All Questions
Tagged with inner-model-theory large-cardinals
57 questions
36
votes
3
answers
3k
views
Latest status 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:
https://...
- 1,312
23
votes
1
answer
2k
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 ...
- 7,125
23
votes
1
answer
3k
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 ...
user46667
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 ...
18
votes
2
answers
2k
views
A “paradox” about the inner model problem
As stated in Woodin, Davis, and Rodriguez - The HOD dichotomy, a longstanding open problem in set theory is to construct a canonical inner model for supercompactness. In general there are various ...
- 18.6k
18
votes
1
answer
822
views
What sets can be unraveled?
A set $X\subseteq\omega^\omega$ is unravelable iff there is a possibly larger set $A$ and a clopen set $Y\subseteq A^\omega$ (with respect to the product topology coming from the discrete topology on $...
- 21.2k
17
votes
3
answers
832
views
Classifying set theories whose standard models sharing the same ordinals are equal
Let's say that a (recursively axiomatizable) set theory $T$ extending ZF is "ordinal-categorical" if, whenever $M$ and $N$ are standard models of $T$ sharing the same ordinals, one has $M = N$. For ...
- 4,985
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
14
votes
1
answer
452
views
What are some good lower bounds on the consistency of the failure of the PCF conjecture?
Shelah's celebrated theorem states that $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{\omega_4}$.
But the conjecture is that $\omega_4$ can be provably replaced by $\...
- 39.7k
13
votes
1
answer
2k
views
What is the evidence for and against the HOD conjecture?
I'm aware that the HOD conjecture is implied by the Ultimate-L conjecture, but I don't know what the evidence is for the Ultimate-L conjecture. On the other hand, I'm aware the evidence against the ...
- 419
13
votes
1
answer
406
views
Getting a model of $\mathsf{ZFC}$ that fails to nicely cover an inner model
Consider the following statement:
$(\dagger)$ $\ $ There is an inner model $M$ such that $M \models \mathsf{GCH}+\square$ and for every countable $X \subseteq \mathrm{Ord}$, there is a countable $Y \...
- 18.5k
12
votes
2
answers
778
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 ...
- 779
12
votes
1
answer
532
views
Why do we need the comparison lemma?
An inner model is a standard transitive (proper class) structure which satisfies all the axioms of ZFC and contains all the ordinals. The simplest and most well-known inner model is Gödel’s $L$, which ...
- 704
12
votes
1
answer
464
views
Does $Π^1_{2n+1} = (Σ^1_{2n+1})^{M_{2n-1}}$?
Every $Π^1_1$ formula $φ$ without free second order variables can be converted into a $Σ^1_1$ $ψ$ such that $φ ⇔ ψ^\mathrm{HYP}$, and vice versa. ($\mathrm{HYP}$ is the hyperarithmetical universe, ...
- 7,545
12
votes
1
answer
477
views
Is there a natural inner model of AD$_\mathbb{R}$?
The question is as in the title, but let me explain a bit.
Assuming a proper class of Woodin cardinals, $L(\mathbb{R})$ satisfies AD (and DC). And $L(\mathbb{R})$ is a very natural inner model. I'm ...
- 21.2k
11
votes
1
answer
429
views
Coding the universe into a real over better core models
One of the most incredible results in modern set theory, due to Jensen, is that given any model of $\sf ZFC$, there is a class forcing which adds a real number $r$ and in the extension $V=L[r]$. ...
- 39.7k
11
votes
2
answers
377
views
Does $0^{\#}$ imply the failure of upward directedness in the set generic universe over $L$?
Question 1. Suppose that $0^{\#}$ exists. Are there set generic filters $g,h$ over $L$ ($g,h \in V$) such that for any $f$ ($\in V$) which is generic over $L$ we have that $L[g] \cup L[h] \not \...
- 1,080
11
votes
1
answer
472
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 ...
- 1,174
10
votes
1
answer
721
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?
- 109
10
votes
0
answers
234
views
Absoluteness of the core model under a proper class of completely Jónsson cardinals
Example 2.4.2 of Larson's The stationary tower (based on Woodin's lecture) describes how we can absorb a generic filter over a set forcing in a definable inner model into a generic extension by $\...
- 3,042
9
votes
1
answer
547
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 ...
- 32.1k
9
votes
1
answer
313
views
Do precipitous ideals "always" come from collapsing?
It's well-known that if $\kappa$ is a measurable cardinal, then there is a poset $\mathbb{P}$ that forces $\kappa$ to carry a precipitous ideal.
Suppose that $\omega_1$ carries a preciptous ideal $I$.
...
- 1,142
9
votes
0
answers
177
views
Inner model of "CH + large cardinals" that satisfies MM?
I've been told that if $V$ has CH and large cardinals, then there is an inner model in which MM holds. The sketch is as follows:
Any $\Sigma^2_1$ sentence that can be forced is already true in such $V$...
- 335
9
votes
0
answers
314
views
Is there a relationship between the $\Omega$-conjecture and choiceless large cardinals?
Woodin’s $\Omega$-conjecture is absolute under set-forcing. One proposal to decide it is that some large cardinal hypothesis might refute it. On the other hand, Woodin is seeking extensions of the ...
