# Questions tagged [large-cardinals]

The large-cardinals tag has no usage guidance.

784
questions

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### Forcing $\neg\square_{\omega_1}$ from a Mahlo cardinal, reference

In Jensen's The fine structure of the constructible hierarchy, it is stated that Solovay proved the consistency of $\neg\square_{\omega_1}$ by collapsing a Mahlo cardinal to $\omega_2$. I was ...

3
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1
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### Large cardinals approached through $\infty$-categories

I am an undergraduate student (rising junior) majoring in philosophy and mathematics. For some time, I have been interested in homotopy type theory and so-called "univalent foundations". On ...

4
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1
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### On a question about ordinals $\xi$ satisfying $j_0(\xi)=j_1(\xi)$ for an $I_3$-embedding $j$

Let $j\colon V_\lambda\to V_\lambda$ be an $I_3$ embedding with the critical sequence $\kappa_n$. Define $j_0=j$, $j_1 = j[j]=\bigcup_{\alpha<\lambda} j(j\upharpoonright V_\alpha)$. My question is ...

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### Consistency upper bounds for $\neg\square_{\aleph_\omega}$

In the introduction of Cummings and Friedman's $\square$ on the singular cardinals the following is written:
Failure of $\square_\lambda$ for $\lambda$ singular is stronger and rather more ...

4
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### Jensen's proof that $\diamondsuit$ holds at subtle cardinals

At the end of these notes by Ronald Jensen (which I found from this question) there is a proof that $\diamondsuit_\kappa$ (diamond principle) holds if $\kappa$ is a subtle cardinal.
By induction on $\...

4
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1
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### Impact of coining $L$ in $\mathcal L_{\omega_1, \omega}$ on which large cardinals it can satisfy?

What happens to the constructible universe $L$ if we build it in the first of infinitary languages $\mathcal L_{\omega_1, \omega}$?
Would the usual limitation of $L$ not satisfying existence of a ...

4
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1
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### Reference request: $\kappa^+$-saturated ideal from iterated Cohen forcing on measurable $\kappa$

In Kunen [1] the author makes the following note: Let $\kappa$ be measurable with normal measure $\mathscr{U}$ in a model of $\mathsf{GCH}$. Let $\mathbb{P}$ be an iteration of $\operatorname{Add}(\...

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### Must there be a proper class of Reinhardt cardinals if there is a Reinhardt cardinal?

A cardinal is Reinhardt if $\kappa$ is the critical point of a nontrivial elementary embedding of $V$ to itself, where $V$ is the class of all sets. As Reinhardt cardinals are inconsistent with $\...

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### Large cardinals beyond choice and HOD(Ord^ω)

Are Reinhardt and Berkeley cardinals (and even a stationary class of club Berkeley cardinals) consistent with $V=\mathrm{HOD}(\mathrm{Ord}^ω)$ ?
It seems natural to expect no, but I do not see a proof....

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### Reference: If $X$ is metrizable, then $X$ is realcompact iff $|X|$ is non-measurable

Note: What I call a measurable cardinal seems to be non-standard among set theorists, and should be called a $\sigma$-measurable cardinal.
I know that a discrete space is realcompact iff its non-...

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### 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 $P_κ(λ)^N$...

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### Why is inner model theory evidence for consistency of large cardinals?

I want to understand the viewpoint that existence of canonical inner model for a large cardinal notion is strong evidence for its consistency. For example, below is Trevor Wilson's answer to What &...

9
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1
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### 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 $I$.
...

9
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### Consistency strength of strongly compact cardinal

Where can I find a proof that strongly compact cardinal has higher consistency strength than Woodin cardinal, or even just strong? Recall that a strongly compact cardinal itself may not be strong, ...

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0
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### What is the strength of adding this de-schematizing inference rule to Ackermann's set theory?

Language: first order logic with equality, membership, and a constant symbol $W$.
Axioms:
Extensionality: $\forall z \, (z \in x \leftrightarrow z \in y) \to x=y$
Comprehension: $\exists x \forall y \,...

