9
votes
1answer
292 views

Inner model in which every uncountable cardinal is large

The following is known: $(*)$ If $0^\sharp$ exists, then any uncountable cardinal is is an inaccessible cardinal (and even more) in $L$. My question is that: Are there any large cardinal ...
15
votes
1answer
889 views

Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...
8
votes
1answer
289 views

What is known about equiconsistency of PFA and existence of supercompact cardinals?

Question: What is the last status of known partial results and new approaches on the problem of equiconsistency of PFA and existence of supercompact cardinals?
25
votes
3answers
1k views

Latest stand of core model theory?

What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004: ...
3
votes
4answers
249 views

What is the role of absoluteness in existence of a non-trivial self elementary embedding on an inner model?

ِDefinition (1): Call an inner model M "rigid" if there are no non-trivial elementary embeddings $j: M\longrightarrow M$. It is possible that $L$ is not rigid, while the problem is open for $HOD$. ...
4
votes
1answer
308 views

Models of Determinacy

Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it ...
4
votes
1answer
151 views

$\omega$-small and properly small premice.

Let $\mathcal M$ be a premouse. $\mathcal M$ is said to be $\omega$-small if and only if whenever we have that $\kappa=crit(E)$ for some extender $E$ on the sequence of $\mathcal M$, then $\mathcal ...
5
votes
3answers
304 views

A question about Mitchell/Steel Fine Structure and Iteration Trees

In chapter 8 of Mitchell's and Steel's FSIT, they prove a central fine structural result, which basically states that if $\mathcal{M}$ is 1-small, $k$-sound, $k$-iterable premouse then the ...
7
votes
1answer
396 views

Why “adding” a single extender cannot give an L-like inner model for say, a strong cardinal?

The constructible universe $L$ is too thin for large cardinals greater than measurable. To build $L$-like inner models for large cardinal, it is natural to think about "adding" the evidences into the ...
13
votes
1answer
846 views

Devlin's “Constructibility” as a resource

It is fairly well-known among set-theorists that Keith Devlin's 1984 book "Constructibility" has flaws in its initial development of fine structure theory. (See Lee Stanley's review 1 of the text for ...
10
votes
1answer
300 views

Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals?

Background I am interested in elementary embeddings from a model of set theory into itself. One way of producing such elementary embeddings is when the model is generated by indiscernibles; this idea ...
6
votes
2answers
428 views

Characterizing elementary embeddings of $L$ and $L_\alpha$ under 0#

Suppose 0# exists. It is clear that every order preserving map from the indiscernibles to the indiscernibles gives an elementary embedding from $L$ to $L$. Furthermore, following lemmas 18.7 and 18.8 ...