Timeline for How "strong" is the existence of a non trivial ultrafilter on $\omega$?
Current License: CC BY-SA 3.0
6 events
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| Dec 31, 2016 at 10:54 | comment | added | Maxtimax | Of course what I'm about to answer is not precise at all (that was also one of the goals of the question; as well as the other question you mentionned in your answer), but I would say they are because they allow to "choose" from various sets, sets that themselves need not have any connection with $\omega$. So they do mention $\omega$, but not in the same way as what I called $\Omega$. Although I kind of feel that countable choice wouldn't be a "general choice principle" because it's restricted to countable sequences, whereas dependent choice is about any relation. But it's not precise at all | |
| Dec 31, 2016 at 9:49 | comment | added | Asaf Karagila♦ | @Maxtimax: So countable choice or dependent choice are not considered "general choices principles" since they do mention $\omega$ in their formulation? | |
| Dec 29, 2016 at 15:48 | vote | accept | Maxtimax | ||
| Dec 29, 2016 at 15:48 | comment | added | Maxtimax | This is indeed what I would call a "general choice principle" (there is no mention of $\omega$, etc.). So this establishes that (with really big quotation marks) "the existence of a nonprincipal ultrafilter over $\omega$, coming from a general choice principle is stricly weaker than $BPI$". Thanks for your answer ! This was mostly the part I was interested in (the other part was some form of generalization, that could be helpful to solve this) | |
| Dec 29, 2016 at 15:39 | comment | added | Asaf Karagila♦ | (You're not making a stupid mistake.) | |
| Dec 29, 2016 at 15:13 | history | answered | Andreas Blass | CC BY-SA 3.0 |