Yair Hayut
  • Member for 8 years, 2 months
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Can $L$ be thin?
Accepted answer
29 votes

Claim: $|V_\alpha \cap L| = |\alpha|$ for every $\alpha \geq \omega$ implies that $0^\#$ exists. Proof: Let's assume, toward contradiction, that $0^\#$ doesn't exist that $|V_\alpha \cap L| = |\...

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Does $\diamondsuit(\kappa)$ provably hold at Woodins or inaccessible Jónssons $\kappa$?
13 votes

This is a partial answer. I will show that if $\delta$ is Woodin then $\diamondsuit_\delta$ holds. Claim: Any Woodin cardinal is subtle. Proof: Let $\delta$ be a Woodin cardinal. Let $\vec{A} = \...

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Can there be an embedding j:V → L, from the set-theoretic universe V to the constructible universe L, when V ≠ L?
13 votes

Assuming $0^\#$ doesn't exist, it's consistent to get a negative answer to those questions: Assume $(2^{\aleph_0})^V > \aleph_2$ and that ${\aleph_n}^V = {\aleph_n}^L$ for at least $n=1,2,3$. I ...

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Partitioning $\omega_1$-branching trees of size and height $\omega_1$
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12 votes

The existence of such trees is independent of ZFC. On one hand, if $CH$ holds then $\omega_1^{<\omega_1}$ (and in fact - any $\sigma$-closed $\omega_1$-branching tree) cannot be partitioned into $...

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Changing cofinalities above supercompact cardinals
11 votes

There is no such forcing that preserves $\lambda^+$. Since $\lambda$ is measurable, $2^{<\lambda} = \lambda$ and therefore $\square_{\lambda,\lambda}$ holds in $V$. Since $\lambda^{+}$ is preserved,...

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Getting a model of $\mathsf{ZFC}$ that fails to nicely cover an inner model
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10 votes

The consistency strength of the failure of $(\dagger)$ is an inaccessible cardinal. Building on the comment of Mohammad, if $\omega_2^V$ is a successor cardinal in $L$ then there is a set $X \subseteq ...

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Does this consequence of measurability in terms of games of length $\omega+1$ imply measurability?
10 votes

This answer addresses only the consistency strength of $\omega+1$-strategically measurable. Claim: If there is a $\omega+1$-strategically measurable cardinal then there is an inner model with a ...

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Is every transitive ZF-model of inaccessible height a truncation of an inner model?
Accepted answer
10 votes

Theorem: Let $\kappa$ be strongly inaccessible in $V$, such that $V \models ZFC$. If $M\models ZF$, then $L(M) \cap V_\kappa = M$. Proof: Let us prove by induction on $\alpha < \kappa$ that $L(M) \...

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Cofinality of $j(\kappa)$ for a measurability embedding $j:V\to M$ with critical point $\kappa$
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10 votes

As Mohammad Golshani remarked, it is possible to control the cofinality of $j(\kappa)$ by iterating the forcing that adds a function $f\colon \kappa \to \kappa$ which is eventually larger than any ...

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singularize the least inaccessible?
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10 votes

In the paper "On Lowenheim-Skolem-Tarski numbers for extension of first order logic", by Magidor and Vaananen, in Theorem 21 they state that it is consistent, relative to the existence of a ...

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cardinality of perfect sets in generalized Baire space
10 votes

For every $\kappa$ of uncountable cofinality there is a tree $T\subseteq 2^{<\kappa}$ such that $[T]=\kappa$. The tree $T$ is the tree of all binary sequences $f\colon \alpha \to 2$, $\alpha <\...

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Can there be an almost-special not-fully-special Aronszajn tree?
10 votes

The answer is yes. Force a generic $\square(\omega_1)$ sequence. This poset, $S$, is $\sigma$-closed, so it doesn't collapse $\omega_1$. By Todorcevic, in the $\omega_1$-tree obtained from the ...

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Questions about Prikry forcing and Cohen forcing
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9 votes

This should be a comment - but it is too long: Assume that $U = U_1 = U_2$. I want to show that $\mathbb{P}_U ^2 \cong \mathbb{P}_U\times \mathbb{C}$ where $\mathbb{C}$ is the Cohen forcing. Let $\{ ...

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absorption of strategically closed posets
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9 votes

If $\mathbb{P}$ is $\kappa$-strategically closed then there is a projection from $Col(\kappa,\mathbb{P})$ onto $\mathbb{P}$. The proof is very similar to the $\kappa$-closed case: Let $\sigma$ be ...