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0
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### Is this modified H. Friedman theory bi-interpretable with ZFC + ORD is Mahlo?

The following theory is a modification of Harvey Friedman $\sf K(W)$ theory.
Language: first order logic with equality, membership, and a constant symbol $W$.
Axioms:
Extensionality: $\forall z \, (z ...

6
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### From HODs to corresponding models of AD

If $M$ is HOD of a model $N$ of $\text{AD}^+ + V=L(P(ℝ))$, what kind of forcing construction in $M$ gives back such an $N$?
HODs for $\text{AD}^+ + V=L(P(ℝ))$ are conjectured (and under anti-large ...

12
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1
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730
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### Can proper classes have different sizes?

I'm presently working in a non-ZF set theory, where there are proper classes. (Think MK or VNBG.) And I'm interested in how to think about the possibility (or impossibility) of proper classes with ...

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### Higher-order equivalence of ordinals

I wonder which ordinals are second-order equivalent, and similarly for other logical equivalences. Let the signature be fixed and include only <. For concreteness, let us first ask for the first ...

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### Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$

I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ ...

5
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### Can a generic ultrafilter over $\mathrm{NS}^+_{\omega_1}$ witness $\omega_1$ is Ramsey-like?

Suppose that $\kappa$ is an appropriate large cardinal (preferably a Woodin cardinal, but possibly something stronger) and let $G$ be a $\operatorname{Col}(\omega_1,<\kappa)$-generic filter over $V$...

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### 1970 question of Reinhardt - how large is this ordinal?

On page 241 of William Reinhardt's paper "Ackermann's set theory equals ZF" (Annals of Math. Logic vol. 2, 1970), question 4.15 is the following:
How large is the first ordinal $\gamma$ ...

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### 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 ...

8
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### What is the least inaccessible cardinal for Tarski-Grothendieck set theory?

Let ordinal $\alpha$ be the least ordinal such that $V_\alpha\models$ Tarski-Grothendieck set theory.
What position does $\alpha$ have in the hierarchy of inaccessible cardinals?

12
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1
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### 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 ...

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### Follow up question: Shelah's "Can you take Solovay's inaccessible away?"

In this answer to the question " Shelah's "Can you take Solovay's inaccessible away?" " the following is stated:
Assume that $\aleph_1$ is not inaccessible in $L$, hence a ...

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### Is $\kappa \rightarrow [\kappa]^2_3$ the same as $\kappa \rightarrow [\kappa]^2_2$ for inaccessible $\kappa$

The principle $\kappa \rightarrow [\kappa]^2_\alpha$ states that whenever we have a coloring $c:[\kappa]^2\rightarrow \alpha$ there is $H \subset \kappa$ of size $\kappa$ s.t. $|c"[H]^2|<\alpha$.
...

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### Ultrafilter projections and critical points of factor maps

Suppose $j : V \to M$ is $\lambda$-supercompactness embedding derived from an $\kappa$-complete normal ultrafilter $U$ on $P_\kappa(\lambda)$, $\lambda$ regular. Suppose $\eta$ is an ordinal such ...

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### What can be the measure of a Vitali set?

Suppose the continuum $\mathfrak{c}$ is real-valued measurable, i.e., there exists a countably additive probabilistic measure on $\mathfrak{c}$ that measures all subsets. Then by the construction on p....

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### Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics

To be clear, I am not a mathematics educated student and I can not follow the details of the technicality of the forcing extension, but I feel that I have a good understanding of the big picture (of ...

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4
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### A Löwenheim–Skolem–Tarski-like property

I am interested in the following Löwenheim–Skolem–Tarski-like property.
Given a cardinal $\kappa$, what (if any) is a property $\phi(x)$ such that if $\phi(\kappa)$ holds, then we can prove the ...

14
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### Are Berkeley cardinals easier to refute in ZFC than Reinhardt cardinals?

