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).

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