All Questions
15 questions
5
votes
0
answers
213
views
Friedman's proof of covering lemma for $L$
There is a two-page proof of the covering lemma for $L$ using $\Sigma_n$ substructures (Theorem 3.10) in Sy Friedman's Fine Structure and Class Forcing, compared to the proof that spans about twenty ...
- 333
5
votes
1
answer
243
views
Inner model with a $\mathit{\Delta}^1_3$-good well-ordering of the reals
The constructible universe $L$ has some nice properties:
$L$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)
For any $\mathit{\Sigma}^1_2$ formula $\varphi(x)$ and a ...
- 2,286
7
votes
0
answers
207
views
Literature on the reals or their gaps in $L[0^\sharp]$?
I'm doing my Bachelor's Thesis on the continuum in $L$ and $L[0^\sharp]$.
In $L$ I study the gaps without new reals (sets of natural numbers) in the hierarchy, as presented in Gaps in the ...
- 421
6
votes
0
answers
214
views
Recent literature on the gaps of reals on $L$ or other inner models?
I'm doing my Bachelor's Thesis on Gödel's constructible universe $L$.
I'm interested in the gaps without new reals (sets of natural numbers) in the hierarchy, as presented in Gaps in the constructible ...
- 421
7
votes
1
answer
335
views
The core model and elementary embeddings
Let $K$ be the core model (below a Woodin cardinal). Let $j \colon K \to M$ be an elementary embedding, where $M$ is well founded. Under which conditions can we conclude that $j$ is an iterated ...
- 5,112
5
votes
1
answer
396
views
GCH implies acceptability
I have been studying the concept of acceptability, particularly in its relation to GCH.
There are many versions of it in the sources I have found, with some slight variations, and some of them are ...
- 2,467
14
votes
3
answers
934
views
Forcings predicted by core model theory $+$$ZFC$ results proved by the method of core model theory
I have two unrelated question.
First question. To motivate the question, let me explain an example. The natural way to force the failure of singular cardinals hypothesis ($SCH$), is to start with a ...
- 32.1k
19
votes
2
answers
9k
views
The Ultimate L in a Nutshell: On Descriptive Articles
Everybody who catches a fleeting glimpse of Woodin's central papers on Ultimate $L$ (i.e. Suitable Extender Models I & II), admits that they aren't so tempting for lazy readers who don't like to ...
3
votes
1
answer
253
views
Definition of $M_n^\sharp(X)$ for arbitrary set $X$
The definition of $M_n^\sharp$ in [OIMT10] is the unique sound, $(\omega,\omega_1,\omega_1+1)$-iterable mouse which is not $n$-small, but all of whose proper initial segments are $n$-small.
What is ...
- 1,170
2
votes
0
answers
149
views
Strong determinacy principles for "small" sets
In the course of a (fun but silly) project I'm working on, I've run into the following class of determinacy notions:
Suppose I have a "reasonably definable" class $\mathcal{C}$ of games (in the usual ...
- 21.2k
7
votes
1
answer
497
views
Core model with $\omega$ Woodin cardinals
In [KwoM] it is proved that the the core model K exists under the assumption that there is no inner model having a Woodin cardinal and satisfying ZFC. Furthermore, they also mention that the result is ...
- 1,170
4
votes
0
answers
142
views
Characterization of $L[T_{2n+1}]$ as a direct limit of mice
I am asking for a reference request/proof sketch for the result of Steel
that characterizes $L[T_{2n+1}]$ as a direct limit of mice.
Given that both $L[T_{2n+1}]$ and $M_{2n}$ have a $\Sigma_{2n+2}$ ...
- 603
15
votes
1
answer
408
views
Consistency strength of $\aleph_2$-Souslin hypothesis
Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis?
Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and $\...
- 32.1k
7
votes
0
answers
294
views
Core model for supercompact cardinals and iteration trees
I have a few somehow related questions:
Question 1. What do we expect for an inner model $\mathcal{K}$ to be a core model for a supercompact cardinal? What properties should it have, and what ...
- 32.1k
5
votes
1
answer
921
views
A Hot Betting On HOD
Remark: This question is based on an open question at the end of a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency".
$HOD$ as an inner model of $ZFC$ lies ...
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