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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1answer
85 views

tree property and arrow notation

How can I see that for an uncountable cardinal $\kappa$ these 2 conditions are equivalent : $\kappa$ has the tree property $\kappa \to (\kappa)^2$
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fine measure/Jech's book misunderstanding of the fine measure notion

I have several questions about a short paragraph in Jech's famous book Set Theory. In the definition of normal fine measure we have $U$ on $P_{<\kappa}(A)$ . But $P_{\kappa}(A)$ is already the set ...
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1answer
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elementary embeddings, $V$ vs. $L$

What is possible from these choices, there is an (nontrivial) elementary embedding $$j:V\to L$$ or There is an elementary embedding $$j:L\to V$$ and second part of my Q. is to what is this consistent ...
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What is the link between interpretability hierarchy and consistency strength

I am trying to understand this definition https://plato.stanford.edu/entries/independence-large-cardinals/#IntHie of Interpretability Hierarchy and how it relates to the concept of Consistency ...
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1answer
61 views

Quick question about inaccessible cardinals

I am trying to find a source that states, it is consistent with ZFC that weakly inaccessible cardinal does not exist. Can I please get some sources? It was quite hard for me to find such. Indeed, I ...
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3answers
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Compact cardinal cannot be successor?

This is a follow-up question to $\kappa$ is compact $\implies$ $\kappa$ is regular. The definition I'm using for "compact" is the same as there. I am trying to show if $\kappa$ is compact, ...
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$\kappa$ is compact $\implies$ $\kappa$ is regular

Definitions The Stanford Encylopedia of Philosophy makes the following definition in their article on Infinitary Logic https://plato.stanford.edu/entries/logic-infinitary/ Let κ be an infinite ...
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105 views

A restricted form of the inner model hypothesis

To keep things relatively simple I'm presenting a somewhat-butchered version of the IMH; for more details, see S.-D. Friedman, Internal consistency and the inner model hypothesis. Throughout, "...
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Doing inverse limit reflection from an $I_0$ embedding to an $I_1$ embedding

Suppose that $j:L(V_{\lambda+1}) \rightarrow L(V_{\lambda+1})$ with critical point $\kappa<\lambda$. Then $\kappa$ is a limit of cardinals $\kappa'<\kappa$ such that $I_1(\kappa',\delta)$ for a $...
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1answer
38 views

Axiom schema of Replacement analogue for Von Neumann stage

I am working without Choice for the time being, so "cardinal" means "well-founded cardinal." Let $\kappa$ be a strongly inaccessible cardinal. I want to show that the Axiom Schema ...
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Exact Functors and Elementary Embeddings

One of the possible equivalent definitions of measurable cardinals defines them to be the critical point of an embedding of the universe $V$ into a transitive class $M$. Recently I learned of Trnková ...
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1answer
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Name of infinite cardinals which has nonprincipal $\sigma$-complete ultrafilters?

The book "General Topology" by Engelking defines non-measurable cardinals as cardinals admitting no nonprincipal $\sigma$-complete ultrafilters. And then it claims that the discrete space of ...
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A proper class of ordinals without an infinite constructible subset

Edit: Mohammad Golshani has answered this question positively on MO. If $0^\sharp$ exists then the $L$-indiscernibles form a proper class of ordinals without any infinite constructible subset, as $0^\...
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How can the direct limit of a rank-into-rank embedding fail to be well-founded?

I have a question about proposition 6.4. of I0 and rank-into-rank axioms by Dimonte 2017. The question is about the direct limit of a directed system of rank-into-rank elementary embeddings defined as ...
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1answer
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Properties of $\aleph_1$ under axioms where $2^{\aleph_0} = \aleph_2$

There are some interesting set-theory axioms under which the cardinality of the continuum, $2^{\aleph_0}$, is equal to $\aleph_2$. One example is Woodin's Strong $\Omega$ Conjecture, though I have ...
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What is the formal definition of a natural instance of non-linearity in consistency strength?

I was inspired to ask this question after watching Joel David Hamkins give a talk about natural instances of non-linearity in consistency strength. Basically, large cardinal axioms seems to be ...
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Possible inconsistency of almost huge cardinals (I hope not)

The paper "Double helix in large large cardinals and iteration of elementary embeddings" by Kentaro Sato 2007 gives the following characterization of almost huge cardinals as lemma 5.4 (note ...
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Why are extenders defined using finite sequences?

My question is inspired by definitions 3.1 and 3.2 (3) of "Double helix in large large cardinals and iteration of elementary embeddings" by Kentaro Sato 2007 but I think my question is ...
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1answer
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Why are inaccessible cardinals/Solovay's model always excluded in measure theory texts?

