# Tagged Questions

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### higher-order reflection

In the first-order context, "reflection" of a formula $\varphi(x)$ below $\kappa$ refers to the the following situation:
There are many ordinals $\alpha<\kappa$ such that for all $a \in ...

**5**

votes

**0**answers

139 views

### A variant of Chang's model with choice

Let $M_n$, $n < \omega$, be a models of $ZFC$ with the same ordinals, closed under countable sequences. Let $\alpha_n$ be an ordinal which is a regular cardinal in $M_n$.
Question 1: Is it ...

**10**

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**0**answers

294 views

### Ideas behind Gitik's solution of PCF conjecture

Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem:
Theorem. Assuming the consistency of infinitely many strong cardinals, one ...

**7**

votes

**1**answer

239 views

### Inaccessible becomes successor of singular

Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and ...

**2**

votes

**1**answer

155 views

### The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$.
One important example of left distributive algebras arises ...

**7**

votes

**1**answer

122 views

### Strongly compact cardinal with bad covering properties

This is a continuation of the question covering properties of strongly compact embedding.
Recall that a cardinal $\kappa$ is $\nu$-strongly compact cardinal if there is an elementary embedding ...

**11**

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**1**answer

281 views

### What is the large cardinal strength of the assertion that every $\kappa$-complete filter on $\kappa$ extends to a $\kappa$-complete ultrafilter?

It is well-known that an uncountable regular cardinal $\kappa$ is strongly compact if and only if every $\kappa$-complete filter on any set extends to a $\kappa$-complete ultrafilter on that set. The ...

**6**

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195 views

### Does Sageev's result need an inaccessible?

In 1981, building on work by Ellentuck in 1974, Sageev showed ("A model of ZF + there exists an inaccessible, in which the Dedekind cardinals constitute a natural non-standard model of arithmetic," ...

**10**

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429 views

### Does Turing determinacy imply full determinacy?

The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined.
In "Turing ...

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2k views

### What “forces” us to accept large cardinal axioms?

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).
Their non-existence is consistent with axioms of usual mathematics.
It is provable that some of ...

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**2**answers

591 views

### Questions about Prikry forcing and Cohen forcing

I have some questions.
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...

**9**

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**1**answer

288 views

### Inner model in which every uncountable cardinal is large

The following is known:
$(*)$ If $0^\sharp$ exists, then any uncountable cardinal is is an inaccessible cardinal (and even more) in $L$.
My question is that:
Are there any large cardinal ...

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**0**answers

184 views

### $\infty$-Borel Determinacy?

An $\infty$-Borel set is a set $X\subseteq\mathbb{R}$ which has an $\infty$-Borel code - a set $r$ coding the construction of $X$ via open sets, complementation, and well-ordered unions (see ...

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246 views

### cardinals below the critical point of a generic embedding

This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension?
To focus on the ...

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**0**answers

563 views

### Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like ...

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**1**answer

242 views

### Woodin Cardinals and Inner Models

I have a few questions I have been thinking about that I could definitely use some insights on:
Question 1. Since a Woodin cardinal is a "local" notion, defined with respect to some rank-initial ...

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**0**answers

105 views

### Does the Lévy collapse obey this nice characterization? [duplicate]

This question is related to an issue in my answer to Monroe Eskew's question on the failure of Cantor-Bernstein for the Lévy collapse.
Question. Is the Lévy collapse $\text{Coll}(\omega,\lt\kappa)$ ...

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884 views

### Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...

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**4**answers

973 views

### What is the definition of a large cardinal axiom?

In different books one can find different implicit definitions for a large cardinal axiom.
My question is that which one of these definitions are more popular or standard amongst set theorists?
Any ...

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votes

**3**answers

631 views

### PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)

We know $PFA$ implies $2^{\aleph_0}=\aleph_2$.
Q1. What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum ...

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242 views

### Are superstrong stronger than strongly compact cardinals? (or vice versa)

In the last part of Kanamori's excellent "The Higher Infinite" there is a small diagram about the strength and consistency strength of some major large cardinal axioms.
Below supercompact cardinals ...

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1k views

### Latest stand of core model theory?

What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004:
...

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348 views

### V=HOD & The Height of the Large Cardinal Tree

As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are ...

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216 views

### $(\kappa,\lambda)$ - Minimal Models & Stronger Version of Rowbottom's Theorem

Definition 1: Let $M$ be a $\mathcal{L}$ - structure and $A\subseteq Dom(M)$. Define:
$Def_{A}(M):=\{X\subseteq Dom(M)~|~\exists n\in \omega~~\exists \varphi (x,y_1,...,y_n)\in ...

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256 views

### $(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$

The notion of minimality in model theory is related to the existence of a gap in the size of definable subsets of a model. Now consider the following generalization:
Definition 1: Let $M$ be a ...

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428 views

### Is the tree of large cardinals linear?

Kanamori in the introduction of his book "The Higher Infinite" says:
"The investigation of large cardinal hypotheses is indeed a mainstream of modern set theory, and they have been found to play a ...

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**4**answers

1k views

### A New Continuum Hypothesis (Revised Version)

Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$.
What happens after exponentiation?
We have the following equation: $2^{N_n}=N_{2^{n}}$.
(Which says: For all finite cardinal $n$ ...

