# 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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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) $\{0, 1\}$-...
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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 $\mathfrak M$ ...
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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-...
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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 ...
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### 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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 $(\kappa,\lambda)$-...
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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 ...
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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'...
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### 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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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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 ...
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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 $\lambda^+$...
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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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 ...
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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 $\aleph_\omega$?
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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)...
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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 ...
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### 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 ...
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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 ...
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### 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 ...
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