Kunen showed that Reinhardt cardinals are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. Berkeley cardinals are stronger than Reinhardt cardinals. You can ...

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### Can Friedman's property fail at or above a supercompact cardinal?

If $\kappa>\omega_1$ is a regular cardinal, $FP_\kappa$ is the assertion that every stationary subset of $\kappa$ consisting of ordinals of countable cofinality has a closed subset of order type $\...

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### How can we control the cardinality of $j(\kappa)$ for $\kappa$ an $\aleph_1$-strongly compact cardinal?

The following question was posted to MathStackExchange (original here). As there were no comments/answers on the original, I have ported it unedited.
I am interested in determining the cardinality of $...

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### Collapsing every cardinal outside the Prikry sequence

All variants of Prikry forcing with collapses that i have been able to find preserve some points outside of the generic sequence (at least the successors). This is done for two reasons, (1) to obtain ...

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### Is Vopěnka's principle inherited by Grothendieck topoi?

I call the Vopěnka's principle:
Every subfunctor of an accessible functor is accessible
but other formulations (which may lose equivalence in weak contexts?) are also interesting to me.
If this is ...

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### Does the consistency of a large cardinal axiom imply the $\omega$-consistency of that axiom?

Let $P$ be some large cardinal property (or indeed any first-order formula in the language of set theory, but lets focus on large cardinals for now). Does the $\omega$-consistency of $\mathsf{ZFC}+P$ ...

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### Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?

Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...

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### Operations on the set of large cardinal axioms

Here's a question from a non-set-theorist, but a sometime-user of large cardinals.
The name Cantor's attic is pretty evocative for the collection of large cardinal axioms: looking through the pages ...

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### Has there been any progress on this open problem about co-well-poweredness of accessible categories?

On the relations between accessible categories and large cardinal axioms, one big example is the following:
Assume the existence of a proper class of strongly compact cardinals. Then every accessible ...

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### Embedding large countable ordinals into the complex plane

Consider large countable ordinals (e.g. $\epsilon_0$ which is not "large", but still interesting).
These are countable sets, so they inject into the complex plane ( or even the real line).
...

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### How much information do we need to guess a large cardinal?

Suppose $\kappa$ is a cardinal and we want to guess if $\kappa$ is a large cardinal, and if so what kind, by looking at the large cardinal status of a selection of cardinals below $\kappa$.
The ...

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### Can MK+"Ord is almost-huge"+MM$^{++}$ be new standard foundations instead of ZFC?

I'll try to explain what this looks like to a non-expert in set theory. First, $MK$ is just a second-order $ZFC$, and there are moments when we would like to use second-order statements, for example, ...

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### 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 ...

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### Large cardinal near inconsistencies

I am looking for examples of results about large cardinals, large cardinal axioms, or other objects of high (or seemingly high) consistency strength that are almost inconsistencies. I am looking for ...

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### stating large cardinal axioms in ZF

Can I ask whether there is a good reference for how to state the standard large cardinal axioms in the context of $ZF$? My concern is that it seems that the usual proof that embeddings defined from ...

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### Cardinality of maximal elements in terms of set-theoretic inclusion in the space $c_0(\mathbb{N})$

Let $c_0(\mathbb{N})$ be the space of real-valued sequences converging to zero.
Here, each element $\{\alpha_n\} \in c_0(\mathbb{N})$ itself is a "set", so that we can think about set-...

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### 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 ...

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### Omega logic, V = Ultimate L and hierarchy of laws collapse

I would like to ask a question about the omega conjecture and its relationship with the V = Ultimate L axiom.
In his lecture Prof. Hugh Woodin have stated that "Assuming the omega conjecture, ...

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### Is determinacy of (some) very long open games consistent?

For $\varphi$ a first-order sentence in the language of set theory and $\kappa$ an ordinal, let $G_\varphi^{\kappa}$ be the game of length $\kappa$ in which players $1$ and $2$ alternately play ...