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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Ordinals that satisfy $\alpha = \aleph_\alpha$ with cofinality $\alpha$
I am searching for a proof that there is an aleph fixed point $\alpha$ with cofinality $\alpha$. I know that this is a weakly inaccessible cardinal and thus cannot be constructed in ZFC (but in some ...
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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 $\...
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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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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\...
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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 &...
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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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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^+(...
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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 ...
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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 ...
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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 $j: V \to M$ ...
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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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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 ...
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Can ZFC be proven to be consistent in this way?
I was reading the wikipedia article of Gödel's incompleteness theorems and I saw that it has been proven that ZFC + "there exists an inaccessible cardinal" (which I'll call "ZFC + IC&...
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A countable language to express the theory of ZFC + “there exists a proper class of $\text{I0}$ cardinals”
Does there exist a countable language (i.e. a language with countably many symbols) $\mathcal{L}$ of set theory such that the theory ZFC + "there exists a proper class of $\text{I0}$ cardinals&...
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Do indecomposable abelian groups of inacessible cardinal rank exist?
In Fuchs' Infinite Abelian Groups, it is proved that there exist indecomposable torsion-free abelian groups of every rank smaller than the first strongly inaccessible cardinal. It is natural to ask: ...
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elementary embeddings $j$ in set theory with $V$ and $M$
I'm confused by a variety of elementary non-trivial elementary embedings $j$ we might have.
There are 9 "syntactical" possiblities;Here $M$ is a transitive model. I'll name them with a wish ...
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What is the complexity (in the Lévy hierarchy) of the axiom $\text{I0}$?
I have read the Wikipedia article on the Lévy hierarchy. It mentions a few large cardinal axioms, but it does not mention any of the rank-into-rank axioms. So I have the following questions:
what is ...
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How would one formulate large cardinals beyond rank into rank?
Rank into rank cardinals seem to push the limits of consistency, and are stronger than (almost?) every other consistent large cardinal. Despite that, It seems to me, and from a few online discussion ...
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Modern/introductory books on partition relations/partition calculus?
Are there any modern books on partition relations for cardinals/the partition calculus? I've seen several sources say it's a very active field of infinitary combinatorics, but the only book on it I ...
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Uniformity of ultrafilter causing contradiction; Hanf-Scott, Rowbottom theorem
I am trying to understand the proof of Theorem 8.30. in the book "The theory of ultrafilters" by Comfort and Negrepontis.
Theorem. If $p$ is a normal ultrafilter on a measurable cardinal $\...
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Ill-founded iterated ultrapowers
I have two related questions.
Can I find an ultrafilter on $\kappa$ which is $\theta$-complete for some uncountable $\theta<\kappa$ but not $\kappa$-complete?
We know that if $\mathcal{U}$ is a $\...
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If $\delta = \kappa^{(n)}$ then $U = D$ (Exercise 19.7, Jech's Set Theory)
Exercise 19.7 of Jech's Set Theory says:
Assume $V = L[D]$. If $U$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$ and if $U \neq D$, then there is a monotone function $f : \kappa \to \...
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Is the first worldly cardinal with respect to "ZFC+ inaccessibles" singular?
Let "$@_{\sf T}$" denote the first cardinal $\kappa$ such that $V_\kappa \models \sf T$.
This I describe as the first worldly cardinal with respect to $\sf T$.
Now, working in $\sf ZFC + @_{\...
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What are the motivations of large cardinal research?
Why do set theorists research large cardinals? Is it about the consistency results, or is there a type of mathematical beauty to large cardinals? If so, are there any examples of beauty in large ...
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Is this theory equivalent to TG set theory?
Is the following theory equivalent to Tarski-Grothendieck set theory?
The language is first order logic with equality, and membership, with axiom schemata:
Specification: $\forall A \exists! x \, \...
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Extendible cardinals
In Kanamori's book on large cardinals, he defines a cardinal $\kappa$ to be extendible if for any $\eta$ there is some $j:V_{\kappa+\eta} \prec V_\zeta$ with crit($j$)=$\kappa$ an $j(\kappa)$> $\...
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Can vector spaces occupy a large cardinal amount of dimensions?
So far I have found that a vector space of uncountably infinite dimensions is mathematically valid, but what about vector spaces that can occupy, say, an inaccessible or Mahlo cardinal amount of ...
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Closure of extender ultrapowers
Some elementary embeddings $j : V \to M$ can be defined as ultrapower embeddings by extenders. Extenders are defined using finite indices, and as I've noted in a previous question, that makes it not ...
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Kanamori on Vopenka's principle
This question is related to this one Equivalence of some statements regarding Vopenka's principle and some $V_\kappa$ for inaccessible $\kappa$
In particular, on page 336 of Kanamori's book on ...
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elementary embeddings and ultrafilters
In Kanamori's book on large cardinals (second edition), on page 300, he is proving 22.4 Proposition (d), where the proposition says that if $U$ is an $\omega_1$-complete ultrafilter over a set $S$, ...
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a small category equivalent to the category of all sets
assuming grothendieck universes, is it possible to have a small category equivalent the category of all sets?
mac lane in categories for the working mathematician assumes the existence of a ...
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What is a 'icarus' set?
I've been hearing about the Icarus set on the internet recently, and I've been wondering what is it? I've searched the internet and there isn't really an explanation for it. The most I know about the ...
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What are benefits of fixing a Grothendieck universe?
Let's assume the Tarski's axiom: For all set $u$, there is a Grothendieck universe $\mathbb U$ such that $u\in\mathbb U$. From now on I will drop "Grothendieck" and just write "universe....
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