So forgive me for a moment, as I am only comfortable with the VERY basics of Lebesgue measure, at the moment. I will be taking a course on it soon, but apart from the definitions and basic properties, ...
6
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1answer
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(Long) extender in its own ultrapower

I've seen stated in numerous places that if $E$ is an extender and $M$ is the ultrapower (of $V$) by $E$, then $E\not\in M$. I understand the proof of this for short extenders, but it's not clear to ...
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1answer
105 views

ineffable cardinals has a stationary subset of inaccessibles

Definition : a cardinal is called ineffable if for every sequence of sets $\left<A_\alpha\ \big|\ \alpha<\kappa\right>$ s.t $A_\alpha \subset\alpha$ there is a set $B\subset\kappa$ s.t $$S=\...
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1answer
161 views

How large is this cardinal?

The function $f(\alpha)$ outputs an uncountable ordinal from an arbitrary ordinal $\alpha$. It is defined as follows: $$f(\alpha) = \left\{ \begin{array}{l} {\omega _1}\quad {\text{if}}\quad\alpha = ...
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1answer
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For an extender $F$ of minimal length witnessing the coherence condition of a Martin-Steel Doddage, $\kappa_F = \iota_F$

This question of mine comes from Woodin's "In Search of Ultimate-L" article on page $36$. Let $\mathcal{E}$ be a Martin-Steel Doddage and $(\alpha, \beta)\in \mbox{dom}(\mathcal{E})$ and ...
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Dual notions of current large cardinals

This question might be characterized as a soft-question, and I am just curious to know if some of the notions below make sense or are consistent. So to kick things off, we may note that the existence ...
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1answer
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Does counting inaccessibles have a fixpoint?

Assume the existence of a proper class of inaccessibles $I$, and let $f : \mathrm{Ord} \to I$ denote the unique strictly increasing class surjection given by Mostowski Collapse. Does $f$ have a ...
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1answer
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Can all structures have "lots of compactness" with respect to expansions by constants?

This is one of those "surely not ..." questions that I embarrassingly can't answer at the moment. Given a structure $\mathcal{A}$ in a language $\Sigma$ and infinite cardinals $\kappa<\...
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1answer
106 views

Product Measure $\mathcal{V}$ of normal measure $\mathcal{U}$ s.t $\mathcal{U}\nleq_{RK} \mathcal{V}$

Let $\kappa$ be measurable and let $\mathcal{U}$ be some measure on it we define a measure on $\kappa\times\kappa$ $$ A\in \mathcal{V} \iff \{\alpha \ \big |\ \{\beta\ \big|\ (\alpha,\beta)\in A\}\in \...
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1answer
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The Rudin-Keisler Order Equivalent Definitions

Definition: Let $\mathcal{U}, \mathcal{V}$ be ultrafilters on $X,Y$ respectively define an order $$ \mathcal{U}\leq_{RK} \mathcal{V} $$ if there is a function $f:Y\rightarrow X$ s.t $$ A\in\mathcal{U} ...
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Computations over a normal Ultrapower

Let $\mathcal{U}$ be a normal measure on $\kappa$ and let $j = j_\mathcal{U} : V\rightarrow M = M_\mathcal{U}$ be the corresponding ultrapower. Prove that $\sup\{j(\alpha)\ \big|\ \alpha<\kappa^+\}...
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2answers
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Example of a nonprincipal incomplete ultrafilter

I am trying to build my intuition on ultrafilters by finding examples of non-implications between these properties: ultra nonprincipal completeness Any filter (on a regular cardinal $\kappa$) ...
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1answer
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Are extendible cardinals $\Sigma_3$-reflective, choicelessly?

Definition. A cardinal $\kappa$ is extendible if for each $\alpha$ there is $\zeta$ and $j:V_{\kappa+\alpha}\prec V_\zeta$ and $\operatorname{crit}j=\kappa$, $j(\kappa)>\alpha$. (There was ...
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1answer
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A tame(ish) fragment of second-order logic

This question is about a tame(?) fragment of second-order logic with the standard semantics $\mathsf{SOL}$, motivated by the Tarski-Vaught test. The general setup is as follows. Given structures $\...
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Information About Transcendence for Large Cardinals

I've been looking into the idea of large cardinals recently, and I found this question in particular to be interesting. Large Cardinals Ordered by Cardinality of Least Instance The most popular answer ...
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1answer
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The consistency of "Ord is Mahlo" and inaccessible correct cardinal

In Cantor's Attic, it is stated that the theories 1 and 2 are equiconsistent. $\mathsf{ZFC}$ + $\delta$ is inaccessible and $V_\delta\prec V$ (the latter expressed as a scheme in the language $\{\...
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1answer
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For a measurable cardinal $κ$, show that $cf(γ)≠κ$ implies $j_U(γ)=\sup\{j_U(δ):δ<γ\}$ ($U$ $κ$-complete ultrafilter, $j_U$ associated embedding) [closed]

For : $κ$ a measurable cardinal, $U$ a $κ$-complete ultrafilter over $κ$ $j_U$ the elementary embedding of $V$ into the ultrapower of $V$ to $U$ How to show that : If $\operatorname{cf}(γ)≠κ$ ...
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Is it possible to formalize iterative reflection in FOL, and what is its strength?

