# Statements that Could be Forced by Ultrapowers

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:

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 Any reference to lecture notes and unpublished papers are also welcome.

• Unfortunately Assaf Shani's thesis is not available for those who are not affiliated to Hebrew University of Jerusalem. (I emailed him a few days ago but received no reply yet). I will be thankful if somebody who has a copy or an access to the full text send me the pdf file via email address available in my profile. Thanks in advance. – Morteza Azad Jan 20 '16 at 18:52
• I disagree with the votes to close - I don't think it's too broad for a useful answer. – Noah Schweber Jan 21 '16 at 15:31
• [As the continuation of my above comment above] A friend who has an access to Assaf Shani's thesis told me that he will send a copy to me as soon as possible. Thus please ignore my previous comment. Thanks for your attention and help. – Morteza Azad Jan 22 '16 at 8:39
• Assaf's masters thesis is now available at math.ucla.edu/~assafshani/Thesis.pdf. – Noah Schweber Jul 31 '16 at 22:42
• @NoahSchweber Thank you for informing me, Noah! – Morteza Azad Oct 23 '16 at 22:33

## 2 Answers

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

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.

• I recommend to see "Combinatorics at $\aleph_{\omega}$" too. It's very interesting paper. – Mostafa Mirabi Jan 21 '16 at 5:29
• @MostafaMirabi Hi Mostafa! By the way is "Combinatorics at $\aleph_{\omega}$" of some model theoretic interest? – Morteza Azad Jan 21 '16 at 6:23
• Weak square is equivalent to the existence of a special Aronszajn tree. The existence of a special $\kappa^+$-Aronszajn tree is equivalent to the existence of a $(\kappa^+, \kappa)$-model for a suitable first order sentence. So they relate to gap-1 cardinal transfer results in model theory. – Mohammad Golshani Jan 21 '16 at 6:34
• @MohammadGolshani (+1) Thanks for reminding this point! Part of such results are covered in Mitchell's paper: "Aronszajn trees and the independence of the transfer property". – Morteza Azad Jan 21 '16 at 6:41
• @MortezaAzad Hi Morteza, No, but this paper has interesting results related to your question. For this reason I recommended that. – Mostafa Mirabi Jan 21 '16 at 7:23