Timeline for Does ZF + BPI alone prove the equivalence between "Baire theorem for compact Hausdorff spaces" and "Rasiowa-Sikorski Lemma for Forcing Posets"?
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| Dec 3, 2021 at 3:44 | comment | added | Samuel G. Silva | And the point that was bothering me, it showed up to be some subtlety. If ZF proves that a poset is countable and also proves that a certain family of dense sets is countable, then the corresponding forcing axiom holds in ZF. But when you go to ZFC there are posets which are countable due to some countable union of countable sets argument, and similarly there are families of dense sets which need some portion of choice to be countable, and things get a little more complicated. | |
| Dec 3, 2021 at 3:36 | comment | added | Samuel G. Silva | Hi Asaf, sorry. My student and myself, we figured out how to write down the argument. Thanks ! | |
| Dec 1, 2021 at 8:49 | comment | added | Asaf Karagila♦ | Did you ever get around to reading my answer again? | |
| May 10, 2021 at 18:17 | comment | added | Asaf Karagila♦ | I'm not using DC in that implication. I'm using BCT. (Also, I don't have Kunen on hand, so it's a bit vague when you keep mentioning 3.1 and 3.2 there.) | |
| May 10, 2021 at 17:48 | comment | added | Samuel G. Silva | But, as I said in the beginning, my reference for such an argument is 3.2 of Chapter II of Kunen, and in this part he refers to 3.1, where the Axiom of Choice is required to define certain functions which will be used (in a closure argument) to construct a p.o. of restricted cardinality. That is, precisey, the point I want to avoid the use of the Axiom of Choice: the use appearing at 3.1 of Kunen in the we-can-work-with-smallers-orders stuff, and this happens in an implication at which DC is at the thesis, not at the hypothesis. | |
| May 10, 2021 at 17:47 | comment | added | Samuel G. Silva | Oh, I see. Thanks. But there is still a problem. The implication where is supposed to be used Boolean algebras (the Stone Space, etc.) is: Baire for Compact Hausdorff spaces --> Rasiowa-Sikorski for posets, so there is no DC to start with in this part (in fact, we want to prove DC at the end of it, since RS and DC are equivalent). | |
| May 5, 2021 at 22:40 | comment | added | Asaf Karagila♦ | Yes. That's an observation due to George Boolos, but also due to several other people (including myself). The one direction is the usual proof, the other is simple: if $T$ is a tree of height $\omega$, $T^*\prec T$ is a countable elementary submodel, then $T^*$ is a countable subtree of height $\omega$, so either $T^*$ has a maximal node which is maximal in $T$ or it has a branch which is a branch in $T$. | |
| May 5, 2021 at 22:38 | comment | added | Samuel G. Silva | ... Wow, is the Principle of Dependent Choices equivalent to Downward Lowenheim Skolem ? This is news to me (I thought Downward Lowenheim Skolem was equivalent to the full Axiom of Choice). I will take a look at that... | |
| May 5, 2021 at 17:45 | history | edited | Asaf Karagila♦ | CC BY-SA 4.0 |
Complete rewrite!
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| May 5, 2021 at 1:03 | history | answered | Asaf Karagila♦ | CC BY-SA 4.0 |