Timeline for Making all cardinals countable and its HOD
Current License: CC BY-SA 3.0
12 events
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| Oct 28, 2013 at 11:56 | history | edited | Mohammad Golshani | CC BY-SA 3.0 |
added 145 characters in body
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| Aug 22, 2013 at 2:47 | comment | added | Andrés E. Caicedo | @EverettPiper Another reason is that there is a very compelling picture of the universe as a (class) forcing extension of its $\mathsf{HOD}$, and the ultimate $L$ analysis would suggest the same ought to hold under appropriate assumptions replacing $\mathsf{HOD}$ with "ultimate $L$". But, in fact, since core-like models are ordinal definable, we expect that (ultimately), this largest core model ought to coincide with $\mathsf{HOD}$ itself. I'm sure there are more serious and sophisticated reasons as well. | |
| Aug 22, 2013 at 2:44 | comment | added | Andrés E. Caicedo | @EverettPiper Here is a weak reason: The model $\mathsf{HOD}$ of natural models of determinacy certainly has this property. This was shown by Steel (and extended by Woodin) for $L(\mathbb R)$, and has since been established for much larger models; in fact, the possibility of such a description of $\mathsf{HOD}$ is key in recent developments in descriptive inner model theory. Under very large cardinals, we expect "higher order instances" of results that hold under determinacy to be true. | |
| Aug 21, 2013 at 12:00 | comment | added | Everett Piper | @AndresCaicedo Would you mind elaborating on the expectation that an initial segment of HOD be "core like"? I ran across a similar sentiment a couple of years ago but I was unable to see why the person making this claim would have this expectation. | |
| Aug 19, 2013 at 21:18 | answer | added | Joel David Hamkins | timeline score: 8 | |
| Aug 19, 2013 at 18:56 | comment | added | Andrés E. Caicedo | @MonroeEskew It seems a natural problem. Under appropriate assumptions, we would expect an initial segment of $\mathsf{HOD}$ (certainly beyond $\mathsf{HOD}\cap V_{\omega_1}$) to be "core like", in particular to satisfy $\mathsf{GCH}$. This is a natural test question to try to clarify what those additional assumptions ought to require. | |
| Aug 19, 2013 at 18:37 | comment | added | Monroe Eskew | What is the motivation for this question? | |
| Aug 19, 2013 at 14:25 | comment | added | Joel David Hamkins | A more fundamental question is what does $\text{HOD}^{V[G]}$ mean exactly? The usual definition of HOD makes use of reflection to levels $V_\theta$ of the von Neumann hierarchy, which are ordinal definable, but since we lack power sets in $V[G]$ we do not have this hierarchy there. So what is $W$ exactly? | |
| Aug 19, 2013 at 9:52 | comment | added | Asaf Karagila♦ | (In my first comment I meant $H(\omega_1)^{V[G]}$, not $V_{\omega_1}$) | |
| Aug 19, 2013 at 9:42 | comment | added | Asaf Karagila♦ | There is also a paper by Roguski called "Hartogs's numbers and axiom of power-set" which is impossible to find online (and the library at HUJI, where the paper can be found, is closed this week). I don't recall the exact construction the author uses, but it might be relevant. | |
| Aug 19, 2013 at 9:39 | comment | added | Asaf Karagila♦ | Perhaps a good test case would be to collapse an inaccessible $\kappa$ like that, and consider $V_{\omega_1}^{V[G]}$. | |
| Aug 19, 2013 at 9:20 | history | asked | Mohammad Golshani | CC BY-SA 3.0 |