Fri Mar 15, 2013
60 Evans Hall, 4:10–6 PM
Farmer Schlutzenberg (UC Berkeley)
Jónsson Cardinals in L(R)
The standard axioms of set theory leave many natural questions undecided. Determinacy axioms and large cardinal axioms give a much more complete picture, one that is natural and compelling. Strong connections are known to exist between these two families of axioms.
The Axiom of Determinacy (AD) states that in any infinite two-player game of perfect information on the natural numbers one player has a winning strategy. AD implies large cardinal axioms. It also contradicts the Axiom of Choice, and even implies all sets of reals are Lebesgue measurable. Therefore AD fails in the full universe of sets, where Choice holds. However, there are important subuniverses where AD may hold, among them the universe L(R) of all sets constructible from the reals. Assuming AD holds in L(R), a detailed analysis of L(R) is possible, extending classical results of descriptive set theory on analytic and Borel sets.
A cardinal is Jónsson iff every structure of that size has a proper elementary substructure of that size. They were introduced by Bjarni Jónsson in 1972. We will survey the above background and sketch a proof that, given AD, there are many Jónsson and Rowbottom cardinals in L(R). Both results use the same key argument. (The Jónsson cardinal result was first announced by Woodin.)