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Non-homogeneous forcing and HOD
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8 votes

It is consistent that the answer is positive and it is consistent that the answer is negative. Claim: There is a generic extension, $V[G]$ by a weakly homogeneous forcing notion in which there is a ...

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Questions about $\aleph_1-$closed forcing notions
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8 votes

The answer to the second question is yes, without any large cardinals assumptions. Claim: if $2^{\aleph_0}$ is singular then every non-trivial $\sigma$-closed forcing of size $2^{\aleph_0}$ collapses ...

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Ordinal-indexed transitive antichain of sets with urelements
7 votes

If there is a subtle cardinal then such $\phi$ doesn't exist. An subtle cardinal is a cardinal $\kappa$ such that for every sequence of sets $\langle S_\alpha \mid \alpha < \kappa\rangle$ such ...

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The role of the rigid relation principle ($RR$) in the Kunen inconsistency
7 votes

Claim: NGB + $\exists j\colon V\to V$ is equiconsistent with NGB + $\exists j\colon V\to V + \neg RR$. In order to prove the left to right direction, we will start with a model of NGB + $\exists j\...

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failure of $\square(\kappa)$ at an inaccessible $\kappa$
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7 votes

The consistency strength of the failure of $\square(\kappa)$ for non-weakly Mahlo inaccessible cardinal $\kappa$ (in particular, for the first strongly inaccessible) is higher than weakly compact. For ...

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Square and stationary reflection
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7 votes

The answer is yes (at least for $\kappa = \omega_1$): Shelah and Harrington showed that you can force that every stationary subset of $S_{\omega}^{\omega_2}$ reflects starting with a Mahlo cardinal. ...

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Set-theoretic geology III: inside the core
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6 votes

Here are a few observations about CORE. Claim: It is consistent that CORE is not pairwise upwards directed. Proof: Let $\mathbb{P}_0$ be the class forcing for the Easton product of the Cohen forcing $\...

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

In my paper "Partial strong compactness and squares", I investigate those questions. The main results of this paper are: If every $\kappa$-complete filter on $\kappa$ can be extended to a $\kappa$-...

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Two questions about higher Souslin trees
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6 votes

About question 1: If $\kappa$ is not weakly compact in $L$, then there is an $\aleph_2$-Suslin tree in $L[G]$. In $L$ there is a $\kappa$-Suslin tree, $T$, and since Mitchell's forcing is $\kappa$-...

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Forcing to "minimally" add new reals.
Accepted answer
6 votes

There is no forcing that satisfies $(*)$. Let $r$ be a new real, $r \in M[G]\cap \mathcal{P}(\omega) \setminus M$. Take $t = \{n < \omega \mid n + 1\in r\}$, and let $a, b\in M$ such that $t = (a\...

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Erdős cardinals and $0^\sharp$
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6 votes

Similarly to Asaf's comment, let $\kappa = \kappa(\omega_1^L)$ (we assume that it exists). Then in $L_\kappa$, there is $\alpha$-Erdős cardinal for every $L$-countable $\alpha$ (since being $\alpha$-...

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Preserving distributivity with finite support products
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5 votes

GCH implies that $\mathbb{Q}_\alpha$ is $\alpha$-distributive in the generic extension by $\mathbb{P}_\alpha$. Note that $\Vdash_{\mathbb{Q}_\alpha} ``\check{\mathbb{P}}_\alpha$ is $\check\alpha$.c....

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Products of Cohen forcings
5 votes

Assuming that there is an $\omega_1$-dense $\sigma$-complete ideal, $I$, on $\omega_1$ (such that every positive set can be partitioned into $\omega_1$ positive sets), we can generalize Joel's ...

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capturing small sets in small factors
Accepted answer
5 votes

Assume that a regular cardinal $\kappa > \omega_1$ has the property that for every $\kappa$.c.c. forcing notion $\mathbb{P}$ and name of set of ordinals $x \in V^{\mathbb{P}}$ smaller than $\kappa$,...

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Countably closed end-extensions of elementary submodels
4 votes

It seems unlikely. Let me use the following consequence of partial global square (which appears in my joint paper with Garti). This partial sequence is consistent with the existence of very large ...

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If $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, what properties does $κ$ have?
Accepted answer
4 votes

Let me start by observing that this property is equivalent to a more standard property: Claim: Let $M\subseteq V$ be a transitive model of $\mathrm{ZFC}$ and let $\mu \in M$ be a cardinal in $V$. The ...

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