Timeline for What are the Possible Large Cardinals of $L[X]$?
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| Oct 5, 2013 at 21:42 | comment | added | Andrés E. Caicedo | The thing is, what an extender codes depends among other things on what we call its length. Very short extenders are essentially ultrafilters, and we do not get more than an $L[\mu]$ model out of them (but this is as it should be). Longer extenders code more. And (to be safe, let's say) for a while at least, they relativize nicely so we can look at constructibility over them. Many of the complications of the sequences $\mathcal E$ are simply not relevant if we do not care about fine structural issues. That said, I cannot think of a reference explicitly addressing this. | |
| Oct 5, 2013 at 21:41 | comment | added | Andrés E. Caicedo | The first topic is covered in Kanamori's The higher infinite. Section 25 is on $\mathcal P_\kappa(\lambda)$, and the specific result on constructibility is exercise 25.8 (which has a generous hint). Extenders are discussed in section 26, but their relation with constructibility is not explored. This is discussed to some extent in Steel's handbook article, for example. Two key points are that you can code significant embeddings with extenders (which Kanamori discusses), and that extenders can be applied to models other than the one the embedding came from. Steel explains this carefully. | |
| Oct 5, 2013 at 20:34 | comment | added | user40919 | And also probably basic: for extenders $E$ on $\kappa$, any general results what $L[E]$ could be, whether it is $L$, $L[\mu]$, or something more complicated? Thank you | |
| Oct 5, 2013 at 20:31 | comment | added | user40919 | Probably very basic, but might you know where I can read about how $\mathcal(P)_\kappa(\lambda)$ ultrafilters added to $L$ as predicates give us back $L$? | |
| Oct 5, 2013 at 20:26 | comment | added | user40919 | Thank you Andres, that is very helpful. I am trying to digest it all. I will definitely take a look at the Steel article and Zeman book. | |
| Oct 5, 2013 at 20:08 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
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| Oct 5, 2013 at 20:01 | history | answered | Andrés E. Caicedo | CC BY-SA 3.0 |