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 present some general properties of proper forcing and give examples of posets with finite conditions that are proper.

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).

Laver functions for supercompact cardinals appear in many forcing constructions, including all known constructions of models of strong forcing axioms. Viale proved that the Proper Forcing Axiom implies the existence of a "generic" Laver function from $\omega_2 \to H_{\omega_2}$. I will discuss his result and some recent work of mine on generic Laver functions.

We finish the presentation of Laver preparation from the last week.