Timeline for Does the generalized $\Delta$-system lemma imply some weak version of the GCH?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Jun 26, 2013 at 19:17 | vote | accept | Santi Spadaro | ||
| May 22, 2013 at 4:24 | comment | added | user10290 | Thank you, Joel. I think I understand this a bit better. | |
| May 18, 2013 at 2:10 | comment | added | Joel David Hamkins | Well, as I said, $\Delta(\omega_2,\omega_2)$ can already be refuted without KH by considering the tree $2^{\lt\omega_1}$, where one has at least $\omega_2$ many branches of size $\omega_1$, but no $\Delta$ system of size 3, and Kurepa trees have nothing to do with it. But I'm not sure why you think this refutes $\Delta(\omega_2,\omega_1)$, and in particular, I dispute your implication in (A). | |
| May 17, 2013 at 20:15 | comment | added | Eran | My question is why KH is not a counter example: A) $\Delta(\omega_2,\omega_1) => \Delta(\omega_2,\omega_2)$ B) KH (using the same arguments as in the proof) provides $\neg \Delta(\omega_2,\omega_2)$ (=>$\neg\Delta(\omega_2,\omega_1)$) C) It is consistent to have ZFC + GCH + KH -------------------------- hence GCH doesn't have to imply $\Delta(\omega_2,\omega_1)$. | |
| May 17, 2013 at 18:48 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Fixed confusion of $\lambda$ with $\eta$.
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| May 17, 2013 at 17:45 | comment | added | Joel David Hamkins | Eran, about the Kurepa tree, I don't see how it helps. A Kurepa tree would seem to be about getting a family of $\omega_2$ many subsets of $\omega_1$ without any $\Delta$-system of size 3. But we already have such a family without any KH hypothesis, just by using paths through the tree $2^{\lt\omega_1}$. Or have I misunderstood your idea? | |
| May 17, 2013 at 17:43 | comment | added | Joel David Hamkins | Eran, I don't think the implication you mention in your comment is quite right. Although in the former case, we see that every very large family of small sets has a large subfamily forming a $\Delta$-system, perhaps a slightly smaller family of small sets may not have a subfamily forming a $\Delta$ system of that size. Indeed, my argument shows that if $2^\delta=\delta^{++}$, but otherwise the GCH holds, then $\Delta(\delta^{+++++++},\delta^+)$ will hold, but not $\Delta(\delta^{++},\delta^+)$, which contradicts your implication. | |
| May 17, 2013 at 17:39 | comment | added | Joel David Hamkins | Erin, my update shows that $\Delta(\delta^{+++++++},\delta^+)$ is equivalent to the assertion that $(\delta^{++++++})^\delta=\delta^{++++++}$, which is equivalent to saying $2^\delta$ is at most $\delta^{++++++}$, a weak form of the GCH. | |
| May 17, 2013 at 17:32 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Proved full equivalence
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| May 17, 2013 at 14:27 | comment | added | Santi Spadaro | Very nice, Joel! It's more than I hoped. | |
| May 17, 2013 at 10:07 | comment | added | Eran | @Joel - isn't this proof goes just as well assuming ZFC + GCH + KH (Kurepa's Hypothesis)? | |
| May 17, 2013 at 10:00 | comment | added | Eran | Erin - $\Delta( \delta ^{+++++++}, \delta^+) => \Delta( \delta ^{++}, \delta^+)$ | |
| May 17, 2013 at 7:22 | comment | added | user10290 | It is interesting! What happens when $\Delta( \delta ^{+++++++}, \delta^+)$ ? | |
| May 17, 2013 at 0:58 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 289 characters in body
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| May 17, 2013 at 0:32 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 640 characters in body; added 175 characters in body
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| May 17, 2013 at 0:23 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |