In this series of two talks I will give an introduction to some of my recent research on the ineffable tree property. The ineffable tree property is a two cardinal combinatorial principle which can consistently hold at small cardinals. My recent work has been on generalizing results about the classical tree property to the setting of the ineffable tree property. The main theorem that I will work towards in these talks generalizes a theorem of Cummings and Foreman. From omega supercompact cardinals, Cummings and Foreman constructed a model where the tree property holds at all of the $\aleph_n$ with $1 < n < \omega$. I recently proved that in their model the $(\aleph_n,\lambda)$ ineffable tree property holds for all $n$ with $1 < n < \omega$ and $\lambda \geq \aleph_n$.

Basic facts on towers and homogeneity systems will be presented.

Basic theory of ultrapowers for towers of measures will be presented.

Given a lightface $\Sigma^1_2$ set of reals A we present the construction of a tree on $\omega\times\omega_1$ such that A is the projection of T. Moreover, the tree T is an element of any transitive model of ZF-PowerSetAxiom that has $\omega_1$ as element.

We will introduce the "lightface" projective hierarchy and examine it both from syntactical and semantical aspect. "Lightface" \Sigma^0_1" sets are effective versions of open sets. We also prove that lightface \Sigma^0_1 sets of reals can be represented as sets of branches of recursive trees, and lithtface \Sigma^1_1 sets can be represented as projections of recursive trees.