The tree property and the failure of SCH at \alpeh_{\omega^2} II

Speaker: 

Dr Dima Sinapova

Institution: 

UCI

Time: 

Monday, January 31, 2011 - 4:00pm

Location: 

RH 440R

The tree property at \kappa^+ states that every tree with height \kappa^+ and levels of size at most \kappa has an unbounded branch. There is a tension between the tree property and the Singular Cardinal Hypothesis (SCH). Woodin asked if the failure of SCH at a singular cardinal \kappa implies the tree property at \kappa^+. Recently Neeman answered this question in the negative. Here we show that this result can be obtained at small cardinals. In particular we will show that given \omega many supercompact cardinals there is a generic extension in which the tree property holds at \aleph_{\omega^2+1} and SCH fails at \aleph_{\omega^2}.

The tree property and the failure of SCH at \alpeh_{\omega^2}

Speaker: 

Dr Dima Sinapova

Institution: 

UCI

Time: 

Monday, January 24, 2011 - 4:00pm

Location: 

RH 440R

The tree property at \kappa^+ states that every tree with height \kappa^+ and levels of size at most \kappa has an unbounded branch. There is a tension between the tree property and the Singular Cardinal Hypothesis (SCH). Woodin asked if the failure of SCH at a singular cardinal \kappa implies the tree property at \kappa^+. Recently Neeman answered this question in the negative. Here we show that this result can be obtained at small cardinals. In particular we will show that given \omega many supercompact cardinals there is a generic extension in which the tree property holds at \aleph_{\omega^2+1} and SCH fails at \aleph_{\omega^2}.

The Tree Property at $\aleph_{\omega+1}$ IV

Speaker: 

Dr Dima Sinapova

Institution: 

UCI

Time: 

Monday, November 29, 2010 - 4:00pm

Location: 

RH 440R

We show that given $\omega$ many supercompact cardinals, there is a
generic extension in which there are no Aronszajn trees at
$\aleph_{\omega+1}$. This is an improvement of the large cardinal
assumptions. The previous hypothesis was a huge cardinal and $\omega$ many
supercompact cardinals above it, in Magidor-Shelah.

The Tree Property at $\aleph_{\omega+1}$ III

Speaker: 

Dr Dima Sinapova

Institution: 

UCI

Time: 

Monday, November 22, 2010 - 4:00pm

Location: 

RH 440R

We show that given $\omega$ many supercompact cardinals, there is a
generic extension in which there are no Aronszajn trees at
$\aleph_{\omega+1}$. This is an improvement of the large cardinal
assumptions. The previous hypothesis was a huge cardinal and $\omega$ many
supercompact cardinals above it, in Magidor-Shelah.

The Tree Property at $\aleph_{\omega+1}$ II

Speaker: 

Dr Dima Sinapova

Institution: 

UCI

Time: 

Monday, November 8, 2010 - 4:00pm

Location: 

RH 440R

We show that given $\omega$ many supercompact cardinals, there is a
generic extension in which there are no Aronszajn trees at
$\aleph_{\omega+1}$. This is an improvement of the large cardinal
assumptions. The previous hypothesis was a huge cardinal and $\omega$ many
supercompact cardinals above it, in Magidor-Shelah.

The Tree Property at $\aleph_{\omega+1}$ I

Speaker: 

Dr Dima Sinapova

Institution: 

UCI

Time: 

Monday, November 1, 2010 - 4:00pm

Location: 

RH 440R

We show that given $\omega$ many supercompact cardinals, there is a
generic extension in which there are no Aronszajn trees at
$\aleph_{\omega+1}$. This is an improvement of the large cardinal
assumptions. The previous hypothesis was a huge cardinal and $\omega$ many
supercompact cardinals above it, in Magidor-Shelah.

Catching slender functions II

Speaker: 

Dr Sean Cox

Institution: 

Munster University, Germany

Time: 

Monday, October 18, 2010 - 4:00pm

Location: 

RH 440R

I will present the proofs of some recent results of Viale
and Weiss. Weiss introduced the notion of a slender function in his
dissertation: roughly, a function $M \mapsto F(M) \subset M$ (where
$M$ models a fragment of set theory) is slender iff for every
countable $Z \in M$, $Z \cap F(M) \in M$; i.e. $M$ can see countable
fragments of $F(M)$. Viale and Weiss proved that under the Proper
Forcing Axiom, for every regular $\theta \ge \omega_2$, there are
stationarily many $M \in P_{\omega_2}(H_{(2^\theta)^+})$ which
``catch'' $F(M \cap H_\theta)$ whenever $F$ is slender (i.e. whenever
$F$ is slender then there is some $X_F \in M$ such that $F(M \cap
H_\theta) = M \cap X_F$). The stationarity of this collection implies
many of the known consequences of PFA; e.g. failure of weak square at
every regular $\theta \ge \omega_2$; and separating internally
approachable sets from sets of uniform uncountable cofinality.

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