## Dictionary

I took a look at the Abraham/Magidor article in the Handbook of Set Theory, and what they discuss in Section 5 of their article is within epsilon of what Shelah uses in Section 1 of Chapter VIII of The Book. There are minor technical differences,but their writing is so much clearer than Shelah’s that I’m tempted to prove the main results of Chapter VIII section 1 using their version of things (minimally obedient universal sequences and ${\kappa}$-presentable models) instead of Shelah’s (“suppose (a)-(e) of 1.2 hold”).

What this means in practical terms is that I’m going to be doing some translation of Section 1 of Chapter VIII into the language of the Handbook and see how well the proofs go through.

Tedious, but probably worthwhile in the interest of making the material accessible!

## 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.

## Quick update

Well, I think hitting the medium-term goals is going to be much harder more interesting than I thought! I’ll start posting pieces of what I know starting next week after our Spring Break is over…