Timeline for Action of infinite symmetric groups on iterated power sets
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2018 at 13:44 | history | edited | user44143 | CC BY-SA 4.0 |
edited to reflect changed question
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| Nov 18, 2018 at 12:15 | comment | added | user44143 | @ThomasForster, the phrase "put all thought of the axiom of choice out of your mind" was not helpful -- it was ambiguous between "avoid" and "don't worry about". I will edit the question. | |
| Nov 18, 2018 at 6:46 | comment | added | Thomas Forster | Yes, k=2 is easy because, as you say, two things belong to the same $G_1$-orbit iff they [and their complements, actually] are equinumerous. However, for the purposes for which i need this result, it is quite important to have a proof that doesn't use AC. And i need it for arbitrarily large $k$. Thank you all for your interest! | |
| Nov 18, 2018 at 4:41 | comment | added | მამუკა ჯიბლაძე | Mmm actually I oversimplified - in general one has leveled graphs that are not necessarily forests. But still I believe such a graph faithfully represents an element of $\mathcal P^k(X)$. Given one such, say $\mathfrak S$, place its elements on the bottom level, elements of $\bigcup\mathfrak S$ above them, elements of $\bigcup\bigcup\mathfrak S$ above those, etc. The relation is just $\ni$. Any embedding of the top level of every such height $k$ leveled graph into $X$ then determines an element of $\mathcal P^k(X)$ provided any two nodes at any level have distinct successor sets above them. | |
| Nov 18, 2018 at 3:53 | comment | added | user44143 | I assume so, but I don't trust my ability to visualize or reason informally about $\cal{P(P(P(X)))}$. | |
| Nov 18, 2018 at 3:41 | comment | added | მამუკა ჯიბლაძე | The same should work for all $k$ too, no? I mean, take two copies of the same leveled forest of height $k-1$ and let them grow differently to the $k$th level (differently means having, say, all cardinalities of all new branches just pairwise different) | |
| Nov 18, 2018 at 2:53 | history | edited | user44143 | CC BY-SA 4.0 |
deleted 2 characters in body
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| Nov 18, 2018 at 2:36 | history | answered | user44143 | CC BY-SA 4.0 |