$T^4_J(\mu)$ and some motivation

\( \newcommand{\wsat}{\mathrm{wsat}} \newcommand{\GH}{\mathrm{GH}} \newcommand{\REG}{\mathrm{REG}} \newcommand{\Ch}{\mathrm{Ch}} \newcommand{\reg}{\mathrm{reg}} \newcommand{\rk}{\mathrm{rk}} \newcommand{\ot}{\mathrm{ot}} \newcommand{\pcf}{\mathrm{pcf}} \newcommand{\tcf}{\mathrm{tcf}} \newcommand{\cf}{\mathrm{cf}} \newcommand{\ran}{\mathrm{ran}} \newcommand{\dom}{\mathrm{dom}} \newcommand{\nacc}{\mathrm{nacc}} \newcommand{\acc}{\mathrm{acc}} \newcommand{\Tr}{\mathrm{Tr}} \)
Last time we defined the cardinals: $$ T^4_J(\mu)=\min\{\sup T^2_{J+(\kappa\setminus A_n)}: A_n\subseteq A_{n+1}\subseteq\kappa=\bigcup_{n<\omega}A_n, A_n\notin J\}. $$ $$ T^5_J(\mu)=\sup\{\sup T^2_{J+(\kappa\setminus A_n)}: A_n\subseteq A_{n+1}\subseteq\kappa=\bigcup_{n<\omega}A_n, A_n\notin J\}. $$ The cardinal $T^4_J(\mu)$ appears in section 6 of Sh410, and there Shelah claims that $T_J(\mu)\leq T^4_J(\mu)$. Unfortunately, it seems that the proof only works if we replace the $\min$ with a $\sup$, and so we're also defining $T^5_J(\mu)$. In this post I wanted to briefly talk about why we want to relate $T_J(\mu)$ to pcf-theoretic cardinals, and then discuss the (potentially wrong) definitions of the above defined cardinals.

I first want to recall that, assuming $J$ is a $\sigma$-based ideal on a cardinal $\kappa$ such that $\kappa^{<\sigma}<\mu$, we know that $T_J(\mu)$ is actually the cardinality of any maximal $J$-adt $\mathcal{F}$, which is a natural thing to be curious about. On the other hand, we have quite a few tools to deal with pcf-theoretic objects, so situating $|\mathcal{F}|$ among pcf-theoretic objects is useful insofar as this potentially allows us to put bounds on $|\mathcal{F}|$. Beyond that, let's suppose that $\kappa\leq \theta<\sigma$, so $J=[\kappa]^{<\theta}$ is a $\sigma$-based ideal on $\kappa$. Let $$ AD_J(\mu)=\sup\{|\mathcal{F}|: (\forall A\neq B\in\mathcal{F})(A,B\in[\mu]^{\kappa} \wedge A\neq_J B )\}. $$ In other words, $AD_J(\mu)$ is the supremum of the cardinality of $\theta$-almost disjoint families of subsets of $\mu$ with size $\kappa$. Note that given any such family $\mathcal{F}$, if we let $f_A: \kappa\to \mu$ be a faithful enumeration of $A$ for each $A\in \mathcal{F}$, then $\{f_A : A\in \mathcal{F}\}$ is a $J$-adt. On the other hand, if $\mathcal{F}$ is a $J$-adt, let $A_f=\{(i,f(i)) : i<\kappa\}$ be the graph of $f$ for each $f\in\mathcal{F}$. Then the family $\{A_f : f\in \mathcal{F}\}$ is a $\theta$-almost disjoint family. Thus, $T_J(\mu)=AD_J(\mu)$ for the above $J$. Here we see that relating $AD_J(\mu)$ to pcf-theoretic objects continues the trend in much of Shelah's work of connecting cardinals related to collections of sets (e.g. covering numbers) to cardinals related to functions (e.g. pseudo powers).

Now, back to those $T^4$ and $T^5$ numbers. One thing I want to note is that neither of the proofs given in section 6 of 410 work (as written) with the given definition of $T^4_J(\mu)$. Another thing I want to talk about is that given ideals $I\subseteq J$ on $\kappa$, we have that $T^2_I(\mu)\leq T^2_J(\mu)$. To see this, suppose that $\lambda$ is weakly $I$-representable as witnessed by the block sequence $\langle E(i) : i<\kappa\rangle$, so $\lambda\leq\max\pcf(\bigcup_{i\in A} E(i))$ for every $A\in I^+$. Since any $J$-positive set is $I$-positive as well, it follows that $\lambda\leq\max\pcf(\bigcup_{i\in A} E(i))$ for any $A\in J^+$ as well and thus that $\lambda$ is weakly $J$-representable. The inequality follows.

With this in mind, now note that if $A_0\subseteq A_1$, then the ideal $J+(\kappa\setminus A_0)$ contains the set $\kappa\setminus A_1$ since $\kappa\setminus A_1\subseteq\kappa\setminus A_0$. Thus, $J+(\kappa\setminus A_1)\subseteq J+(\kappa\setminus A_0)$ and so $T_{J+\kappa\setminus A_1}(\mu)\leq T_{J+\kappa\setminus A_0}$. But this means that given a sequence of sets $A_n\subseteq A_{n+1}\subseteq\kappa=\bigcup_{n<\omega}A_n$ with each $A_n\notin J$, the cardinal $$ \sup \{T^2_{J+(\kappa\setminus A_n)}:n<\omega\} $$ is just $T^2_{J+\kappa\setminus A_0}$. This means that $$ T^4_J(\mu)=\min\{\sup T^2_{J+(\kappa\setminus A_n)}: A_n\subseteq A_{n+1}\subseteq\kappa=\bigcup_{n<\omega}A_n, A_n\notin J\}=\min\{T^2_{J+\kappa\setminus A}: A\notin J\}. $$ $$ T^5_J(\mu)=\sup\{\sup T^2_{J+(\kappa\setminus A_n)}: A_n\subseteq A_{n+1}\subseteq\kappa=\bigcup_{n<\omega}A_n, A_n\notin J\}=\sup\{T^2_{J+\kappa\setminus A}: A\notin J\}. $$ I'm not entirely sure if this is a mistake on my part in not reading the definition correctly or getting an inequality reversed, or an oversight in Sh410. Either way, I don't know what the correct definition of $T^4_J(\mu)$ should actually be in order to make all of the claims in section 6 go through. With that said, my goal is to now repair the proof of the inequality $T_J(\mu)\leq T^4_J(\mu)$ in that section and use that to figure out what teh right version of $T^4_J(\mu)$ should be given the proof.