Questions tagged [large-cardinals]
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604
questions
10
votes
1answer
267 views
Elementary embeddings and replacement
Let $\alpha\not= 0$ be such that for every $\beta<\alpha$ there is $\beta<\gamma<\alpha$, where $V_\gamma$ is an elementary substructure of $V_\alpha$. In other words, $V_\alpha$ is a limit ...
5
votes
0answers
144 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(...
11
votes
1answer
288 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 $V=L[r]$. ...
4
votes
1answer
239 views
Compactness number for a fragment of second-order logic
Previously asked and bountied without response at MSE.
This question is a companion to this one, about a tame(?) fragment of second-order logic with the standard semantics, $\mathsf{SOL}$, motivated ...
7
votes
1answer
244 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 ...
6
votes
2answers
311 views
Analogues of worldly cardinals for (an unusual version of) second-order $\mathsf{ZFC}$
Let $\mathfrak{ZFC}(\mathsf{SOL})$ be the theory in second-order logic (with the standard semantics) gotten from $\mathsf{ZFC}$ by modifying the Separation and Replacement schemes to apply to ...
7
votes
0answers
174 views
Determinacy of symmetric games
Is it consistent that for all ordinals $α$ and $λ$ and infinite regular cardinals $κ$, games on $V_λ$ with game length $κα$ and $\mathrm{OD}(\mathrm{On}^κ)$ payoff that depends only on the set of all ...
3
votes
1answer
289 views
Cardinality of infinite towers of Alephs - can tower be more than countable?
Lets define function T as
$$T(0) = \aleph_0$$
$$T(1) = \aleph_{\aleph_0}$$
$$T(2) = \aleph_{\aleph_{\aleph_0}}$$
etc
No finite tower of alephs can reach the first inaccessible cardinal
My questions ...
3
votes
0answers
113 views
Complexity of $Σ_n$ theory of $H(ω_2)$ under MM$^{++}$
Under Woodin's $\mathbb{P}_\text{max}$ axiom (which is implied by MM$^{++}$), what is the complexity of the $Σ_n$ theory of $H(ω_2)$? Same question for $Σ_n(I_\text{NS})$ (i.e. using the ...
0
votes
0answers
130 views
Proper class of nested rank into rank embeddings
I propose the following large cardinal axiom:
There exists a proper class of cardinals $\lambda$, such that for each $\lambda$, there exists a rank-into-rank embedding $j: V_\lambda \rightarrow V_\...
2
votes
0answers
159 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 ...
9
votes
2answers
320 views
Large cardinals without replacement
Let $ZC$ be Zermelo set theory with choice, which differs from $ZFC$ in omitting the axiom scheme of replacement. EDIT: I think I want to include foundation in the axioms, which apparently isn't ...
5
votes
0answers
158 views
Some charactrization of Mahlo cardinals
Following the well-known characterization of supercpmpact cardinals by Magidor, in our paper we have defined the notion of a $\kappa$-Magidor model, for supercompact cardinal $\kappa$.
I defined a ...
73
votes
10answers
8k views
Reflection principle vs universes
In category-theoretic discussions, there is often the temptation to look at the category of all abelian groups, or of all categories, etc., which quickly leads to the usual set-theoretic problems. ...
4
votes
0answers
162 views
Sequences of sequences of sequences and elementary embeddings
Suppose that $\kappa$ is the critical point of $j\colon V\to M$, and suppose that $\mathcal F=(F_\alpha\mid\alpha\leq\kappa)$ is a sequence such that for every limit ordinal $\alpha$, $F_\alpha$ is a ...
0
votes
0answers
221 views
Can this Ackermann like set theory formulated without adding a primitive of set-hood reach the consistency of ORD is Mahlo?
The following is a theory that uses a reflection principle similar to Ackermann but on a size notion that is definable in the language of set theory, my question is about its consistency strength ...
5
votes
0answers
152 views
Which very large cardinals are preserved under Woodin's forcing for $\mathsf{AC}$?
Woodin showed that we can force $\mathsf{AC}$ if there is a proper class of supercompact cardinals while preserving supercompacts, by forcing Easton-support iteration of $\operatorname{Col}(\kappa,<...
