thanks to User:Usagiop. The current "Proof 1" due to User:GFauxPas seems, to me, unsalvageable. There is a reference to an exercise in Real Analysis by Folland. I checked the exercise, which gives no hint for the proof except that $\MM$ contains an infinite family of disjoint sets. This is used in the proof by Usagiop, so for all we know there might not be an essentially different proof? Shouldn't the working proof be put before the error ridden, at the very least? –St.nerol (talk) 10:11, 7 September 2023 (UTC)

Can we get a consensus here? --prime mover (talk) 10:20, 7 September 2023 (UTC)

Yes, I only fixed the definition of $F_i$ and copied the rest, because I don't like to overwrite someone's proofs. But feel free to modify or rearrange my stuff. --Usagiop (talk) 17:34, 7 September 2023 (UTC)

I put a note on the talk page of GFauxPas. It seems unlikely that anyone else can or will fix his version of the proof. If he doesn't reply within some reasonable time, I think it contributes nothing of value and should be deleted. St.nerol (talk) 20:09, 9 September 2023 (UTC)

I put a deletion proposal for Proof 1. (User:GFauxPas does not seem to be active.) −St.nerol (talk) 08:04, 19 September 2023 (UTC)

done --prime mover (talk) 00:45, 1 December 2023 (UTC)