Dense ideals from determinacy II

Speaker: 

Trevor Wilson

Institution: 

UCI

Time: 

Monday, April 29, 2013 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

Under the Axiom of Determinacy, the least uncountable cardinal omega_1 behaves like a large cardinal. We will present a theorem due to Woodin saying that from a strong form of determinacy, namely AD_R + "Theta is regular," one can force the Axiom of Choice together with the statement "there is an omega_1-dense ideal on omega_1." In essence, omega_1 retains a trace of its "large cardinal" nature that is consistent with AC.

Dense ideals from determinacy I

Speaker: 

Trevor Wilson

Institution: 

UCI

Time: 

Monday, April 22, 2013 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

Under the Axiom of Determinacy, the least uncountable cardinal omega_1 behaves like a large cardinal. We will present a theorem due to Woodin saying that from a strong form of determinacy, namely AD_R + "Theta is regular," one can force the Axiom of Choice together with the statement "there is an omega_1-dense ideal on omega_1." In essence, omega_1 retains a trace of its "large cardinal" nature that is consistent with AC.
 

ISP, guessing models, and PFA III

Speaker: 

Christoph Weiss

Institution: 

UCI

Time: 

Monday, April 15, 2013 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

We complete our introduction to the principle ISP and its relatives as well as their connections to supercompact cardinals and the proper forcing axiom. As a consequence of our analysis we give a proof that all known forcing constructions of models satisfying PFA require very large cardinals.

Stationary reflection, mouse reflection, and strategy reflection

Speaker: 

Ralf-Dieter Schindler

Institution: 

Uni Muenster (Germany) and UC Berkeley

Time: 

Monday, February 25, 2013 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

Set theory studies reflection principles of different forms. The talk will discuss the role of stationary reflection and threadability in the core model induction. I will not presuppose any serious knowledge of inner model theory, though.

Combinatorial consequences of strongly compact and supercompact cardinals II

Speaker: 

Christoph Weiss

Institution: 

UCI

Time: 

Monday, February 11, 2013 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

We survey several well-known direct consequences of very large cardinal axioms.  In particular we intend to cover SCH (Solovay), the failure of the approachability property (Shelah), and the failure of Not So Very Weak Square (Foreman--Magidor).  If time permits, we will discuss a characterization of strong compactness due to Ketonen or the tree property (Magidor-Shelah).

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