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Questions tagged [large-cardinals]

Large cardinals are such cardinals whose existence cannot be proved within ZFC, and requires stronger axioms to be added to ZFC, they are often used to measure the consistency strength of a certain statement in the language of set theory.

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Absoluteness of inaccessible cardinals

I'm studying large cardinals and I'm hoping to fully understand the proof that says ZFC is not able to prove the existence of inaccessibles (given ZFC is consistent, of course). I've already fully ...
Darsen's user avatar
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Aleph sequence fixed points and the least inaccessible [duplicate]

Using the properties of the least inaccessible cardinal (being a regular fixed point of the aleph function) I was able to prove that the least inaccessible $\kappa$ is the $\kappa$-th fixed point of ...
Arianit Niti Gashi's user avatar
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Use of (weak forms of) AC for elementary embeddings proof

I encounter this issue when going through equivalent characterizations of measurable cardinals. For completeness, let me reproduce the statement: For ordinal $\kappa$, the following are equivalent. ...
Raczel Chowinski's user avatar
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What is $j_\mathcal{U}(\kappa)$ if $2^{\kappa}<j_\mathcal{U}(\kappa)<(2^{\kappa})^{+}$?

Let $\mathcal{U}$ a non-principal $\kappa$-complete ultrafilter on $\kappa$ and let $j_\mathcal{U}\colon V\to M$ its associated elementary embedding of the universe $V$ in $M=Ult_\mathcal{U}(V)$. In ...
Alberto Caccavale's user avatar
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On the Singular Cardinal Hypothesis

I'm trying to find the proof of this result. If for each $\lambda\geq2^\omega$, $\lambda^\omega\le\lambda^+$, then the SCH holds. I'm not sure where to look. So if you have any info about this, please ...
Selena's user avatar
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If $\mathsf{GCH}$ fails on a measure one set must it fail at $\kappa$?

Suppose $U$ is a normal measure on $\kappa$. It is well known that if $\{\alpha<\kappa:2^\alpha=\alpha^+\}\in U$ then $2^\kappa=\kappa^+$. Question: does $\{\alpha<\kappa:2^\alpha=\alpha^{++}\}\...
Lxm's user avatar
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Every weakly compact cardinal $\kappa$ is the $\kappa$th inaccessible cardinal

I'm currently reading Jech's Set Theory book, and in particular, while reading chapter 9, he mentions right after Lemma 9.9 that "We shall prove in Chapter 17 that every weakly compact cardinal $\...
Num2's user avatar
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6 votes
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A metrizable space is realcompact iff it has non-measurable cardinality?

A space is realcompact if its a closed subspace of an arbitrary product of real lines, with product topology. A cardinal $\kappa$ is called measurable if there exists a (countably additive) $\{0, 1\}$-...
Jakobian's user avatar
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How to show that the application of two elementary embeddings is an elementary embedding?

Let $\mathcal L$ be a language of first-order logic. Given two structures $\mathfrak M$, $\mathfrak N$ for $\mathcal L$ with domains $M$, $N$ respectively, an elementary embedding from $\mathfrak M$ ...
C7X's user avatar
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Is this property of Mahlo cardinals correct?

Quoting from a paper discussing large cardinals: for any ρ that is a Mahlo cardinal, there must also exist ρ smaller cardinals (call them κ, with all κ < ρ) that are k-inaccessible, hyper k-...
Theorem's user avatar
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What is the relation between the first and second inaccessible cardinals? [closed]

I know how the first inaccessible transcends the other cardinals by being regular and strong limit but what does a second inaccessible mean.I had two possibilities in mind: Either: It is inaccessible ...
Arianit Niti Gashi's user avatar
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1 answer
78 views

How to prove consistency with choice for large cardinal extensions?

How can we know if an extension of $\sf ZF$ by some large cardinal property that results in a consistency strength beyond $0^{\#}$ is compatible with choice or not? I mean the easiest way to know if ...
Zuhair's user avatar
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Why is strongness of a cardinal a different notion than Reinhardtness?

A cardinal $\kappa$ is Reinhardt if there is a nontrivial elementary embedding $j:V\to V$ such that $\kappa$ is the critical point of $j$. Kunen proved (using choice) that this large cardinal axiom is ...
C7X's user avatar
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If there is a large cardinal, can GCH also hold?

Let $P$ be a statement saying there is large cardinal of some kind. For example, $P$ can be one of There is a weakly inaccessible cardinal. There is a Mahlo cardinal. There is a weakly compact ...
fantasie's user avatar
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If $\kappa$ is weakly inaccessible, then it is the $\kappa$th element of $\{\alpha: \alpha =\aleph_\alpha\}$

This is an exercise from Kunen: Exercise I.13.17 If $\kappa$ is weakly inaccessible, then it is the $\kappa$th element of $\{\alpha: \alpha =\aleph_\alpha\}$. If $\kappa$ is strongly inaccessible, ...
Alphie's user avatar
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Are two Grothendieck universes, such that one is not a subset of another, possible?

