## Some Assumptions

I have been contemplating exactly where to begin the discussion of these matters, and I think that I want to go back to Chapter VIII of The Book, and look at the first section of it because the arguments there keep appearing in later works.

In this post, I’m going to write down some assumptions [(a)-(e) of Claim 1.2] that will be used in the next few posts.

The Assumptions

We assume the following

1. ${\mathfrak{a}}$ is a set of regular cardinals satisfying ${|\mathfrak{a}|^+<\min(\mathfrak{a})}$

2. For every ${\mathfrak{b}\subseteq\mathfrak{a}}$, we let ${\overline{f}^{\mathfrak{b}}=\langle f^{\mathfrak{b}}_\alpha:\alpha<\max{\rm pcf}(\mathfrak{b})\rangle}$ satisfy
• ${f^{\mathfrak{b}}_\alpha\in\prod\mathfrak{a}}$,
• ${\bar{f}^{\mathfrak{b}}}$ is strictly increasing modulo ${J_{<\max{\rm pcf}(\mathfrak{b})}[\mathfrak{a}]}$,
• if ${\alpha<\max{\rm pcf}(\mathfrak{b})}$ and ${|\mathfrak{a}|<{\rm cf}(\alpha)<\min(\mathfrak{a})}$, then for each ${\theta\in \mathfrak{b}}$ we have

$\displaystyle f^{\mathfrak{b}}_\alpha(\theta)=\min\{\bigcup_{\beta\in C}f^{\mathfrak{b}}_\beta(\theta): C\text{ club in }\alpha\}. \ \ \ \ \ (1)$

• for every ${f\in\prod\mathfrak{b}}$ and ${\alpha<\max{\rm pcf}(\mathfrak{b})}$, there is a ${\beta>\alpha}$ such that ${f (everwhere)

3. ${\chi}$ is a sufficiently large regular cardinal, ${<_\chi}$ is a well-ordering of ${H(\chi)}$

4. ${\bar{N}=\langle N_i:i\leq\delta\rangle}$ is an increasing continuous sequence of elementary submodels of ${\langle H(\chi), \in, <_\chi\rangle}$ such that
• ${|N_i|<\min(\mathfrak{a})}$
• ${\langle N_j:j\leq i \rangle \in N_{i+1}}$
• ${|N_i|+1\subseteq N_i}$
• ${\bar{f}=\{\bar{f}^{\mathfrak{b}}:\mathfrak{b}\subseteq\mathfrak{a}\}\in N_0}$

5. ${\mathfrak{a}\subseteq N_0}$ and ${|\mathfrak{a}|<{\rm cf}(\delta)\leq\delta<\min(\mathfrak{a})}$.

I’m going to do some “dictionary work” to get the official names for such objects. I just want to check which terms have become standard, and I’ll use the Abraham/Magidor Handbook article as the final word.