Questions tagged [large-cardinals]
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772
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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
0
answers
178
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
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0
answers
173
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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_\...
3
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0
answers
208
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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
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2
answers
549
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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 ...
6
votes
0
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268
views
Some characterization of Mahlo cardinals
Following the well-known characterization of supercompact cardinals by Magidor, in our paper we have defined the notion of a $\kappa$-Magidor model, for supercompact cardinal $\kappa$.
I defined a ...
91
votes
10
answers
14k
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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
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173
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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
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0
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274
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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
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0
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207
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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
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1
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170
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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 ...
9
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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
0
answers
317
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
1
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263
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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
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1
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393
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 ...
16
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1
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763
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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
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265
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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
1
answer
170
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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
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1
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314
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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
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168
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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
1
answer
582
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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
1
answer
359
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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
1
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361
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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
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2
answers
1k
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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
1
answer
527
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
0
answers
411
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(\...
17
votes
2
answers
2k
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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
1
answer
225
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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 ...
13
votes
1
answer
550
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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
1
answer
3k
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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
1
answer
298
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
0
answers
101
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
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0
answers
202
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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 ...
17
votes
1
answer
749
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
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0
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224
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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
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0
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432
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
1
answer
371
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
1
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263
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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 ...
22
votes
1
answer
894
views
How badly can the GCH fail globally?
It's known that we can 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
1
answer
222
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
1
answer
225
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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}(...
5
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0
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320
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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 ...
5
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0
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194
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Can a weakly inaccessible non-weakly-Mahlo cardinal carry a $\kappa$-complete, $\kappa^+$-saturated ideal?
An ideal $I$ on a regular cardinal $\kappa$ is said to be $\mu$-saturated if whenever a family $\langle S_\alpha \mid \alpha<\lambda\rangle$ of subsets of $\kappa$ is such that each $S_\alpha\notin ...
5
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0
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245
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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 $◊$ ...
12
votes
1
answer
452
views
How to understand the interface of the consistency strength hierarchy, reverse mathematics, and proof-theoretic ordinal analysis?
I am aware of three major "hierarchies" of mathematical theories, but I don't know how to relate these hierarchies to one another. Here are the hierarchies I have in mind:
Consistency strength. My ...
0
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0
answers
135
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What's the consistency strength of resemblance + global failure of the continuum hypothesis?
Let $T$ be a theory formalized in first order logic with equality and membership and the additional primitive constant symbol $W$, with the following axioms:
Extensionality: $\forall z (z \in x \...
9
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0
answers
249
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$\Sigma^2_2$ absoluteness and $\diamondsuit$
This question touches on recent questions concerning the role of $\diamondsuit$ and the Continuum Hypothesis. In particular, I would like to know more information about a conjecture of Woodin on the ...
11
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0
answers
453
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$\Sigma^2_1$ and the Continuum Hypothesis
This is a follow up to Will Brian's answer to this recent question. In particular, quoting Brian:
"In fact, Paul Larson has pointed out to me that the statement "$\phi$ and $\phi^{-1}$ are conjugate"...
14
votes
2
answers
415
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Consequences of existence of a certain function from $\omega_1$ to $\omega_1$
In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any countable elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is ...
10
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1
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536
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Is the product of commuting ultrafilters an ultrafilter?
If $U$ is a filter on $X$ and $W$ is a filter on $Y$, their product is the filter $U\times W$ on $X\times Y$ generated by rectangles $A\times B$ where $A\in U$ and $B\in W$.
In certain circumstances ...