**June 19**

**Boban Velickovic**,
University of Paris 7

**Strong guessing models**

The notion of a guessing model introduced by Viale and Weiss. The principle ${\rm GM}(\omega_2,\omega_1)$ asserts that there are stationary many guessing models of size $\aleph_1$ in $H_\theta$, for all large enough regular $\theta$. It follows from ${\rm PFA}$ and implies many of its structural consequences, however it does not settle the value of the continuum. In search of higher of forcing axioms it is therefore natural to look for extensions and higher versions of this principle. We formulate and prove the consistency of one such statement that we call ${\rm SGM}^+(\omega_3,\omega_1)$.

It has a number of important structural consequences:

- the tree property at $\aleph_2$ and $\aleph_3$

- the failure of various weak square principles

- the Singular Cardinal Hypothesis

- Mitchell’s Principle: the approachability ideal agrees with the non stationary ideal on the set of $\text{cof}(\omega_1)$ ordinals in $\omega_2$

- Souslin’s Hypothesis

- The negation of the weak Kurepa Hypothesis

- Abraham’s Principles: every forcing which adds a subset of $\omega_2$ either adds a real or collapses some cardinals, etc.

The results are joint with my PhD students Rahman Mohammadpour.