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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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Can the cardinality of the set of all intervening cardinals between sets and their power sets be always singular?

Is the following known to be consistent relative to some large cardinal assumption? $\forall \kappa [\kappa >2 \to \kappa < \kappa^* < 2^\kappa \wedge singular(\kappa^*)]$ where $\kappa$ is ...
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Are normal, nonprincipal ultrafilters necessarily closed?

Let $U$ be a normal (closed under diagonal intersections), nonprincipal (doesn't contain singletons) ultrafilter on some uncountable cardinal $\kappa$. Q. Is $U$ $<\kappa$-complete (closed under ...
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Measurable Cardinals are Mahlo Cardinals

I am new to set theory and have been working through the proof that every measurable cardinal is Mahlo on page 135 of Jech's text. With the help of Asaf's comments (Measurable $\rightarrow$ Mahlo), I ...
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Query about whether a strengthening of I2 is consistent

Can you consistently have a $j:V\rightarrow M$ with $\mathrm{crit}(j)=\kappa$ and $\delta$ the least ordinal greater than $\kappa$ with $j(\delta)=\delta$ and $V_{\delta}\subseteq M$ and $(V_{\delta+1}...
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Why not weaken reducibility in $K_2(W)$ instead of sub-world separation?

In this theory $K_2(W)$ (page 7) Harvey Friedman argue for weakening Sub-world Separation into SS-. He did that in order to evade making $W$ transitive. But he could have done that by simply reverting ...
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Is the least inaccessible cardinal equivalent to the first aleph fixed point? [duplicate]

Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous ...
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The transitive collapse of an ultraproduct of $V$ is an inner model

So this is the situation: Let $U$ be an ultrafilter over a set $S$ and let $M_U$ be the transitive collapse of $\text{Ult}(V, U)$. $M_U$ is an inner model of ZFC. Thus far i have used Los's theorem ...
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Can elementary embeddings make ordinals jump over strongly inaccessible cardinals?

Let $j:V\to M$ be an elementary embedding. Assume $\xi\geq crit(j)$ is strongly inaccessible, and $\alpha<\xi$. Is it possible that $j(\alpha)\geq\xi$ ? If yes - does the additional assumption $...
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Is Reflection consistent with Resemblance?

The following theory is a class theory that combines two principles that of reflection and resemblance, informally it says that the class $V$ of all sets resembles a set $W$ that stands as a sub-world ...
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intuition for PFA

Kunen (Set theory 2011) says on the page 307: The Proper Forcing Axiom (PFA) is the assertion that $MA_{\mathbb P}(\aleph_1)$ holds for all proper $\mathbb P$. My question is, what kind of axiom ...
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amenable ultrafilters

Suppose $M$ is a transitive model of ZFC-powerset. If $\kappa \in M$ is a cardinal and $U$ is an ultrafilter on the boolean algebra $\mathcal P(\kappa)^M$, we say $U$ is amenable to $M$ if whenever $\...
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“Weak” Ramsey conditions for cardinals

Ok, so these questions just popped into my head and I can't seem to figure it out: Ramsey's theorem tells us that for any $n,r\in\omega$ and any $f:[\omega]^{n}\rightarrow r$, exists an infinite set $...
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What is the consistency strength of Ackermann + the following cardinals to ordinals isomorphism?

This is a try to salvage the attempt written in the posting: "Can we derive a large cardinal axiom by a principle of isomorphism between cardinals and ordinals?" Here we change the base theory to ...
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Can we derive a large cardinal axiom by a principle of isomorphism between cardinals and ordinals

[EDIT, this posting had been answered to the negative, However it couldn't be deleted, so I've written a salvage for it in the posting titled "What is the consistency strength of Ackermann + the ...
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Why is the existence of large cardinals believed to be true?

I am in the middle of watching a video of a presentation given by W. Woodin about the continuum hypothesis. This is not really something I know about, but I am confused by one of the slides (at 22:23 ...
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Measurable cardinals admit homogeneous set

I'm trying to prove that if $\kappa$ is a measurable cardinal with a normal ultrafilter $U$, then for every $f : [\kappa]^{< \omega} \to \gamma$, where $\gamma < \kappa$, there exists $H \in U$ ...
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Should measurable cardinals be regular?

