Timeline for Philosophy of forcing and ctm
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 31, 2022 at 2:46 | comment | added | Timothy Chow | Let us continue this discussion in chat. | |
| Jan 31, 2022 at 2:38 | comment | added | Kushi | But isn't it possible that $\omega_1^M$ is countable? And Corollary 18.5 of Jech says if zero sharp exists then $\mathcal{P}(\mathbb{N})\cap L$ is countable, so definitely $\mathcal{P}(\mathbb{N})\cap L_\alpha$ is countable for any $\alpha$. | |
| Jan 31, 2022 at 2:34 | comment | added | Timothy Chow | @Kushi If $M$ is uncountable and standard and satisfies AC, then $M$ contains all countable ordinals, so ${\cal P}({\mathbb N}) \cap M$ will be uncountable. | |
| Jan 31, 2022 at 2:23 | comment | added | Kushi | I understand how ctm works and how forcing is useful for proving relative consistency statement. By the way why does any uncountable standard model satisfy CH? Say we are doing Cohen forcing, then isn't the existence of generic filter guaranteed as long as $\mathcal{P}(\mathbb{N})\cap M$ is countable (while $M$ could be uncountable)? | |
| Jan 31, 2022 at 2:16 | comment | added | Timothy Chow | As for countability itself, it does play an important role in the standard proof that forcing works. The way you get a generic filter is that you list all the forcing conditions and hit them one by one (a la Rasiowa-Sikorski). Cohen talks about this in his article, "The Discovery of Forcing." He mentions, for example, that in any uncountable standard model of ZFC, CH holds. So in some sense, countability has to enter the picture somewhere, if you're trying to prove the independence of CH. | |
| Jan 31, 2022 at 2:15 | comment | added | Timothy Chow | @Kushi I'm not sure what you mean when you say you "don't feel good about ctm." If the concern is that they seem artificial, then that doesn't matter for the purposes of proving consistency. Something is consistent if there's a model, no matter how strange the model may seem. | |
| Jan 31, 2022 at 2:02 | comment | added | Kushi | Technically forcing uses ctm, and I don't feel good about ctm, so I'm struggling to find a way to interpret it. A more concrete question is 3 (whether there are important statements which has ctm and yet provably does not have uncountable transitive model). | |
| Jan 31, 2022 at 1:20 | history | answered | Timothy Chow | CC BY-SA 4.0 |