Timeline for How wealthy are canonical inner models?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| May 2, 2020 at 18:24 | comment | added | Asaf Karagila♦ | @Todd: Aha, that's nice. Does that also satisfy the "generic is different" criteria, so that generic diamonds are different? | |
| May 2, 2020 at 17:59 | comment | added | Todd Eisworth | Another random thought. Replace your diamond sequences by allowing countably guesses at each $\delta$ instead of one. Then you can distinguish diamond sequences by whether or not they are "somewhere" a $\diamond^*$-sequence, that is, whether there is a stationary set $S$ where any set is guessed "almost always". Once you've got a $\diamondsuit^*(S)$ sequence, any run-of-the-mill single-guessing $\diamondsuit$ sequence on $S$ is built from it in some sense. This would push towards $L$ having only a single type of $\diamondsuit$ sequence, built from the hierarchy in standard way. | |
| May 2, 2020 at 7:17 | comment | added | Asaf Karagila♦ | @Todd: I imagine that this family is somehow related to the "agree modulo permutations on a club"? | |
| May 2, 2020 at 3:49 | comment | added | Todd Eisworth | A $\diamondsuit$-sequence gives you an almost disjoint family of stationary subsets of $\omega_1$ in a natural way. Does that family capture any essential information about the $\diamondsuit$ sequence? Do you get any mileage out of declaring two $\diamondsuit$ sequences equivalent if those a.d. families generate the same ideal, for example? Just random late night thoughts.. | |
| Apr 29, 2020 at 20:35 | comment | added | Asaf Karagila♦ | @Andreas: That's interesting, I can see why that would be true. Maybe someone has a suggestion of a reasonable notion of equivalence, and then we can settle this... | |
| Apr 29, 2020 at 19:54 | comment | added | Andreas Blass | Since you asked about the situation in $L$, I'll conjecture that, for any "reasonable" notion of equivalence, $L$ will have $\aleph_2$ inequivalent diamond-sequences and that this should be provable by tweaking standard constructions of one diamond sequence in $L$. (Here "reasonable" is intended to mean something like "combinatorial" and to exclude relative constructibility.) | |
| Apr 29, 2020 at 19:39 | comment | added | Asaf Karagila♦ | @Andrés: One way is naturally to consider the relative constructibility order on diamond sequences. But then trivially we get that $L$ admits one class. Maybe that's okay... | |
| Apr 29, 2020 at 19:33 | comment | added | Asaf Karagila♦ | @Andrés: Indeed that was my thinking. It might be that somehow I'm wrong. Maybe any reasonable notion of sameness results in either "everything is equivalent" or "there is a maximal number of diamonds". | |
| Apr 29, 2020 at 19:30 | comment | added | Andrés E. Caicedo | @Asaf I fear that we need to clarify the notion first before much can be said. I was hoping that wealth could be expected because of combinatorial consequences: in this model we can build this and that diamond sequences, the first gives us A, the second B, and "there does not seem to be a reasonable way" of getting B from the first or A from the second. | |
| Apr 29, 2020 at 19:26 | comment | added | Asaf Karagila♦ | (Note that this is predicated on a notion of "distinct" that is satisfied by pairwise generic sequences.) | |
| Apr 29, 2020 at 19:24 | comment | added | Asaf Karagila♦ | @Andrés: Correct me if I'm wrong, but we can add a generic diamond sequence by forcing with its initial segments. Now iterate... Or take a side-by-side product, I suppose, for at least $\aleph_2$ different diamonds. | |
| Apr 29, 2020 at 19:22 | comment | added | Andrés E. Caicedo | Let's start with: Is there a candidate (forcing) model where there is some evidence that one should reasonably expect wealth? | |
| Apr 29, 2020 at 19:19 | comment | added | Asaf Karagila♦ | Well, I suspected that much. But then, what would be a good way to identify "essentially the same" diamonds? | |
| Apr 29, 2020 at 19:17 | comment | added | Yair Hayut | I think that equivalence up to a club is still too strong (you still have $2^{\aleph_1}$ many such diamond sequences from one). Take $X \subseteq \omega_1$ and ask the diamond to guess some coding of the pair (A, X), where $A \subseteq \omega_1$. By taking the guess for the A-part, in ordinals in which we guessed the $X$ part correctly, we get a diamond sequence. For any $X$ you get a different diamond sequence and the non-trivial parts of those sequences can agree only on a bounded piece. | |
| Apr 29, 2020 at 18:19 | comment | added | Asaf Karagila♦ | I just want to clarify that diamonds in canonical inner models are ethical diamonds, i.e. diamonds that were not obtained by forcing. | |
| Apr 29, 2020 at 18:13 | history | asked | Asaf Karagila♦ | CC BY-SA 4.0 |