Are two Grothendieck universes, such that one is not a subset of another, possible? It depends on the context: i suspect it is not provable in ZFC or in MK.(i use the second one for my purposes) The ...
georgy_dunaev's user avatar
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$\kappa$ is $\kappa$-supercompact iff measurable?

$\kappa$ is measurable if $\kappa$ is the critical point of a non trivial elementary embedding $j:V \rightarrow M$ of $V$ into a transitive class $M$ (with $j$ a class function on $V$). Given cardinal ...
P W's user avatar
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What are examples of models for which the Continuum Hypothesis is true/false?

I'm not a set theorist so pardon my improper language. I'm trying to make sense of the unprovability of the Continuum Hypothesis. What I've come to understand is this: since set theory is broad and ...
Evyenia Coufos's user avatar
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Is the extender embedding from an I1 embedding necessarily an I2 embedding

If we suppose that $\kappa$ is the critical point of an $I_1$ elementary embedding $j$ and that $\lambda$ is the supremum of the critical sequence of $j$ and we consider the $(\kappa,\lambda)$-...
Rupert's user avatar
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Does worldly cardinal exist if $\mathsf{ZFC}$ is consistent?

A worldly cardinal is a cardinal $\kappa$ such that $V_\kappa$ is a model of $\mathsf{ZFC}$. Please forgive me if this is very silly, but if $\mathsf{ZFC}$ is consistent (so there exists a model of $\...
Jianing Song's user avatar
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How can we control the cardinality of $j(\kappa)$ for $\kappa$ an $\aleph_1$-strongly compact cardinal?

This question has been cross-posted to MathOverflow, found here. I am interested in determining the cardinality of $j(\kappa)$ when $j\colon V\to M$ is an elementary embedding arising from an ...
Calliope Ryan-Smith's user avatar
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The Axioms of 'Fictional Googology'

There have been questions on whether proper classes have cardinality (some say yes). However, I have my own axioms about it. Firstly, we define the 'cardinality' of a proper class as the conglomerate ...
3-1-4-One-Five's user avatar
3 votes
1 answer
122 views

Supercompacts are Woodin limits of Woodins

I was trying to prove a couple of implications between large cardinal properties, but there is one I am not able to do, and I don't even have an idea on how to approach it. The implication is: if $\...
alvoi's user avatar
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Why does $k_{\mathcal E}$ fixes elements in $x\in V_\gamma^{M_{\mathcal E}}$?

I am reading section 26 of Kanamori's book The Higher Infinite. This section is about extenders, and I am struggling to understand what are the roles of these specific variables occurred in the ...
fantasie's user avatar
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3 votes
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Are weakly compact cardinals preserved in arbitrary inner models?

It's well-known that if a cardinal is weakly compact, then it is weakly compact in $L$. Seems natural to ask if weak-compactness is preserved for arbitrary inner models. Since I've never heard this, I'...
spaceisdarkgreen's user avatar
2 votes
1 answer
102 views

Measurable Cardinals, Elementary Embeddings, and building up to transitive collapse

I am looking at the proof that $\kappa$ a cardinal is measurable iff it is the critical point of an elementary embedding from $V$ to an inner model $M$, specifically the forward direction. I ...
abetray's user avatar
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What's wrong with this proof that $\omega$-strong cardinals are $(\omega+1)$-strong?

I would like to modify extender construction by relaxing the restriction that indices have to be finite in order to control closure properties, but I have run into an apparent contradiction. Most of ...
Arvid Samuelsson's user avatar
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Why are strong cardinals tall?

Cantor's Attic defined tall cardinals as follows: A cardinal $\kappa$ is $\theta$-tall iff there is an elementary embedding $j:V \to M$ into a transitive class $M$ with critical point $\kappa$ such ...
Arvid Samuelsson's user avatar
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189 views

What is the power set of an inaccessible cardinal?

I've been reading about set theory and the difference between small and large cardinals. since taking the power set of small cardinals (alephs) allows us to create larger cardinals/alephs I know that ...
Adithya's user avatar
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Existence of sharps from a measurable

(Edited for more specific question) I'm looking for a generalization of proof of the implication between "there exists a measurable cardinal $\kappa$" and "$0^\sharp$ exists". The ...
alvoi's user avatar
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Would undecidability in ZFC of the Laver Table conjecture imply its truth/falsity?

The Laver Table conjecture is the conjecture that the first-row periods of Laver Tables are unbounded. Richard Laver proved in ZFC + rank-into-rank that it holds, but it is not known whether it is ...
user107952's user avatar
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1 vote
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What does Solovay's theorem about the singular cardinal hypothesis above strongly compact cardinals say in level-by-level terms?

A theorem by Solovay says that if $\kappa$ is strongly compact then the singular cardinal hypothesis holds above $\kappa$. But if we only assume that $\kappa$ is $\lambda$-compact (edit: or $\lambda^+$...
Arvid Samuelsson's user avatar
1 vote
2 answers
222 views

Which Concepts to master before studying about large cardinals?