0
votes
1answer
139 views
Coding a function $g:\kappa\to V_{\zeta+1}$ by an element of $V_{\zeta+1}$
Note: this is cross-posted from MSE.
This question is about the following remark (modified to be self-contained), found in Donald Martin's book on determinacy, page 340. The context is proving ...
8
votes
0answers
589 views
What are the known implications of “There exists a Berkeley cardinal”?
Inspired by this question: What are the known implications of "There exists a Reinhardt cardinal" in the theory "ZF + j"?
Definitions:
$\delta$ is Berkeley iff for every $\alpha\...
6
votes
0answers
297 views
Temporary destruction of measures in intermediate models
It is a well-known theorem that if $\kappa$ is measurable, then there is a generic extension in which $\kappa$ is no longer weakly compact, but we can force its weak compactness back and recover the ...
6
votes
1answer
218 views
Tree property at weak inaccessibles
Suppose $\kappa$ is a weakly inaccessible cardinal with the tree property. What can we say about the height of $\kappa$? Is it a weakly-hyper-Mahlo of some sort? Does it enjoy some kind of ...
4
votes
1answer
350 views
Does $H\vDash AC$
The set $H_\kappa$ of sets hereditarily of cardinality less than $\kappa$ is defined as $H_\kappa=\{x||tc(x)|\lt\kappa\}$. What if we define the set $H=H_{Ord}$ of sets hereditarily of cardinality ...
15
votes
1answer
514 views
Can $Ord$ have nontrivial second-order elementary self-embeddings?
I forgot to mention originally: this was motivated by this old MSE question.
It's not hard to show in $\mathsf{ZFC}$ that there are nontrivial elementary embeddings from $(Ord; \in)$ to itself - or ...
4
votes
0answers
195 views
Reflection principles justifying $I2$ and larger cardinals
Consider the language of set theory together with the constant smybols $\langle \Omega_\alpha|\alpha\in Ord\rangle$. Let's add to $ZFC$ the axiom that for every $\alpha$,$n$ there is some $j: V_{\...
5
votes
1answer
142 views
Finding many subsets of $V_{\lambda+2}$ stable under $j:V\prec V$
Working over $\mathsf{ZF}$ with an embedding $j:V\prec V$ with a critical point $\kappa$. Take $\lambda=\sup_{n<\omega}j^n(\kappa)$. (You may assume $\mathsf{DC}_\lambda$ if you need, but I am not ...
8
votes
1answer
226 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?
6
votes
0answers
136 views
Preserving supercompactness in intermediate forcing extensions
Let $\kappa$ be supercompact in $V$. Let $\mathbb{P}$ be one of the standard forcing notions (or an iteration of such), and for simplicity assume that $\mathbb{P}$ is ${<}\kappa$-directed closed (e....
4
votes
1answer
477 views
What is the consistency strength of almost $\omega$-huge cardinals?
What is the consistency strength of a cardinal $\kappa$, such that there is some $j: V\prec M$ such that $M^{\lt j^\omega(\kappa)}\subseteq M$; in other words, for every cardinal $\lambda\lt\delta$, $...
12
votes
1answer
248 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 \...
6
votes
1answer
285 views
How strong is “all up-classes are infinitarily definable”?
Working in MK (or some other not-too-strong class theory if you prefer), say that an up-class is a class of structures $\mathfrak{X}$ which is definable in $V$ (allowing parameters from $V$) and such ...
3
votes
2answers
950 views
Inconsistency of Reinhardt cardinals in ZF+DC
As I'm just a layperson I don't understand the technicalities involved, but does the paper New Large Cardinal Axioms and the Ultimate-L Program, by Rupert McCallum (arXiv:1812.03837) prove the ...
2
votes
1answer
301 views
Limit of Mahlo cardinals
What cardinal is the limit of this fundamental sequence?
{The first Mahlo cardinal, the first 1-Mahlo cardinal, the first hyper-Mahlo cardinal, the first hyper-hyper-Mahlo cardinal, the first hyper-...
8
votes
0answers
389 views
Choice function for elementary embedding $j: V_\lambda\prec V_\lambda$
Work in ZF. Let $\lambda$ be strongly inaccessible. Let $\kappa$ be such that for every $\alpha\lt\lambda$, there is some $j: V_\lambda\prec V_\lambda$, with critical point $\kappa$, such that $j(\...