- 18.6k
8
votes
2
answers
478
views
Consistency strength of being strong cardinal and indestructible under collapses
What is the consistency strength of the following statement:
$\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed ...
- 32.1k
8
votes
1
answer
339
views
Inner model theory without choice
How much of the inner model project can be constructed without assuming the axiom of choice? I.e. which large cardinals provably have canonical inner models not assuming choice?
- 419
8
votes
1
answer
580
views
What consistency results follow the assumption: $\forall\alpha(\aleph_{\alpha+1}\nleq2^{\aleph_\alpha})$?
In a recent question on Math.SE it was asked whether or not For every infinite cardinal $\mathfrak m$ there is no $\aleph$ number, $\kappa$, such that $\mathfrak m<\kappa<2^{\mathfrak m}$.
By ...
- 39.7k
8
votes
0
answers
187
views
Intuition for branch uniqueness in inner model theory
In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage?
At the level ...
- 7,545
8
votes
0
answers
240
views
Universally meager spaces and large cardinals
Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...
- 18.6k
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
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
6
votes
1
answer
325
views
Elementary chains in forcing extensions of $M_1$
Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic ...
- 2,180
6
votes
0
answers
240
views
Are initial segments of coherent measure sequences coherent?
This question is about the "old-fashioned" coherent sequences, in the style of Mitchell
Mitchell, W. (1974). Sets constructible from sequences of ultrafilters. Journal of Symbolic Logic, 39(...
- 2,389
6
votes
0
answers
344
views
Inner models with all sets generic
Question: Under large cardinal axioms, what is the intersection of all inner models $M$ of ZFC such that every set in $V$ is set-generic over $M$?
Every set belongs to a generic extension of HOD, and ...
- 7,545
6
votes
0
answers
176
views
Breaking determinacy with forcing, and then fixing it
While forcing is usually presented over models of ZFC, it works equally well over models of ZF (or even less). However, the general theory of forcing becomes much stranger (much like the general ...
- 21.2k
5
votes
2
answers
639
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 ...
- 1,174
5
votes
1
answer
242
views
Uniqueness of countable version of $L[U]$?
Suppose $\alpha$ is a countable ordinal and $U_0,U_1,\kappa$ are such that $L_\alpha[U_i] \models \mathrm{ZFC} + U_i$ is a normal ultrafilter on $\kappa$. Does $U_0 = U_1$?
The argument for ...
- 18.6k
5
votes
2
answers
351
views
Proper class of Woodins and $\textsf{AD}_{\mathbb R}$-hypothesis
The $\textsf{AD}_{\mathbb R}$-hypothesis is the statement that there is a $\lambda$ which is both a limit of Woodins and a limit of ${<}\lambda$-strongs. Are there any results relating the ...
- 1,170
5
votes
1
answer
622
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 ...
- 3,804
5
votes
1
answer
265
views
Consistency strength of lifting through a lot of collapsing
What is the consistency strength of the following situation?
$j : V \to M$ is an elementary embedding definable from parameters in $V$, with critical point $\kappa$.
$\mathbb P$ is a forcing that ...
- 18.6k
5
votes
3
answers
520
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 $k+1$-...
- 3,804
5
votes
0
answers
192
views
"Very $L$-like" models, part 1: large cardinals
(The original version of this question was much narrower and less natural; but see the edit history if interested.)
Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ ...
- 21.2k
5
votes
0
answers
276
views
Absoluteness and the scale property for $Π^2_2$ or $Σ^2_2$
Under the diamond principle $◊$ and large cardinal axioms, which of the two pointclasses $Π^2_2$ or $Σ^2_2$ is expected to have the scale property?
Because conditional $Σ^2_2$ absoluteness under $◊$ ...
- 7,545
5
votes
0
answers
304
views
Symmetry between V and HOD
Is it consistent that the set of ordinal definable real numbers is countable, but for every $y∈\mathrm{OD}∩ℝ$, every true $Σ_2^{V,y}$ statement holds in $\mathrm{HOD}$?
Note that $Σ_2^V$ is the best ...
- 7,545
4
votes
1
answer
293
views
Are generators of extenders cardinals?
Say $E$ is a $(\kappa,\lambda)$-extender on some model $\mathcal M$ of set theory, and consider the ultrapower embedding $i:\mathcal M\to\text{Ult}(\mathcal M,E)$. Now recall that a generator of $E$ ...
- 1,170
4
votes
1
answer
198
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 J^{...
- 3,804
4
votes
0
answers
260
views
Universe V = Ultimate L inside set theoretic multiverse
Good day to you all,
I would like to ask a question about relation between Prof. H. Woodin V = Ultimate L and a concept of set theoretical multiverse as proposed by Prof. Hamkins.
If V = Ultimate L ...
- 441
4
votes
0
answers
179
views
Inner model theory using indiscernibles
Has an inner model theory been developed on the basis of indiscernibles rather than measures? Is there a reasonable formalization at the level of overlapping extenders?
Fine-structural models beyond $...
- 7,545
4
votes
0
answers
236
views
Does absoluteness imply a club dichotomy?
My question is about two types of consequence of large cardinals, considered over ZFC on their own.
First, we have statements of the form, "The club filter on $\omega_1$ is an ultrafilter when ...
- 21.2k
3
votes
4
answers
439
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 $HOD$. ...
user36136