# Questions tagged [large-cardinals]

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547
questions

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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}(...

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65 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 ...

**5**

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165 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 ...

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70 views

### Would iterating reflection and resemblance results in increment of consistency strength?

Question: Can we have a reverse hierarchy of sets resembling $V$?
That is a sequence of universes $W_\alpha$ where each is an element of the prior one, and such that each universe reflects ...

**5**

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154 views

### 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 ...

**4**

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121 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 $◊$ ...

**8**

votes

**1**answer

221 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 ...

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118 views

### 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 \...

**7**

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151 views

### $\Sigma^2_2$ 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 ...

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293 views

### $\Sigma^2_1$ 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"...

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81 views

### Are there minor tweaks of hereditary replacement that can prove large cardinal properties?

Hereditary replacement: if $\phi(x,y)$ is a formula in which only symbols $x,y$ occur free, and those never occur bound, and in which symbol $B$ never occur; then: $$\big{(}\forall A [\forall x \...

**14**

votes

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335 views

### 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 ...

**9**

votes

**1**answer

316 views

### 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 ...

**10**

votes

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303 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, ...

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255 views

### Is there a second-order expression of '$\kappa$ is Reinhardt' in $NGB$, where $\kappa$ is a cardinal?

In their paper, "Generalizations of the Kunen inconsistency", (Annals of Pure and Applied Logic 163 (2012) 1872-1890, doi:10.1016/j.apal.2012.06.001, arXiv:1106.1951), Hamkins, Kirmayer, and ...

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213 views

### General theory of the reals in Solovay-like models

Solovay's model is a famous model of $\sf ZF$ where we start in $L$ with $\kappa$ inaccessible, and we collapse all the ordinals below $\kappa$ to be countable, without collapsing $\kappa$ itself, and ...

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75 views

### What's the strength of capturing set theory in labeled Mereology?

To Atomic General Extensional Mereology + Bottom, add a primitive one place partial function symbol $L$, signifying "the label of", an add axioms:
Distinctiveness: $Lx=Ly \to x=y$
Labels: $\forall x (...

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108 views

### A parsimonious large cardinal axiom

The ordering of large cardinals by consistency strength is well known.
I was wondering what one can say regarding an ordering by direct implication.
In particular, I am looking for is a parsimonious ...

**6**

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208 views

### $0^\#$ in weak theories vs large cardinals in $L$

To better understand the transition from large cardinal axioms consistent with the constructible universe $L$ to large cardinal axioms transcending $L$, I am looking for natural equiconsistencies ...

**3**

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149 views

### Elementary self-embeddings conservative over ZFC

Question: Is the following theory conservative over ZFC? And if not, what is its strength?
Language: $∈$, $j$ (unary function symbol)
Axioms:
1. ZFC (without separation and replacement for formulas ...

**3**

votes

**1**answer

166 views

### The intersection of all normal ultrafilters on a measurable cardinal

Suppose $\kappa$ is a measurable cardinal. Let $W$ be the intersection of all normal ultrafilters on $\kappa$.
I am interested in a precise characterization of the filter $W$.
One sure way to ...

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votes

**3**answers

1k views

### Necessary use of large cardinals in mathematics [duplicate]

There are some statements, whose consistency (or the consistency of their negation) require the existence of large cardinals (in the sense that if the statement (or its negation) is consistent, then ...

**10**

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253 views

### Seeing what gets Harvey Friedman's “tangible incompleteness” principles into large cardinal territory

I'm trying to wrap my head around some of Harvey Friedman's recent, unpublished work on his tangible incompleteness project, and I'm trying to see the link between his "tangible statements" (...

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vote

**1**answer

86 views

### Complexity of a proper class of extendibles

If consistent, is existence of a proper class of extendible cardinals provably equivalent to a $Σ^V_5$ statement?
Recall that in ZFC, a cardinal $κ$ is extendible iff for every $λ>κ$ there is an ...

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**1**answer

284 views

### What is the consistency strength of adding this ordinal reflection scheme on top of Ackermann set theory?

Axiom scheme of Ordinal Reflection: if $\phi$ is a formula that doesn't use the symbol $V$, whose parameters are among $x_1,..,x_n$; then: $$\forall x_1 \in V,\dotsc,\forall x_n \in V: \phi(On) \to \\\...

**6**

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380 views

### When does “sufficient genericity” actually suffice?

Fix a forcing notion $\mathbb{P}$. Say that a formula $\varphi(x)$ with parameters is $\mathbb{P}$-enforceable if there is some countable set $\mathcal{D}$ of dense sets in $\mathbb{P}$ such that for ...

**6**

votes

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513 views

### Very large axiom of choice

let me say that I am not a set theorist, but I have to settle up some things in category theory and I need your help.
What I'd like to do is, in some way, use axiom of choice for proper classes.
I ...

**5**

votes

**1**answer

394 views

### Can the category of partial orders be fully embedded in the category of linear orders?

Q(1): Can the category of partial orders be fully embedded in the category of linear orders?
Vopěnka's principle, or VP, is a very intriguing axiom with many equivalent forms and consequences ...

**2**

votes

**1**answer

197 views

### (ZC + $\Sigma_2$ replacement + inaccessible cardinal) equiconsistent with (ZFC + inaccessible cardinal)?

Randall Holmes has made a quite convincing argument against the fact that the full axiom schema of replacement should be considered as “intuitively obvious”—even though he does believe ZFC to be ...

