Timeline for Condensation for L[U]
Current License: CC BY-SA 2.5
9 events
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| Oct 10, 2010 at 21:17 | history | edited | Andrés E. Caicedo | CC BY-SA 2.5 |
Added a remark explaining why there are several versions of condensation.
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| Oct 10, 2010 at 12:06 | comment | added | Andrés E. Caicedo | ... (i.e., have the form $L[D]_\beta$ for some $D$ such that $D\cap L[D]_\beta\in L[D]_\beta$ looks like a normal measure on some cardinal $\gamma$), and are not initial segments of $L[U]$ (meaning, are not $L[U]_\gamma$ for any $\gamma$) or of $<_{L[U]}$. | |
| Oct 10, 2010 at 12:06 | comment | added | Andrés E. Caicedo | Hi Eran. All it is claimed in the relevant part of 19.3 is that the canonical well-ordering of $L[U]$ behaves as usual, meaning each constructible-initial-segment $L[U]_\alpha$ is an initial segment. This is obvious from how the order $<_{L[U]}$ is defined. But (letting $L[U]$ be the 'least' model for a measurable, meaning, choosing the measurable of least possible size and $U\cap L[U]$ normal in $L[U]$) there are many premice which look like such a thing ... | |
| Oct 9, 2010 at 7:56 | comment | added | Eran | In Jech Set Theory, Theorem 19.3 (page 340 6'th line from the top) he claims that each premice is an initial segment for $<_{L[D]}$ | |
| Oct 9, 2010 at 2:16 | comment | added | Andrés E. Caicedo | 3.3.(b) is not difficult, actually. (And anyway, Kanamori refers to specific sections in Devlin's "Constructibility" and Moschovakis's "Descriptive Set Theory", and I'm pretty sure it can also be found somewhere in Jech's.) In any case, a more subtle issue is that we cannot get for this version of the construction of L that the small L[U] like model you get by condensing is actually an initial segment of L[U]. I'm running late to catch a plane, but I hope I'll remember (once I'm over the jet lag) to write a remark on this issue. | |
| Oct 8, 2010 at 19:27 | comment | added | Eran | OK, so the proof relies on existence of this sentence sigma_1 from 3.3(b) (then by elementarity every submodel of L[U] belongs to the hierarchy). Now - where can I find this sentence (Without using as a parameter L[U]) ? does this follow from the well ordering of L[U]? | |
| Oct 8, 2010 at 18:43 | comment | added | Andrés E. Caicedo | Hi Eran. Then definitely the argument in the middle of the proof of Lemma 20.2 (with the reference to 3.3.(b)) is the fastest route. | |
| Oct 8, 2010 at 18:38 | comment | added | Eran | Dear Andres, By L[U] I mean a k-model (k measurable) as is defined in Kanamori's section 20. I do refer to the "the non-fine structural version". The fine-structure version is indeed available in few places (best is Devlin) but I would like to know if there is a simpler one, without going into the morass of the J_alphas. Something along the lines of the L-version? | |
| Oct 8, 2010 at 18:23 | history | answered | Andrés E. Caicedo | CC BY-SA 2.5 |