All Questions
Tagged with set-theory inner-model-theory
118 questions
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 ...
- 18.6k
4
votes
0
answers
206
views
Fine structure without choice
In set theory, are there approaches to fine structure that give fine-structural models that do not satisfy the axiom of choice?
We can build fine-structural models above a given set (such as $\mathbb ...
- 12.9k
6
votes
0
answers
179
views
$Δ^1_3$ reals in transitive models
Every real number is $Δ^1_2$ in some $ω$-model of ZFC\P. I am looking for analogous statements for $Δ^1_3$ definability and transitive models, where the situation is more subtle.
What is the ...
- 9,902
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
8
votes
0
answers
153
views
In the construction of $K^c$ for sequences of measures, why do we only add measures of cofinality $\omega_1$?
$\require{cancel}$
I have a question about the definition of $K^c$ for sequences of measures in Mitchell's "The Covering Lemma" chapter in the Handbook. I will give the definition he gives ...
- 315
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://...
- 24.2k
2
votes
1
answer
255
views
Why can't $L_\beta$ contain a real coding a well-ordering of order-type $\beta$, when $\beta$ is a gap ordinal?
In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal is defined as follows
$\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\cap \mathcal{P}(\omega) = \emptyset$
Their ...
- 2,031
3
votes
0
answers
200
views
Weak extender models for supercompactness without choice
Assume ZFC and a supercompact cardinal $κ$. Is it consistent that there is a weak extender model $N⊨\text{ZF}$ for supercompactness of $κ$ such that the axiom of choice and well-ordering of $P_κ(λ)^N$...
- 7,545
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 $I$.
...
- 8,099
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
2
votes
0
answers
144
views
The strongest reflection principle that does not violate covering lemmas
#-generated reflection, or Indiscernible-generation, is considered to be the strongest reflection principle that does not violate the covering lemma in L. [1]
Is there a way to extend this success to ...
- 6,367
3
votes
0
answers
240
views
The most powerful inner model and a $\Delta^2_1$ well-ordering of the reals
With the current research, it seems that we are in a position to get extremely powerful absoluteness theorems (like $\Sigma^2_0$-absoluteness, $\Sigma^2_1$-absoluteness, $\Sigma^2_2$, $\diamondsuit_G$,...
- 695
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 ...
- 2,143
8
votes
1
answer
413
views
Precipitous ideal and inner model
Assume $\kappa$ is measurable, $U$ is its unique normal measure, $V=L[U]$. We levy collapse $\kappa$ to make it become $\omega_1$.
If we don't have the inner model condition, then we only know that $\...
- 9,902
2
votes
0
answers
342
views
What's the definition of a mouse in Mitchell's handbook article "the covering lemma"?
In the book "handbook of set theory", in the chapter "the covering lemma", definition 3.24, Mitchell defines what is mouse.
However he did not give any definition of $\mathcal{U}_\...
- 4,359
2
votes
1
answer
268
views
Inner model for KP and a Well-Ordering of the Reals
It is well known that Gödel proved the following theorem:
$\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)
So:
Is there an inner model for KP/Z/....
- 9,902
8
votes
0
answers
178
views
What is the exact consistency strength, if known, of the precipitousness of $I_{\text{NS}}$ on a successor of a singular cardinal $\kappa$?
The question is in the title. Jech states in his book on page 696 that the consistency strength is "in the region of Woodin cardinals," which is frustratingly imprecise. I tried to find a ...
- 751
8
votes
2
answers
300
views
Destroying the iterability of $M_1^\#$
Suppose $M_1^\#$ exists and is $\omega_1$-iterable.
Is it consistent that we can go to a generic extension $V[G]$ where $M_1^\#$ is no longer $\omega_1$-iterable?
Or "worse" $M_1^\#$ is no ...
- 9,902
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
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 ...
- 8,099
1
vote
0
answers
266
views
Is Jensen's covering lemma meaningful in a platonist's view?