I am a undergraduate student interested in large cardinals since I've watched a youtube video called "How to count past infinity", and the concept of inaccessible cardinal that is introduced ...
Seungjun Wang's user avatar
2 votes
0 answers
146 views

Some questions about the Hyperuniverse Program

The Hyperuniverse Program, founded by Sy D. Friedman, intends to produce new second-order axioms of set theory which appropriately formalize "the universe is maximal" in one of a few ways. A ...
C7X's user avatar
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"Proof" that "ZFC + there exists an inaccessible cardinal" is consistent

I have a proof that this theory is consistent using this theory itself. I want to know what's wrong with this proof: "ZFC + there exists an inaccessible caridnal" proves that "ZFC is ...
Ryder Rude's user avatar
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3 votes
1 answer
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Inaccessible cardinals under a Mahlo [closed]

Is it true the statement that: if $\kappa$ is a Mahlo cardinal, then for all $\alpha<\kappa$ there is a cardinal $\lambda$ with $\alpha<\lambda<\kappa$ and $\lambda$ contains $\lambda$-many ...
alvoi's user avatar
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1 answer
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Are there any proposed operations to actually construct an inaccessible set?

When we postulate the smallest infinite set, we define it using an interative process involving iterations on the empty set. When we postulate the existence of a set of continuum cardinality, we again ...
Ryder Rude's user avatar
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10 votes
1 answer
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Collapse $\aleph_{\omega_1}$ to $\aleph_\omega$.

Is there a forcing notion that collapses $\aleph_{\omega_1}$ to $\aleph_\omega$ while preserving every cardinal below $\aleph_\omega$?
user123's user avatar
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1 vote
2 answers
179 views

When substructures become elementary substructure

It is known that given $M\subseteq N$ structure and a substructure such that $M\equiv N$, we do not necessarily have $M\prec N$. Similarly we can have $M_0\subseteq M_1\subseteq M_2$ such that $M_0\...
Gito Shalom's user avatar
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Question about Silver forcing theorem proof

I'm trying to understand the proof of the following Theorem: If there is a supercompact cardinal $\kappa$, then there exists a generic extension where $\kappa$ is a measurable cardinal and $2^\kappa &...
lowcard's user avatar
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standard model of $\mathbb{N}$ and true $\Pi^0_1$ sentences

Does the fact that provability of some true $\Pi^0_1$ sentences is equivalent to the existence of particular (I do not know from the top of my head to which ones, does someone know how they are called)...
user122424's user avatar
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2 votes
1 answer
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Class of compact cardinals implies every accessible category is co-wellpowered

As the title says, I'm looking for the reference that the existence of a class of compact cardinals implies every accessible category is co-wellpowered. It has been stated in Adamek-Rosicky (Locally ...
interregno's user avatar
7 votes
1 answer
203 views

A weak version of $0^\sharp$

While thinking about how to introduce $0^\sharp$ to students, the following weak analogue occured to me: Say that a transitive structure $\mathcal{A}=(A;\in, ...)$ - that is, a transitive set possibly ...
Noah Schweber's user avatar
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1 answer
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How do I independently study foundational math relate to current developments?

Today I spent hours reading about recent developments regarding the Continuum Hypothesis and Set Theory. Where would I even start with these topics? My undergraduate professors all considered ...
Planetluvver 's user avatar
2 votes
0 answers
93 views

rank into rank embedding exercise in Kanamori

I was trying to solve exercise 24.5 in Kanamori's book, which is the following: Suppose $j:V_\delta \prec V_\delta$ and $k:V_\delta \prec V_\delta$. Then $j^+(k):V_\delta \prec V_\delta$ and crit($j^+(...
nana's user avatar
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Question about the Kunen Inconsistency?

(Why is Kunen inconsistency at the top of Cantor's upper attic?) I have seen statements about the Kunen Inconsistency being thrown around like that it is the 'upper-bound' or 'limit' for how ...
SI J's user avatar
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2 answers
306 views

confusion about models, the universe and inaccessible cardinals

(Why cumulative hierarchy of Sets is not model of ZF) As seen from this post, it is proven that V is a structure that satisfies ZFC and is model of it. As we know an inaccessible cardinal k implies Vk ...
SI J's user avatar
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4 votes
0 answers
220 views

What's wrong with this proof that every rank-into-rank cardinal is a limit of rank-into-rank cardinals?

Theorem 11.10 of Double helix in large large cardinals and iteration of elementary embeddings by Kentaro Sato proves that if $\kappa$ is the critical point of an I2 elementary embedding $j: V \to M$ ...
Arvid Samuelsson's user avatar
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283 views

What are the Implications of having VΩ as a model for a theory?

A model in laymen's terms is a mathematical structure (sets when within set theory) that satisfies a theory (or alternatively a set of axioms). For example the model Vω is a model of a theory of ...
SI J's user avatar
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1 vote
0 answers
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Closure of extender ultrapowers 2

As I've noted in a previous question, the definition of an extender seemingly makes it impossible to ensure that the ultrapower by an extender is closed under countable subsets. That's inconvenient ...
Arvid Samuelsson's user avatar

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