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 continue with basic information on proper forcing.

Basic facts about proper forcing will be presented.

The classical result of Silver -- construction of the model where the Singular Cardinal Hypothesis fails -- will be presented. The emphasis is on presenting Easton suport iteration and extension of elementary embedding to a generic extension of the universe, which is the key ingredient of the entire construction.

In the 80s Gitik proved the following theorem: For every real $x$ and every club $D \subseteq [\omega_2]^\omega$, there are $a,b,c \in D$ such that $x \in L(a,b,c)$. An immediate corollary of Gitik's theorem is: if $W$ is a transitive $ZF^-$ model of height at least $\omega_2$ such that $W$ is missing some real, then the complement of $W$ is stationary in $[\omega_2]^\omega$ (Velickovic strengthened Gitik's Theorem to show that the complement of such a $W$ is in fact projective stationary, not just stationary). I will present Gitik's proof and, if time permits, discuss some recent applications due to Viale and me.