The typical applications of fine structure theory are finding out the lower bounds of consistency strength of axiom systems. In such a proccess, we also constructs many combinatorial objects in core ...
- 1,490
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
1
answer
202
views
Are there premice that are $\omega_1$-iterable but not $(\omega_1+1)$-iterable?
For (hopefully) simplicity, let a premouse be defined coarsely as in Martin and Steel's 1994 paper, Iteration Trees.
Is (or is it consistent that) there is a premouse that is $\omega_1$-iterable but ...
- 9,902
7
votes
1
answer
222
views
Does $\mathit{Aut}(\mathbb{R};+)$ have a copy in $L(\mathbb{R})$ granting large cardinals?
Throughout, work in $\mathsf{ZFC}$ + large cardinals (let's say a proper class of Woodin limits of Woodins but I'm happy to go higher if that would help).
Let $\mathcal{R}=(\mathbb{R};+)$ be the ...
- 9,902
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 ...
- 17.7k
17
votes
3
answers
1k
views
Minimum transitive models and V=L
Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$?
You may assume that ZFC has transitive models. ...
- 7,545
2
votes
0
answers
195
views
"Very $L$-like" models, part 2: combinatorics
Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite ...
- 21.2k
7
votes
0
answers
314
views
Is there a "nice" inner model for $\mathsf{ZF}$ + a Dedekind-finite infinite set of reals?
Below, given a formula $\varphi$ which $\mathsf{ZF}$ proves defines a set of reals and an inner model $W$, I'll write "$\varphi^W$" and "$L(\varphi^W)$" for "$\{x:W\models\...
- 21.2k
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 $\...
- 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
4
votes
0
answers
180
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
3
votes
0
answers
152
views
Why are the sharps of sets of big ordinals not in $\mathcal{P}(\omega)$?
In his talk A Condensed History of Condensation, Welch presents the following recursive sharp function, that is total when all sharps exist:
\begin{align*}
\# \colon ON &\to \mathcal{P}(ON) \\
\...
- 421
3
votes
1
answer
241
views
Do all limit $\alpha \in \omega_1^L$ satisfy $L_\alpha \models V=HC$?
In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal and the start of a gap are defined as follows
$\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\bigcap \mathcal{P}(\...
- 2,031
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
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 ...
- 2,381
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 ...
- 2,381
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
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 ...
- 8,099
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 $E$ ...
- 236k
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 ...
- 39.7k
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
15
votes
1
answer
985
views
Does inner model theory seek canonical models for large cardinals?
Like the author of this question, I have heard that a main goal of inner model theory is building canonical inner models for large cardinals. My questions are: (a) Is this accurate? (b) If so, in ...
- 924
3
votes
2
answers
249
views
Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions
I am curious about the existence of inner models of $\mathrm{ZFC}$ in conjunction with forcing axioms, under assumptions inconsistent with such theories. For example:
can we prove under any extension ...
- 8,099
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 $V=L[r]$. ...
- 32.1k
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 ...
- 9,902
3
votes
0
answers
249
views
Independence through forcing vs generic collapses
Are there known statements in $V_{ω+ω}$ independent through forcing after $\mathrm{Col}(ω,<κ_1)*\mathrm{Col}(κ_1,<κ_2)*\mathrm{Col}(κ_2,<κ_3)*...$ where $κ_1<κ_2<κ_3<...$ are ...
- 7,545
3
votes
1
answer
325
views
If we have a class like $L$ but allowing a set number of unbounded quantifiers, is it strict superset of $L$?
The definition of $L$ only permits bounded quantifiers. If we allow a certain number of unbounded quantifiers, does this result in a strict superset of $L$? For example:
$$
\operatorname{Def}^{\...
- 6,353
8
votes
0
answers
280
views
Inner models from highly saturated ideals
Solovay proved that a measurable cardinal is equiconsistent with the existence of an extension of Lebesgue measure to a full measure on $P(\mathbb R)$. The inner model direction is relatively simple. ...
- 18.6k
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?
- 4,839
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 \...
- 5,112