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votes

**1**answer

170 views

### Failure of GCH at indescribable cardinals

Can $\Pi^m_n$ indescribable cardinal be the first one where $\text{GCH}$ fails?
Hauser showed in
Hauser,K.: Indescribable cardinals and elementary embeddings.
J. Symb. Logic 56, 439457 (1991)
that ...

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249 views

### What is the role of absoluteness in existence of a non-trivial self elementary embedding on an inner model?

ِDefinition (1): Call an inner model M "rigid" if there are no non-trivial elementary embeddings $j: M\longrightarrow M$.
It is possible that $L$ is not rigid, while the problem is open for $HOD$. ...

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197 views

### Are measures of a measurable cardinal measurable? (Edited and Updated Version)

Update: Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers.
Definition: Let $\kappa$ be a measurable cardinal. Define:
...

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322 views

### Is the inclusion version of Kunen inconsistency theorem true?

The relations $\in$ and $\subsetneq$ seem so similar in some sense. For example they are equal on ordinal numbers. So there is a natural question about their possible similar behaviors on the ...

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votes

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255 views

### Are larger large cardinals less expressible?

First note to the following well known theorems:
Theorem (1): The notion of "$x$ is a strongly inaccessible cardinal" is first order expressible and $\Pi_{1}$.
Theorem (2): ...

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224 views

### Vopenka's Principle for non-first-order logics

(For simplicity, the background theory for this post is NBG, a set theory directly treating proper classes which is a conservative extension of ZFC.)
Vopenka's Principle ($VP$) states that, given any ...

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262 views

### Singular successors without large cardinals

Assuming the axiom of choice we have that successor cardinals are regular. However as one of the first examples of uses of forcing show, it is consistent relative to $\sf ZF$ that $\omega_1$ is ...

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363 views

### Is Prikry forcing minimal?

Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb ...

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535 views

### What are Moschovakis cardinals?

The question is exactly that of the title: what are Moschovakis cardinals?
Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency ...

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482 views

### Failure of diamond at large cardinals

What is known about the failure of $\Diamond_{\kappa}$ (diamond at $\kappa$) for $\kappa$ (the least) inaccessible, (the least) Mahlo and (the least) weakly compact.
Remark. The problem of forcing ...

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**1**answer

275 views

### Indescribability of cardinals and categoricity of $V_\kappa$

If $\kappa$ is an inaccessible cardinal then $V_\kappa$ is a model of $\sf ZFC_2$ ($\sf ZFC$ with a second-order replacement axiom).
If there are many inaccessible cardinals then there are many ...

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685 views

### On the large cardinals foundations of categories

(This question was posted on math.SE over two weeks ago, but received no answer. I am therefore posting it here as well.)
It is well-known that there are difficulties in developing basic category ...

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**1**answer

256 views

### Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal

Is anything known about the consistency strength of the following statement?
$\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ such that $a \cap ...

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votes

**1**answer

214 views

### Consistency of many Erdos cardinals

Does anyone have a reference for the consistency strength of saying that the Erdos cardinal $\kappa(\alpha)$ exists for all $\alpha$? Or of various weakenings about how far such cardinals extend into ...

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156 views

### Versions of large cardinals with target model in a generic extension

(I ran into this question while thinking about the (Strong) Inner Model Hypothesis: see http://www.jstor.org/stable/4093051 or http://arxiv.org/abs/0711.0680.)
A measurable cardinal is a cardinal ...

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543 views

### Kunen's inconsistency result

A well-known result of Kunen says that there is no non-trivial elementary embedding $j: V \rightarrow V.$ There are several proofs of this theorem (see Kanamori, The higher infinite). I wonder to know ...

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402 views

### What can the degrees of constructibility be?

If $r, s\in\mathbb{R}$, we say $r$ is constructible relative to $s$ - and write $r\le_cs$ - if $r\in L[s]$. Modding out by the induced equivalence relation $\equiv_c$, we get a partial order, the ...

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298 views

### Homogeneous Namba-like forcing

Let $\kappa \ge \aleph_3$ be a regular cardinal that is countably closed ($\alpha^\omega < \kappa$ for every $\alpha < \kappa$.) I'm mostly interested in the case that $\kappa$ is strongly ...

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266 views

### Elementary Embeddings and Relative Constructibility

Suppose $$j:M\prec N$$ is a non-trivial elementary embedding. Under what conditions on the sets (classes?) $M$ and $N$ (or even the critical point of $j$) does $j$ extend to an elementary embedding ...

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276 views

### Relation between $\neg \square(\kappa)$ and the tree property at $\kappa$.

If $\kappa$ is an inaccessible cardinal then the tree property at $\kappa$ is equivalent to weak compactness of $\kappa$, which implies that $\square(\kappa)$ fails---that is, that every coherent ...

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630 views

### Can measures be added by forcing?

The Lévy-Solovay theorem says that small forcings do not create measures. J.D. Hamkins has generalized this to a larger class of forcings called gap forcings. I would assume this cannot be ...

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361 views

### Can $\omega_1$ be supercompact?

Is "ZF + $\omega_1$ is supercompact" consistent relative to "ZFC + there is a supercompact cardinal"?
In particular, if $\delta$ is supercompact, does it remain so in $V(\mathbb{R} \cap V[G])$ where ...

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863 views

### What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?

Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that's unclear, in this ...