14
votes
1answer
753 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 ...
7
votes
1answer
190 views
Weakly homogenously Souslin sets and the measurability of $\omega_1$
I found this intriguing remark at the end of Woodin's Supercompact cardinals, sets of reals, and weakly homogeneous trees (1988):
The assertion that every set of reals, in $L(\mathbb{R})$, is the ...
9
votes
0answers
229 views
Iterating Neeman's forcing
In the paper, "Forcing with sequences of models of two types," (MR3201836), Neeman claims that, using a supercompact and a weakly compact above, one can force with his pure side conditions ...
34
votes
1answer
3k views
Is the theory Flow actually consistent?
Recently the paper
Adonai S. Sant'Anna, Otavio Bueno, Marcio P. P. de França, Renato Brodzinski, Flow: the Axiom of Choice is independent from the Partition Principle, arXiv:2010.03664
appeared on ...
7
votes
1answer
229 views
Lowenheim-Skolem numbers for SOL + correctness quantifiers
For a logic $\mathcal{L}$, say that a cardinal $\kappa$ is $\mathcal{L}$-correct iff every satisfiable $\mathcal{L}$-theory of size $<\kappa$ has a model of size $<\kappa$. First-order ...
2
votes
0answers
80 views
Consistency strength of iterated classes
Adding classes into a set theory like ${\bf ZFC}$ to get a theory like ${\bf MK}$ adds some consistency strength, but less than even a single inaccessible cardinal since $\kappa$ being inaccessible ...
6
votes
0answers
172 views
Collapse successor of singular while preseving supercompactness
Suppose $\kappa$ is a supercompact cardinal. Is it possible to find a forcing which collapses $\kappa^{+\omega+1}$ to $\kappa^{+\omega}$ (all those $\kappa^{+n}$'s are preserved) while the ...
11
votes
0answers
304 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 $...
1
vote
0answers
176 views
Weakly berkeley cardinal
Define $\kappa$ as $\Sigma_n$-weakly berkeley cardinal if for any transitive set $M$ that includes $\kappa$ exist elementary embedding $j:M\rightarrow M$ save only $\Sigma_n$ formulas and critical ...
4
votes
0answers
248 views
Are hyper-Berkeley cardinals equiconsistent with club Berkeley cardinals or with Berkeley cardinals?
Let's define cardinal $\kappa$ as hyper-Berkeley if for any transitive set $M$ such that $\kappa\in M$ there exists an elementary embedding $j: M\prec M$ with
fixed point $\lambda$ and $\text{crit}j\...
10
votes
1answer
324 views
The stationary reaping number $\mathfrak{r}_{cl}$
Let $\kappa$ be at least inaccessible (but measurable is what I am primarily interested at the moment). Let $x,y \in [\kappa]^\kappa$ both be stationary. We say that $y$ stationary-splits $x$ iff $x \...
5
votes
1answer
245 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 ...
19
votes
1answer
650 views
How badly can the GCH fail globally?
It's known that we have global failures of GCH---for example, where $\forall \lambda(2^\lambda = \lambda^{++})$---given suitable large cardinal axioms.
My question is whether we can have global ...
3
votes
1answer
192 views
smallest ordinal $\alpha$ such that $L \cap P(L_\alpha)$ is uncountable
Let $V$ denote the von Neumann universe and $L$ Gödel's constructible universe. For any set $X$, let $P(X)$ denote the power set of $X$.
Assume that $0^\sharp$ exists (and ZFC).
What is the smallest ...
6
votes
1answer
206 views
Can we recover an inner model of CH after forgetting some generic information?
Suppose $\kappa$ is an inaccessible cardinal. Let $G \times H$ be $\mathrm{Col}(\omega_1,{<}\kappa) \times \mathrm{Add}(\omega,\kappa)$-generic over $V$. Let $X \subseteq \kappa$ be $\mathrm{Add}(...
0
votes
0answers
79 views
At which large cardinal property Ackermann set theory + finitization rule would stop?
By the finitization rule I mean a rule that inputs a schema in the $V$ world and outputs a single statement in the $V$ world that serves to capture that schema!
So in this sense we'll have for every ...
5
votes
0answers
215 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 ...
