Statements that Could be Forced by Ultrapowers

Question

Ultrapower of a structure is a very flexible mathematical creature in comparison with the ground structure and its ordinary products. Depending on the nature of ground structure and the good properties of ultrafilter, it is often manageable to give useful properties to ultrapower.

Interestingly in some papers and theses, ultrapower of a forcing notion $\mathbb{P}$ by an ultrafilter $U$ coming from a special large cardinal $\kappa$ as the index set (e.g. $\kappa$ measurable) appeared as a useful tool for dealing with different types of consistency results (e.g. consistency of some relations between cardinal characteristics). For instance see the following references:

• Diego Alejandro Mejia, Template iterations with non-definable ccc forcing notions.

• Assaf Shani, Ultrapowers of forcing notions.

• Saharon Shelah, Two cardinal invariants of the continuum ($\mathfrak{d} < \mathfrak{a}$) and FS linearly ordered iterated forcing. (No. 700 in Shelah's Archive)

• Anda Ramona Tănasie, The splitting number and some of its neighbors.

Intuitively one can tame the bad properties of $\mathbb{P}$ in its ultrapower by throwing the bad points out of the ultrafilter and keeping the others intact. Also in ultrapower forcing usually good properties of $\mathbb{P}$ (e.g. chain condition) could be preserved under ultrapower (e.g. using a $\kappa$-complete ultrafilter over index $\kappa$) while such properties might be affected by usual forcing products.

Here I would like to ask about more examples of using ultrapower of a forcing notion as a forcing notion, for obtaining consistency results in different realms of set theory, in order to get a better idea of the type of consistency statements which could be obtained via ultrapower forcing.

Question. What are references for examples of consistency results obtained by ultrapower forcing? Any reference to lecture notes and unpublished papers are also welcome.

Answer 1

Cichoń's maximum, arXiv:1708.03691, uses 4 consecutive ultrapowers of a finite support ccc iteration.

Answer 2

Though not exactly an ultrapower forcing, but there are forcings of the form $\prod_{i<\kappa}\mathbb{P}_i/ I,$ for some ideal $I$ on $\kappa$ (equivalently we can replace $I$ with its dual filter) that are useful sometimes. For example consider the following theorem:

Theorem. Suppose that $\kappa$ is a singular strong limit, and $( \tau_n | n < ω)$ is an increasing sequence of regular cardinals with limit $κ$. Then $\prod_n Col(τ_n, κ^+)/f inite$ adds weak square at $κ$.

For the proof see ``Dima Sinapova and Spencer Unger, Combinatorics at $\aleph_{\omega}$, Annals of Pure and Applied Logic 165: 996-1007, 2014''. The above plays important role in some other papers by Sinapova and Unger.