23 votes
Accepted

What is the modal logic of outer multiverse?

I've noticed that recently you have asked a few questions about my work, and so let me thank you; you are kind to take an interest. This particular question can be seen as part of the subject of set-...
Joel David Hamkins's user avatar
20 votes
Accepted

Taking a proper class as a model for Set Theory

What is shown in the cases you mention is not that the model is a model of ZFC, made as a single statement, but rather the scheme of statements that the model satisfies every individual axiom of ZFC, ...
Joel David Hamkins's user avatar
18 votes

A “paradox” about the inner model problem

Inner model theorists use the word "canonical" to explain the problem in intuitive terms, it is indeed a vague problem, though it is as precise as anything in the region of superstrong ...
Grigor's user avatar
  • 904
16 votes

Forcings predicted by core model theory $+$$ZFC$ results proved by the method of core model theory

Question 1: A good example is Woodin's extender algebra. One reference describing the discovery of the extender algebra is the introduction to Neeman's book The Determinacy of Long Games. I am ...
Gabe Goldberg's user avatar
15 votes
Accepted

Why do we need the comparison lemma?

If you take a step back and squint your eyes, inner model theory is basically the theory of a big measuring stick that measures the vague notion of "strength of natural set theoretical theories&...
Andreas Lietz's user avatar
14 votes

The Ultimate L in a Nutshell: On Descriptive Articles

The latest issue of the Bulletin of Symbolic Logic has a long article by Woodin on Ultimate L: WOODIN, W. (2017). IN SEARCH OF ULTIMATE-L THE 19TH MIDRASHA MATHEMATICAE LECTURES. The Bulletin of ...
Todd Eisworth's user avatar
13 votes

A “paradox” about the inner model problem

The issue is Thesis 2. The notion of a canonical inner model is vague, but I don't think that when inner model theorists use the term they mean to suggest that the model is obtained by iterating a ...
Gabe Goldberg's user avatar
12 votes

The Ultimate L in a Nutshell: On Descriptive Articles

You may also look at the following: 1) How Woodin changed his mind: new thoughts on the continuum hypothesis. 2) Tutorial outline: suitable extender sequences. Computational prospects of infinity. ...
Mohammad Golshani's user avatar
12 votes
Accepted

Ordinal-definable witnesses to the perfect set property?

Let $A$ be the set of those reals that are not OD. Then $A$ is OD (since I've just defined it), and it's uncountable (since its complement, being a well-orderable set of reals, must be countable ...
Andreas Blass's user avatar
12 votes
Accepted

Is $V=\textsf{HOD}\not\Rightarrow\textsf{GCH}$ consistent?

Yes, one can produce a model of $ZFC+V=HOD$ in which the $CH$ fails. This should follow from Consistency results about ordinal definability. In fact, I think the arguments of my paper HOD, V and ...
Mohammad Golshani's user avatar
11 votes

Consistency strength of $\aleph_2$-Souslin hypothesis

Answer to 1, without CH: Mitchell and Silver, 1973: Weakly compact is an upper bound. Answer to 1, with CH: Laver and Shelah, 1981: Weakly compact is an upper bound. Shelah and Stanley, 1982: ...
saf's user avatar
  • 1,156
11 votes
Accepted

Does $Π^1_{2n+1} = (Σ^1_{2n+1})^{M_{2n-1}}$?

Your conjecture is true. We assume PD throughout. The proof I see requires citing a number of facts from inner model theory and descriptive set theory. First, it uses Woodin's theorem characterizing ...
Gabe Goldberg's user avatar
11 votes
Accepted

Coding the universe into a real over better core models

For measurable cardinals, the answer is yes and is due to Sy Friedman. See Coding Over a Measurable Cardinal. There is some difficulty to extend the result to the context of Woodin cardinals, see ...
Mohammad Golshani's user avatar
10 votes
Accepted

Is it inconsistent for a model of set theory to contain its own first order theory?

