Timeline for Is the axiom $\Diamond\Box\varphi\to\Box\Diamond\varphi$ in c.c.c. forcing potentialism equivalent to the productivity of c.c.c. forcing?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Feb 9, 2018 at 22:31 | comment | added | Not Mike | One of many, since $\mathsf{MA}_{\aleph_1}$ is equivalent to $\diamondsuit \sigma_A \implies \sigma_A$ for every $\Sigma_1$ formula $\sigma_A$ with parameter $A\subset \omega_1$ and $\Sigma_1$ statements are upward absolute. However, this isn't all that helpful since in the context of $\mathsf{MA}_{\aleph_1}$ the statement $\diamondsuit \sigma$ implies the existence of a powerfully-c.c.c. witness (since being powerfully-c.c.c. is downward absolute; one must necessarily already exists.) | |
| Feb 9, 2018 at 0:53 | comment | added | Joel David Hamkins | That makes it a railway switch, in the terminology of jdh.hamkins.org/set-theoretic-potentialism-ws2018. | |
| Feb 7, 2018 at 20:47 | comment | added | Not Mike | This seemed worth pointing outing: for every c.c.c. partial-order $\mathbb{P}$ of size $\aleph_1$. The statement $\varphi(\mathbb{P}):= $"$\mathbb{P}$ is $\sigma$-centered", has the fun property that one of the statements: $\diamondsuit \square \varphi(\mathbb{P})$ or $\diamondsuit \square \neg\varphi(\mathbb{P})$ always hold. (since $MA_{\aleph_1}$.is always a c.c.c. extension away, and satisfies either $\varphi$ or $\neg\varphi$.) | |
| Feb 6, 2018 at 14:01 | answer | added | Not Mike | timeline score: 6 | |
| Feb 6, 2018 at 9:37 | comment | added | Joel David Hamkins | @NotMike That is very interesting, but could you post a fuller argument? I think it is fine to post as an answer, even though it isn't answering my question, since it is clearly related. | |
| Feb 6, 2018 at 4:57 | comment | added | Not Mike | @MohammadGolshani If I'm not mistaken, the statement "for every finite family of c.c.c. graphs $K_1, \ldots K_n \subset [\omega_1]^2$ there exists an uncountable $H \subset \omega_1$ with $[H]^2 \subset K_1 \cap \ldots K_n$", follows from $(.2)_{ccc}$. This would entail the failure of $CH$. | |
| Feb 6, 2018 at 4:01 | comment | added | Mohammad Golshani | Does the axiom scheme $(.2)_{ccc}$ imply the failure of $CH$? | |
| Feb 4, 2018 at 15:24 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
updated slides link, improved some writing
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| Feb 3, 2018 at 16:41 | comment | added | Joel David Hamkins | Ha! Meanwhile, one of the points I made in my lectures was that this modal vocabulary is capable of expressing sweeping general principles about the object theory, in this case, about ccc forcing. | |
| Feb 3, 2018 at 16:14 | comment | added | YCor | My first reaction reading your question was: "does my browser have an encoding bug"? :) | |
| Feb 3, 2018 at 15:43 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 211 characters in body
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| Feb 3, 2018 at 15:29 | history | asked | Joel David Hamkins | CC BY-SA 3.0 |