I know that there are two very similar questions here, one asking if the condition of $\kappa$ being a regular cardinal in the definition of $\kappa$-complete filter is really needed and another ...
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Does this ordinal (built by using ZFC + ordinal many large cardinals to attain yet larger ordinals) have a name?

Consider the following pair of functions: For α an ordinal, let $\nu(\alpha)$ be the least ordinal $\kappa$ such that $(V_{\kappa}, \in)$ is a model of ZFC in which there are $\alpha$-many ...
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Does exist a singular cardinal $\lambda$ which $\kappa < \lambda \implies 2^{\kappa}<\lambda?$ [duplicate]

I'm reading about inacessible cardinals. I don't know anything about them, I just know basic set theory, cardinal operations etc., but I am too curious to leave this question that now I'm (probably) ...
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1answer
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Can there be a proper class of $\alpha$-inaccessibles for any ordinal $\alpha$?

A cardinal $\kappa$ is $\alpha$-inaccessible if $\kappa$ is inaccessible and such that for any ordinal $\beta < \alpha$, the set of $\beta$-inaccessibles less than $\kappa$ has cardinality $\kappa$....
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Group theoretic Characterizations of Large Cardinals

Vopenka's principle is characterized (in the category theoretical definition) by no large full subcategory of a locally presentable category being discrete. In other words, proper classes of types of "...
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Huge vs. compact cardinal

How is it possible and how does one prove that the least huge cardinal is less then the least compact cardinal (if both exist) but at the same time huge has higher consistency strength then the ...
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Ultrapower of V by D

Let $D$ be a nonprincipal $\kappa$-complete ultrafilter on $\kappa$. The following are equivalent: (i) $D$ in normal (ii) In the ultrapower $Ult_D(V)$, $\kappa=[d]$ where $d$ is the diagonal ...
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2-huge cardinal, the most usual definition [closed]

How are 2-huge cardinals usually defined? Is the definition contained in Jech's set theory book (the millennium edition)? Which page?
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Mahlo operation, consistency border [closed]

Can a (relatively consistent) cardinal notion be given so that its usual Mahlo operation is (probably at least) not consistent?
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Large Cardinals and Diophantine Equations (Penelope Maddy)

Professor Penelope Maddy remarks without elaboration in her famous 'Believing the Axioms' essay that 'It should be mentioned that the Axiom of Inaccessibles also has a few extrinsic merits. It ...
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Unique lifts for small forcing extensions (set theory)

I am taking a directed research course in set theory and I am having a lot of trouble, here is the problem I'm still having trouble with: Problem: Prove that every embedding $j: V \rightarrow M$ in $...
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Ramsey property and linear orders on $\kappa$

I have been trying to solve to prove the following statement: Let $\kappa$ be an uncountable cardinal. The following are equivalent: Every linear order of cardinality $\kappa$ has a ...
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Equivalence of the existence of some ultrafilter and some elementary embedding

I'm in the proof of showing, that the consistency of 2-huge cardinals implies the consistency of hyper-huge cardinals and it needs this equivalence: Let $\kappa,\kappa',\lambda,\lambda'$ be cardinals....
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1answer
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Equi-consistency of inaccessible cardinals [duplicate]

I have trouble understanding this wiki page, concerning inaccessible cardinals. First "These statements are strong enough to imply the consistency of ZFC". I guess what they mean by that is if $\...
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1answer
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What axiomatic set theories say that large cardinals exist [closed]

The title is my question. I'm curious since I can't seem to find any axiomatic set theory that say that large cardinals exist. Another thing I’d like to know is that if there are any axiomatic set ...
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Classify this Large Cardinal

If $\kappa$ is a strongly inaccessible cardinal, then let $\Gamma(\kappa)$ be the number of strongly inaccessible cardinals below $\kappa$. If we call $\lambda$ the smallest fixed point of $\Gamma$, ...
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Proving the categoricity of an extension of ZFC2

In Asaf's answer to this question: Proper-class-many categorical extensions of ZFC2 he confirmed that it is possible to categorically characterize a model of second-order ZFC by adding an axiom ...
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Why does an M-Ultrafilter on $\kappa$ imply that $\kappa$ is weakly compact?

so far I´ve only been reading on this forum, this is my first question. I am trying to understand Kunens article on iterated ultrapowers from 1970 (Some applications of iterated ultrapowers) and ...
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1answer
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Is the Mahlo ordinal the first cardinal unreachable using inaccessibility and diagonalisation?

