The study of canonical inner models for large cardinal hypotheses, with particular attention to their fine structure theory, and iterability issues.

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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
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1answer
902 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 ...
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1answer
295 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?
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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: ...
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4answers
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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 $HOD$. ...
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556 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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Consistency strength of saturated ideals on $[\lambda]^{<\kappa}$

In the Handbook of Set Theory, Foreman notes that in models constructed by Magidor from a huge cardinal, there exists a normal ideal on $[\omega_2]^{<\omega_1}$ that is $\omega_3$-saturated. Other ...
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A Question Regarding the Relation Between 0-sharp and Koepke's Bounded Truth Predicate.

In Jech's SET THEORY (a very early edition to which I have access), it is shown that the existence of 0-sharp implies the existence of a truth definition for the constructible universe L. Does the ...
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1answer
312 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
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1answer
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$\omega$-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 ...
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3answers
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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
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1answer
399 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 ...
3
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1answer
345 views

The Covering Lemma for L[U]

Hi, The covering lemma for L[U] (minimal k-model) is stated in Mitchel's handbook chapter: "Assuming zero-dagger does not exist then covering holds for L[U] or there is a sequence C ⊆ κ, which is ...
13
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1answer
864 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
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1answer
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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
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2answers
431 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 ...