First, let me point out as the others have that if there are large cardinals, then indeed we expect this situation. For example, if there is a worldly cardinal, a cardinal $\kappa$ for which $V_\kappa\...
Joel David Hamkins's user avatar
10 votes

Taking a proper class as a model for Set Theory

Yes, that is true. But note that in its nature statements like $\operatorname{Con}(T)$ are meta-theoretic statements. So when we say that $V$ is a model of $\sf ZF$, we mean that in the meta-theory it ...
Asaf Karagila's user avatar
  • 38.2k
10 votes
Accepted

Getting a model of $\mathsf{ZFC}$ that fails to nicely cover an inner model

The consistency strength of the failure of $(\dagger)$ is an inaccessible cardinal. Building on the comment of Mohammad, if $\omega_2^V$ is a successor cardinal in $L$ then there is a set $X \subseteq ...
Yair Hayut's user avatar
  • 5,192
10 votes
Accepted

Inner model theory without choice

Addressing the first question: I should argue that Choice comes in almost at the beginning of the inner model project, if we regard proving Covering Lemmata as an integral part of that project: one ...
Philip Welch's user avatar
  • 4,919
10 votes
Accepted

Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions

All you have to have is an inner model where some set that it thinks is large, is actually countable. To answer the first question, ZF+AD implies that $0^\sharp$ exists, so there exists a generic for ...
Monroe Eskew's user avatar
  • 18.1k
10 votes

Minimum transitive models and V=L

This is not a full answer, but I found it interesting to notice that if we relax the c.e. requirement somewhat, then there is a sweeping positive answer. Theorem. Every complete theory extending ZFC + ...
Joel David Hamkins's user avatar
9 votes

Reals which must, can't or might be added by forcing

The paper When is a given real generic over L? by Fabiana Castiblanco and Ralf Schindler characterizes all reals which are generic over $L$.
Mohammad Golshani's user avatar
9 votes
Accepted

Uniqueness of countable version of $L[U]$?

Andrés's comment can be turned into an answer, but there is a slight subtlety. Suppose $M$ is a countable iterable model of ZFC + $V = L[U]$. Let $\kappa$ be its measurable cardinal and $\alpha$ be ...
Gabe Goldberg's user avatar
9 votes
Accepted

Minimum transitive models and V=L

Yes, I claim you can in fact get one whose minimum model is a set forcing extension of a segment of $L$. Let $L_\alpha$ be least modelling ZFC. Let $\mathbb{P}=\mathbb{P}^{L_\alpha}$ be Jensen's ...
Farmer S's user avatar
  • 8,752
9 votes
Accepted

Precipitous ideal and inner model

No. Suppose otherwise. Let $G$ be the Levy collapse generic, and $D$ be the generic for forcing with the nonstationary ideal after that. Since $\mathrm{Ult}(L[U,G],D)$ is wellfounded, letting $i_D:L[U,...
Farmer S's user avatar
  • 8,752
8 votes

Is it inconsistent for a model of set theory to contain its own first order theory?

No, this is not a problem. If $U$ is a transitive set with $\mathcal{P}(\omega) \subseteq U$ then $U$ contains the real $\{\ulcorner\sigma\urcorner \mid U \vDash \sigma\}$. So, for example, $V_\alpha$ ...
François G. Dorais's user avatar
8 votes
Accepted

How verminous are mice?

Looking at your example of mouse, it seems you are going to use, and are asking about, the ‘old-fashioned’ fine-structural mouse, as used originally by Dodd and Jensen in their papers, and by Dodd in ...
Philip Welch's user avatar
  • 4,919
8 votes
Accepted

Absoluteness for the Chang model

Corollary 3.1.7 in Larson's Stationary Tower notes states that if $\delta$ is a Woodin limit of Woodin cardinals, then no forcing in $V_\delta$ can change the theory of the Chang model, even with real ...
Cameron Zwarich's user avatar
8 votes
Accepted

The core model and elementary embeddings

Some remarks: By Schindler's paper "Iterates of the core model", if $j:V\to N$ is elementary ($N$ transitive) and $N$ is closed under $\omega$-sequences, and $k:K\to K^N$ is the restriction ...
Farmer S's user avatar
  • 8,752
8 votes
Accepted

Are there premice that are $\omega_1$-iterable but not $(\omega_1+1)$-iterable?

It is consistent (relative to large cardinals). There is an example given in Example 3.6 here. For a brief summary: the model is the minimal proper class mouse $S$ such that $\mathbb{R}^S$ is closed ...
Farmer S's user avatar
  • 8,752
8 votes
Accepted

Inner model with a $\mathit{\Delta}^1_3$-good well-ordering of the reals

The situation is a bit more complicated than you might hope because of the periodicity phenomena in the projective hierarchy. For odd $n$, assuming $\mathbf{\Delta}^1_{n-1}$-determinacy, the set $Q_n$ ...
Gabe Goldberg's user avatar
7 votes
Accepted

Is there a natural inner model of AD$_\mathbb{R}$?

A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le_W A$, where $\le_W$ denotes Wadge reducibility, then $B\...
Andrés E. Caicedo's user avatar

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