# Some Cardinal Invariants from SH410

In Section 6 of Sh410, Shelah defines several related cardinal invariants, one of which makes an appearance in a proof of the revised GCH (the one in the Abraham-Magidor chapter of the handbook). I want to use this space to clear up some of the definitions.

Def: Let $I$ be an ideal on some set $X$. Then $I$ is $<\sigma$-based if, whenever $A\subseteq X$ is such that $A\in I^+$, we also have $[A]^{<\sigma}\cap J^+\neq\emptyset$.

Our working assumptions for this post here are the following:

• $J$ is an ideal on a cardinal $\kappa$.
• $\kappa$ is not the union of countably-many sets from $J$.
• $J$ is $<\sigma$-based, where $\sigma=cf(\sigma)>\aleph_0$.
• $\mu$ is a cardinal with $\kappa^{<\sigma}<\mu$.

Def: Say a cardinal $\lambda$ is representable if there is a collection $\{E(i) : i<\kappa\}$ of finite subsets of $(\kappa^{<\sigma},\mu]\cap\mathrm{Reg}$ such that, for any $A\in J^+$, we have $\lambda=\max pcf(\bigcup_{i\in A}E(i))$.

Def: Say a cardinal $\lambda$ is weakly representable if there is a collection $\{E(i) : i<\kappa\}$ of finite subsets of $(\kappa^{<\sigma},\mu]\cap\mathrm{Reg}$ such that, for any $A\in J^+$, we have $\lambda\leq\max pcf(\bigcup_{i\in A}E(i))$.

Def: $T^2_J(\mu)=\sup\{\lambda : \lambda\text{ is weakly representable}\}$

Def: $T^3_J(\mu)=\sup\{\lambda : \lambda\text{ is representable}\}$

It’s clear here that $T^2_J(\mu)\geq T^3_J(\mu)$ simply because the supremum is being taken over more cardinals. It turns out that these cardinals are actually equal to each other, but now that I look at the proof, I can’t really make heads or tails of it. For now, I’ll go ahead and assume that the proof is correct and see if I can piece together what’s going on later.

Anyway, we have another cardinal invariant which appears, the definition of which I want to take some time to consider. Basically, the definition is probably incorrect, given the proof of theorem 6.1, so I want to repair it in a way that makes sense.

Def: (incorrect version) $T^4_J(\mu)=\min\{\sup\{T^2_{J+\kappa\setminus A_n} : n<\omega\} :\kappa=\bigcup A_n\wedge A_n\subseteq A_{n+1}\wedge A_n\in J^+\}$

This definition looks pretty horrible, but let’s take a look at what’s going on. First, let’s replace $T^2_{J+\kappa\setminus A_n}$ with its $T^3$ counterpart to make things easier. Now, recall here that

$J+(\kappa\setminus A_n)=\{B\subseteq\kappa : B\cap A_n \in J\}$

Then, note that if $B\cap A_{n+1}\in J$ then certainly $B\cap A_n\in J$ and so we have that the sequence of ideals is decreasing. Now, let’s suppose that $\lambda$ is representable by $J+(\kappa\setminus A_{n+1})$, and fix a representation $\{E(i) : i<\kappa\}$ of $\lambda$. Then for any $A$ which is positive with respect to $J+(\kappa\setminus A_{n+1})$, we have that $\lambda=\max pcf(\bigcup_{i\in A}E(i))$. Now, since $J+(\kappa\setminus A_n)$ is larger (hance has fewer positive sets), it follows that $\lambda$ must be representable by $J+(\kappa\setminus A_{n})$. In other words:

$T^3_{J+(\kappa\setminus A_n)}\geq T^3_{J+(\kappa\setminus A_{n+1})}$.

Hence, that sup is achieved by $T^3_{J+(\kappa\setminus A_0)}$. So, as written that definition seems somewhat strange. On the other hand, what would make sense is asking for a min instead of a sup, since we get a decreasing sequence of ordinals. Finally, my advisor, Todd Eisworth, pointed out that the proof of Theorem 6.1 doesn’t actually go through as written and requires that first min actually be a sup. Now, looking at the proof, it remains intact if we actually take the definition of $T^4_J(\mu)$ to be:

Def: (correct version) $T^4_J(\mu)=\sup\{\min\{T^2_{J+\kappa\setminus A_n} : n<\omega\} :\kappa=\bigcup A_n\wedge A_n\subseteq A_{n+1}\wedge A_n\in J^+\}$.

This definition actually makes much more sense, and it makes the proof of theorem 6.1 go through. So, my suspicion is that the above is the correct definition of $T^4_J(\mu)$.