I'll largely present this posting informally, because I'm not certain if it can be done formally in the language first order logic with identity and membership. What I want to do is to iterate ...
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The reflection axiom and large cardinals?

Extensionality: as in Zermelo set theory Separation: as in Zermelo set theory Reflection: if $\phi$ is a formula having all of its free variables among $\vec{x}$ that doesn't use the symbol $v$ ; and ...
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2answers
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Are those two theories about universes equivalent?

If we define a universe as a well founded extensional transitive set that is closed under power, union, and non-bigger than, formally this is: $\mathbf U (X) \iff \forall a \in X: \\ \forall m \in a (...
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1answer
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Is BZC inconsistent with Reinhardt cardinals

A Reinhardt cardinal is defined as the critical point of a nontrivial elementary embedding $j: V\rightarrow V$ from the universe $V$ to itself, and is known to be inconsistent with the axiom of choice ...
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To which large cardinal property this inaccessible cardinal is equal?

I want to define a "$t^\zeta$-unreachable" where $t^\zeta$ stands for a tuple of size $\zeta$. I'd call that tuple as the index of reachability of the cardinal in question. I'll start with $...
2
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1answer
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About a detail of the proof of $\mathsf{GCH}$ via iterated ultrapower

I am reading the proof of $L[U] \models \mathsf{GCH}$ given on Mitchell's Beginning Inner Model Theory at the handbook, but I confronted a detail how to fill the details. Mitchell's proof goes as ...
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1answer
127 views

What led Hausdorff to consider the possibility of "unerreichbar" or "weakly inaccessible" cardinals?

I would like to know how it was that Felix Hausdorff came to consider the possibility of, if not indeed necessity of, what he was the first to call "unerreichbare" or "unreachable" ...
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1answer
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Showing elements of an ultrapower have the form $j(f)([id]_{\mathcal{U}})$

I am in a set theory class and we just started talking about ultrapowers. My professor mentioned that, if $\kappa$ is a measurable cardinal and $\mathcal{U}$ is a nonprincipal $\kappa$-complete ...
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0answers
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Which large cardinal this theory stops at proving its existence?

The language of this theory is first order predicate calculus with extra-logical primitive symbols of $``=; \in, W"$, where $``W"$ is a constant symbol. Axioms: those for identity theory + ...
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Coding a function $g:\kappa\to V_{\zeta+1}$ by an element of $V_{\zeta+1}$

Update: this question has been answered on MathOverflow. This question is about the following remark (modified to be self-contained), found in Donald Martin's book on determinacy, page 340. The ...
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1answer
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Is it true ZF can't prove there is a choice function on the Turing degrees?

It is possible to show there is no Borel choice function on the Turing degrees. In fact we can even prove there is no Turing invariant Borel injection from $2^\omega$ to $2^\omega$, which sort of says ...
7
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1answer
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Consistency of non-trivial elementary embedding $j: Ord \to Ord$

What is the consistency strength of the existence of an elementary embedding $j: Ord\to Ord$ from the class of ordinal to itself? For example, if there is an elementary embedding $j: V\to M$, then $j\...
7
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2answers
340 views

What is wrong with this proof that inaccessible cardinals are inconsistent

Disclaimer: I believe this proof is wrong, but I'm asking because I can't find what's wrong with it, which means I must have some basic misunderstanding of the concepts involved. First, some ...
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2answers
133 views

What is wrong with this proof there is no $\omega$-th worldly cardinal

Call a cardinal $\kappa$ worldly iff $V_\kappa\vDash ZFC$. Let $\kappa_\alpha$ be the $\alpha$th worldly cardinal, i.e. the least worldly cardinal such that $\{\beta\lt\kappa_\alpha|\beta\text{ is ...
2
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1answer
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Which sets admit a countably additive measure defined on their powerset?

I want to know when a cardinality $\kappa$ admits a (countably additive) probability measure on its powerset that gives probability zero to subsets of cardinality strictly less than $\kappa$. I am ...

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