Timeline for Do $X$ and $Y$ have the same cardinality if their families of finite subsets do?
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Jul 4, 2023 at 18:31 | vote | accept | Salvo Tringali | ||
| Jul 4, 2023 at 3:57 | answer | added | Guozhen Shen | timeline score: 16 | |
| Jul 3, 2023 at 17:33 | answer | added | Farmer S | timeline score: 11 | |
| Jul 3, 2023 at 14:34 | comment | added | godelian | @MikhailKatz The original question did not indicate that this was intended to be thought in ZF, and in ZFC it is indeed a trivial well-known fact (as even the OP suspected), whence my vote to migrate to Maths Sack Exchange. Now the question has changed | |
| Jul 3, 2023 at 14:25 | comment | added | Calliope Ryan-Smith | Not a solution, but I think that if $A$ is Dedekind-infinite and $|A^2|\geq^*|\mathscr{P}(A)|$ then we have that $A^2$, $[A]^{{<}\omega}$, $[A^2]^{{<}\omega}$, and $\mathscr{P}(A)$ all surject onto each other, while only some inject into each other. | |
| Jul 3, 2023 at 14:07 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
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| Jul 3, 2023 at 14:06 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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| Jul 3, 2023 at 14:03 | comment | added | Mikhail Katz | This seems like an interesting question (without choice as per the comment above). I don't understand why people are voting to close it. @SalvoTringali you should modify your question to reflect the fact that it is about ZF, not ZFC. | |
| Jul 3, 2023 at 14:02 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
improved on the question (hopefully!)
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| Jul 3, 2023 at 2:27 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Clarified title/grammar
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| Jul 2, 2023 at 22:23 | comment | added | Andrés E. Caicedo | But I don't know right away what the situation is without choice. | |
| Jul 2, 2023 at 22:13 | comment | added | Andrés E. Caicedo | The answer is yes under choice. | |
| Jul 2, 2023 at 21:59 | comment | added | Salvo Tringali | @MichaelHardy I'm not asking whether $|X| = |Y|$ implies that the family $\mathcal P_{\rm fin}(X)$ of finite subsets of $X$ has the same cardinality as the family $\mathcal P_{\rm fin}(Y)$ of finite subsets of $Y$. I'm rather asking whether $|\mathcal P_{\rm fin}(X)| = |\mathcal P_{\rm fin}(Y)|$ implies that $|X| = |Y|$. | |
| Jul 2, 2023 at 21:42 | review | Close votes | |||
| Jul 3, 2023 at 14:32 | |||||
| Jul 2, 2023 at 21:36 | comment | added | Salvo Tringali | @NikWeaver Thanks! And a ref for the statement that "An infinite set has, in ZFC, the same cardinality as the collection of its finite parts" is essentially Corollary 8.13 in Chapter 0 of Hungerford's Algebra. | |
| Jul 2, 2023 at 21:11 | comment | added | Nik Weaver | I think this is just the fact that (in ZFC) any infinite set has the same cardinality as the set of all of its finite subsets. So yes, if the sets of finite subsets of $X$ and $Y$ have the same cardinality then so do $X$ and $Y$. | |
| Jul 2, 2023 at 21:09 | comment | added | Michael Hardy | $\ldots\,$the mapping $Z\mapsto\{ f(x):x\in X\} \subseteq Y$ is a bijection between the set of all finite subsets of $X$ have the same cardinality as the set of all finite subsets of $Y.$ Conclude that those both have the same cardinality. $\qquad$ | |
| Jul 2, 2023 at 21:07 | comment | added | Michael Hardy | You state yourself that the question of whether $X$ and $Y$ have the same cardinality is undecidable in ZFC, then you make it part of your question. The lesser part of your question seems to be: If $X$ and $Y$ have the same cardinality, then does the set of all finite subsets of $X$ have the same cardinality as the set of all finite subsets of $Y.$ That one has an easy affirmative answer: If $X$ and $Y$ have the same cardinality, then there is a function $f:X\to Y$ that is injective and surjective. For finite sets $Z\subseteq X,$ show that$\,\ldots\qquad$ | |
| Jul 2, 2023 at 21:03 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
deleted 23 characters in body
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| Jul 2, 2023 at 20:58 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
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| Jul 2, 2023 at 20:51 | history | asked | Salvo Tringali | CC BY-SA 4.0 |