I read the weakly Mahlo ordinal is weakly inaccessible , hyper-weakly inaccessible, hyper-hyper-weakly inaccessible, (1@α)-weakly inaccessible, and so on as far as you diagonalize. But is it the ...
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In what sense are inaccessible cardinals inaccessible?

Another title for this question could be: where do inaccessible cardinals live? It may be that this question does not make any sense. So I will try to explain what I mean. I think of the ZFC axioms ...
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Equivalent formulations of Woodin cardinals

Consider the following properties for a given ordinal $\delta$: For all $f \colon \delta \to \delta$ there is an elementary embedding $$ j \colon V \to M, M \text{ transitive} $$ such that $f " \...
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Consistency strength of the Large cardinals is almost linear

Is there a known and strong reason why the large cardinal hierarchy is almost linear w.r.t. the consistency strength?
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Showing that $\kappa$ is weakly compact in ultrapower.

Suppose $\kappa$ is measurable. In trying to show that there are many weakly compact cardinals below it, I quickly reduced the problem to showing that $\{\alpha < \kappa : \alpha$ is weakly compact ...
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Regarding Extenders [closed]

I'm having a hard time proving the following claim: Let $j\colon M\rightarrow N$ be an elementary embedding (between inner models) with $\operatorname{crit}(j)=\kappa$. Let $\kappa<\lambda$. ...
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Definition of an $\omega$-huge cardinal

Martin and Steel's paper "A proof of projective determinacy" defines an $\omega$-huge cardinal to be an I2 cardinal but more recently you see it being defined to be an I1 cardinal. Is there a standard ...
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1answer
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Inaccessible side-effects in MK

In MK (Kelley-Morse) class theory, if i add an axiom that any cardinal except $On$ has an inaccessible greater than it (ie. essentially a Tarski/Grothendiek universe axiom), does that compel me to ...
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Proof that the cofinality of the least worldly cardinal is $\omega$

Prove in $\text{ZFC}$ that if there is some ordinal $\alpha$ with $V_\alpha \models \text{ZFC}$, then $\text{cf}(\beta)=\omega$ where $\beta$ equals the least such $\alpha$. Here is what I came up ...
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1answer
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Inaccessibility in L vs. Inaccessibility in ZFC

We know that we can prove GCH using the axiom of constructibility, and so that implies the the continuum is regular and uncountable, and therefore weakly-inaccessible. However, is this "inaccessible" ...
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Does ZFC + the Axiom of Constructibility imply the nonexistence of inaccessible cardinals?

In ZFC set theory, does adding Constructibility imply the nonexistence of inaccessible cardinals like it does the nonexistence of other types of higher cardinals?
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Normality of some generic ultrafilter

My question is about some proof details in Lemma 6.4 in Gitik's chapter from the Handbook of Set Theory. On the sequel suppose that, $j:V\longrightarrow M$ stands for the elementary embedding ...
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1answer
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If $G$ is $P$-generic over $V$ and $G^*$ is $j''P$-generic over $M$ then $j$ can be extended to $V[G]$.

I've seen the following result: For elementary $j$, if $G$ is $P$-generic over $V$ and $G^*$ is $j(P)$-generic over $M$ and $j''G\subseteq G^*$ then $j$ can be extended to elementary $j:V[G]\...
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Possibility of preserving the ultrafilter on $\mathcal{P}_{\kappa}(\lambda)$ in V[G] after forcing with a <$\kappa$ directed closed poset?

Suppose we have the ultrafilter definition of a supercompact cardinal. Now, we know if $\kappa$ is supercompact and indestructible by any <$\kappa$ directed closed forcing poset in V then $\kappa$ ...
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1answer
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Target of a superstrong embedding

Let $j\colon V \to M$ be a superstrong elementary embedding (i.e. $M$ is transitive, $j\neq id$, and $V_{j(\kappa)} \subseteq M$ where $\kappa$ is the critical point of $j$). Is $j(\kappa)$ ...
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Can we find $\mu<\kappa$ with $2^{\mu^{(n)}}<\sup_{n<\omega}\mu^{(n)}$?

Let $\kappa$ be inaccessible. Can we find $\mu$ with $cf(2^\mu)<\kappa$ and $2^{\mu^{(n)}}<\sup_{n<\omega}\mu^{(n)}$ for every $n<\omega$?