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67 views

### Is having multiple embedding proper class hierarchies consistent?

Question 1: Is it consistent to extend "MK - Extensionality + weak Extensionality (as in NFU) - Infinity - limitation of size + set existence+ axiom of subsets (that is: a subclass of a set is a set)" ...

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99 views

### What's the consistency strength of this theory of Stretchable Hierarchies?

Working in Morse-Kelley set theory:
A hierarchy is defined as a class that is the union of sets uniquely indexed by ordinals, called as stages, such that each stage is the power set of the ...

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votes

**1**answer

117 views

### Is the principle of indifference of hierarchical construction consistent? What's its consistency strength?

Sometimes when one tries to capture the abstract aspect of some notion that is intuitively considered as being truly of abstract nature, in set theoretic terms, then this can extend the theory in a ...

**2**

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**1**answer

196 views

### Possible inconsistency related to embeddings $j: M\prec V$

In the paper
Vickers, J.; Welch, P. D., On elementary embeddings from an inner model to the universe, J. Symb. Log. 66, No. 3, 1090-1116 (2001). ZBL1025.03049.
it is stated to that if $Ord$ is ...

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**2**answers

294 views

### Why do ordinal collapsing functions use regular cardinals?

Inaccessible cardinals are defined as regular strong limit cardinal, and weakly inaccessible cardinals as regular weak limit cardinal. These cardinals are used by some ordinal collapsing functions. My ...

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36 views

### Why do ordinal collapsing functions use regular cardinals? [duplicate]

Inaccessible cardinals are defined as regular strong limit cardinal, and weakly inaccessible cardinals as regular weak limit cardinal. These cardinals are used by some ordinal collapsing functions. My ...

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144 views

### Is there no anti-foundational theory exists Reinhardt and hold Global Choice?

J. D. Hamkins proved in "The foundation axiom and elementary self-embeddings of the universe" that, working in $ZFGC^− +BAFA$, there are nontrivial automorphisms and elementary embeddings of the ...

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140 views

### Can second order ordinal arithmetic be extended to the same extent as ZFC?

In a prior posting, I've posed the idea of reducing set theory to an extended kind of second order ordinal arithmetic $\small \sf`` 2 oO A"$. The idea was to have a domain of ordinals and sets of ...

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262 views

### What would cohomological localization be good for?

An open problem in algebraic topology is whether arbitrary cohomological localizations of simplicial sets (or, equivalently, topological spaces) can be proven to exist in ZFC. It's provable in ZFC ...

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votes

**1**answer

286 views

### A weak (?) form of Shelah cardinals

The following definition of a large cardinal property combines parts of the definitions of "Shelah cardinal" and "Woodin cardinal":
A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \...

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**1**answer

281 views

### Proving independence with large cardinals?

Suppose I want to prove some statement S is independent of ZFC.
Now instead of the usual approach of making models, I do the following:
- Take two large cardinal axioms L1 and L2
- Prove that ZFC + L1 ...

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votes

**4**answers

914 views

### “Bootstrapping” an unbounded class of inaccessible cardinals

The "richness principle" of set theory asserts roughly that "everything that happens once should happen an unbounded number of times".
An example would be the existence of an unbounded class of ...

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156 views

### Stationary sets and $\kappa$-complete normal ultrafilters

Let $\kappa$ be a measurable cardinal, and let $u$ be a normal $\kappa$-complete ultrafilter over $\kappa$. It is a standard easy fact that every closed unbounded set must belong to $u$ (notice that ...

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**1**answer

196 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 ...

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**1**answer

110 views

### What is the strength of claiming that the class of all $V_\kappa$ stages that are $H_\kappa$ when $\kappa$ is regular, is inaccessible?

[EDIT] This posting had been edited to assert that we are speaking about regular mutual stages.
Let $H_{\kappa}$ be the set of all sets that are hereditarily strictly smaller in cardinality than ...

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votes

**1**answer

286 views

### Upward reflection of rank-into-rank cardinals

Rank-into-rank cardinals have the rather intriguing property that they reflect upwards. I would be interested to know how far the upward reflection goes:
1) Does "There exists a rank-into-rank ...

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90 views

### Can the cardinality of the set of all intervening cardinals between sets and their power sets be always singular?

This is a question that I've posted to Mathematics Stack Exchange, that was un-answered.
To re-iterate it here:
Is the following known to be consistent relative to some large cardinal assumption?
$...

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136 views

### Generalized graph-minor theorem?

Consider the following generalized graph-minor theorem:
GM($κ,λ$): Given any collection $S$ of $κ$ simple undirected graphs each with less than $λ$ vertices, there are distinct graphs $G,H$ in $S$ ...

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157 views

### Proving that being an inaccessible cardinal is absolute, for $V_\kappa$, where $\kappa$ is inaccessible?

I'm going through the proof that if $\kappa$ is inaccessible then $V_\kappa \vDash \mathrm{ZFC}$ and how thus we have $\mathrm{ZF} \nvdash \text{"There exist inaccessible cardinals"}$.
So the last ...

**6**

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203 views

### Generic saturation of inner models

Say that an inner model $M$ of $V$ is generically saturated if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses ...

**10**

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259 views

### Does the statement 'there exists a first-order theory $T$ with no saturated models' have any set theoretic strength?

Exercises 6, 7, and 8 in section 10.4 of Hodges' big model theory textbook contain an outline of a proof of the consistency of the following statement
There exists a countable first-order theory $T$...