We present some basic applications of the Proper Forcing Axiom: Square inaccessibility of all cardinals above \omega_1, and the tree property at \omega_2.
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.
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).
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).
We prove a theorem of Woodin that, assuming $\mathsf{ZF} + \mathsf{AD}+ \theta_0 < \Theta$, every $\Pi^2_1$ set of reals has a semi-scale whose norms are ordinal-definable. The consequence of $\mathsf{AD}+\theta_0 < \Theta$ that we use is the existence of a countably complete fine measure on a certain set, which itself is a set of measures. If time permits, we outline how "semi-scale" can be improved to "scale" in the theorem using a technique of Jackson.
We define a hierarchy of normal fine measures \mu_\alpha on some set
X_\alpha and discuss the consistency strength of the theory (T_\alpha) =``AD^+ + there
is a normal fine measure \mu_\alpha on X_\alpha." These measures arise naturally from
AD_R, which implies the determinacy of real games of fixed countable length. We
discuss the construction of measures \mu_\alpha on X_\alpha from AD_R (in this
context, \mu_0 is known as the Solovay measure). The theory (T_\alpha) is strictly
weaker than AD_R in terms of consistency strength. However, we show that (T_\alpha) is
equivalent to the determinacy of a certain class of long games with
\utilde{\Pi^1_1}-payoff (and <\omega^2-\utilde{\Pi^1_1}-payoff).