(\text{thin})$ isomorphic?

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Dec 22, 2023 at 10:55	comment	added	YCor		Actually this argument is very natural because if trying to prove that the answer is positive, essentially the only thing to show is that every decreasing sequence of nonzero element has a nonzero lower bound (Parovicenko characterization of BAs isomorphic to $2^\omega/$fin under CH). Once one tries to do so, one sees that the density provides an obvious obstruction.
Dec 22, 2023 at 3:35	comment	added	Iosif Pinelis		One may say that "the famous construction of Hausdorff" is a species of the diagonal argument. Indeed, if the elements of the set $A_0\cap\cdots\cap A_n$ are $b_{n,0}<b_{n,1}<\cdots$, then $A=\{b_{0,0},b_{1,1},\ldots\}$.
Dec 21, 2023 at 21:42	comment	added	Michael Hardy		ok, Got it. $\qquad$
Dec 21, 2023 at 21:38	comment	added	Joel David Hamkins		...the least element of $A_0$. This is the index when you enumerate it in order $a_0,a_1,a_2,$ and so forth. But if you want to start at $n=1$, the construction will also work fine.
Dec 21, 2023 at 21:35	comment	added	Michael Hardy		So the $0\text{th}$ element of $A_0$ is . . . $\qquad$
Dec 21, 2023 at 21:33	comment	added	Joel David Hamkins		@MichaelHardy I start my enumerations with 0.
Dec 21, 2023 at 21:32	comment	added	Michael Hardy		Or "for every $n$ except $0$"? $\qquad$
Dec 21, 2023 at 21:32	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 1318 characters in body
Dec 21, 2023 at 20:28	comment	added	Dominic van der Zypen		Thanks for this really elegant argument!
Dec 21, 2023 at 20:27	vote	accept	Dominic van der Zypen
Dec 21, 2023 at 17:45	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 43 characters in body
Dec 21, 2023 at 16:38	comment	added	Joel David Hamkins		I added a brief explanation. See also my essay on the orders of infinty at infinitelymore.xyz/p/the-orders-of-infinity.
Dec 21, 2023 at 16:37	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 163 characters in body
Dec 21, 2023 at 16:34	comment	added	Iosif Pinelis		Could you please explain what "the famous construction of Hausdorff" is or give a reference to it?
Dec 21, 2023 at 16:27	history	answered	Joel David Hamkins	CC BY-SA 4.0