Timeline for The poset of k-small downward-closed subposets of a poset P is k-filtered when k is a regular cardinal?
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10 events
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| Aug 1, 2010 at 16:12 | comment | added | Joel David Hamkins | The smallest non-regular infinite cardinal is $\aleph_\omega$, which is a countable union of the sets $\aleph_n$, each of which are smaller than $\aleph_\omega$. But every infinite successor cardinal is regular. | |
| Aug 1, 2010 at 15:41 | comment | added | Harry Gindi | I meant what I wrote, but everything works out fine anyway. If you look at the edit history, I was confused for a moment, because I noticed, as you said, that the union of those downward-closed subsets will be downward closed. The thing I was reading wasn't very clear, so I tried to change it to make it less trivial. Anyway, this was the answer I was looking for. I didn't know that regularity is this property (small union of small sets will be small), and, perhaps as a sign of my lack of knowledge of set theory, I thought that all infinite cardinals enjoyed this property. | |
| Aug 1, 2010 at 14:06 | comment | added | Joel David Hamkins | In general, what you stated in your question is fine: if $\kappa$ is regular, then the union of fewer than $\kappa$ many $\kappa$-small downward closed subsets will still be a $\kappa$-small downward closed subset. (But this is not what you wrote in symbols, since your are missing a downarrow in the codomain of $A_i$. If you are mapping just into $P_\kappa(P)$, then you do need to take the downward closure, and the issue of my last paragraph kicks in. | |
| Aug 1, 2010 at 13:59 | comment | added | Harry Gindi | $\kappa$-small downward-closed subsets, rather. | |
| Aug 1, 2010 at 13:57 | comment | added | Harry Gindi | Re: Your edit: Alright, well it seems that everything works out fine in this case anyway, since the poset $P$ is specifically a $\kappa$-good tree (well-founded order, and the downward closure of any element $\alpha$ is $\kappa$-small, so $P$ is the $\kappa$-filtered union of its $\kappa$-small subsets. | |
| Aug 1, 2010 at 13:53 | comment | added | Joel David Hamkins | The cofinality definition is equivalent, since a smaller cofinality means there is a small increasing sequence of ordinals below $\kappa$, which would be a counterexample to the characterization I give. | |
| Aug 1, 2010 at 13:47 | vote | accept | Harry Gindi | ||
| Aug 1, 2010 at 13:47 | comment | added | Harry Gindi | Why doesn't the Wikipedia page have that definition?! It has some silly thing about "a cardinal is called regular if it is equal to is own cofinality", which is completely opaque and unhelpful. Thank you very much! | |
| Aug 1, 2010 at 13:46 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
added 306 characters in body
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| Aug 1, 2010 at 13:40 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |