Timeline for What are some good lower bounds on the consistency of the failure of the PCF conjecture?
Current License: CC BY-SA 4.0
16 events
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| Apr 20, 2019 at 22:55 | review | Close votes | |||
| Apr 21, 2019 at 0:36 | |||||
| Feb 13, 2019 at 4:07 | answer | added | Mohammad Golshani | timeline score: 8 | |
| Feb 12, 2019 at 15:07 | comment | added | Asaf Karagila♦ | @Todd: Yes, and I am just wondering what are the current known bounds on the failure at $\aleph_\omega$. Do we know it at least implies the many strong cardinals needed for Gitik's work? (Or, as I initially hoped before your last comment, do we know that it requires even more?) Thanks for the two comments by the way! | |
| Feb 12, 2019 at 15:05 | comment | added | Todd Eisworth | With regard to the specific question, my understanding is that it's still not clear if the obstacle to moving things down to $\aleph_\omega$ is a matter of insufficient technology, or something deeper lurking beneath the surface. | |
| Feb 12, 2019 at 14:19 | comment | added | Todd Eisworth | Shelah has a lot of results that treat pcf assumptions themselves kind of like large cardinal statements. A typical result might show that a combinatorial statement implies a pcf statement, and from the pcf statement one can force the combinatorial statement, with the pcf statement being something currently intractable. | |
| Feb 12, 2019 at 13:20 | comment | added | Andrés E. Caicedo | Yes, that one.$ $ | |
| Feb 12, 2019 at 9:16 | comment | added | Asaf Karagila♦ | @Andrés: Do you mean the one from 2002 about PCF theory and Woodin cardinals? | |
| Feb 11, 2019 at 19:23 | comment | added | Andrés E. Caicedo | This is a good question. I don't think we know much about it yet. Something along the lines of the Gitik-Schindler-Shelah paper seems to be the state of the art. | |
| Feb 11, 2019 at 18:08 | history | edited | Asaf Karagila♦ | CC BY-SA 4.0 |
added 246 characters in body
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| Feb 11, 2019 at 18:02 | comment | added | Asaf Karagila♦ | @Will: Yes, but this is just $2^{\aleph_\omega}=\aleph_{\omega+2}$. We are talking about significantly larger gaps here. Good bounds include Woodin cardinals, or proper class of strong cardinals, or a sequence of $\omega_1+1$ strong cardinals, or whatever. I'm understand there is some ambiguity in "good lower bound", but obviously Gitik's initial result about SCH is not that. | |
| Feb 11, 2019 at 18:01 | comment | added | Will Brian | What are the best bounds you know already? From Gitik's work on SCH it follows that $2^{\aleph_0} > \aleph_{\omega+1}$ (with $\aleph_\omega$ a strong limit) is equiconsistent with a measurable cardinal $\kappa$ of Mitchell order $\kappa^{++}$. I'm guessing you already know this? It's probably worth mentioning in the question. It's hard to know what you would call a "good" bound without first knowing what you might consider an "everybody-already-knows-that" bound. | |
| Feb 11, 2019 at 17:15 | history | edited | Asaf Karagila♦ | CC BY-SA 4.0 |
edited title
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| Feb 11, 2019 at 17:14 | comment | added | Asaf Karagila♦ | YCor, I'm all in favor of informative titles, but I'm not sure what would be more informative? The PCF Conjecture is a fairly common term. | |
| Feb 11, 2019 at 17:14 | comment | added | YCor | A title informative about the topic (not only for specialized people) would be useful | |
| Feb 11, 2019 at 17:11 | comment | added | Asaf Karagila♦ | Of course, any corrections to what I understand is also welcomed. | |
| Feb 11, 2019 at 17:11 | history | asked | Asaf Karagila♦ | CC BY-SA 4.0 |