Spencinar #2: Martin's Maximum and Weak Square

Today John Susice gave an excellent presentation about a paper of Cummings and Magidor entitled "Martin's Maximum and Weak Square". Since the content of the talk was pretty close to the paper (which is well-written and available online), I won't blog about the precise details of the proof. Instead, I'll try to summarize what I see are the main points, and offer some soft speculation on future possibilities. I realize that what I write below might be unreadable, but I hope it conveys my general understanding of things, which might be helpful in a different way from careful writeups of proofs.

First, let me explain the two title principles. Martin's Maximum (MM) is a very strong forcing axiom, i.e., a principle which, for posets belonging to certain classes, asserts the existence of filter of each poset which meets every family of dense sets up to a certain size. We informally call this filter, meeting the appropriate choice of dense sets, a pseudogeneric. Martin's Axiom is perhaps the most well-known example of a forcing axiom; MM is the analogue of this for the class of posets whose generic extensions preserve the stationarity of ground model subsets of $\omega_1$ (the class of posets is usually simply called stationary preserving), and we must meet any $\omega_1$ many dense sets at a time. John made an amusing remark and said the name "non-forcing axiom" might be more appropriate, since the pseudogeneric objects they provide exist in the ground model, without any actual forcing taking place.

Weak square is a combinatorial principle introduced by Jensen. Generally, square principles assert the existence of sequences $\langle C_\alpha: \alpha<\lambda^+\rangle$ so that each $C_\alpha$ is club in $\alpha$, and there is some coherence between the $C_\alpha$. The "weak" part means that we will also allow multiple clubs at each level. I won't get into it too much, because we will really be working with a consequence of weak square: the existence of good scales.

Let $\lambda$ be a singular cardinal, and $\langle \lambda_i : i<\mathrm{cf}(\lambda)\rangle$ a cofinal sequence of regular cardinals less than $\lambda$. In this case, a scale is a sequence of functions $\langle g_\alpha: \alpha<\lambda^+\rangle$ in $\prod_{i<\mathrm{cf}(\lambda)} \lambda_i$ which is increasing and unbounded in the eventual domination ordering. Shelah proved that these scales exist for any singular cardinal (for some careful choice of $\langle \lambda_i : i<\mathrm{cf}(\lambda)\rangle$).

A good point of the scale is just an $\alpha_0<\lambda^+$ with cofinality $>\mathrm{cf}(\lambda)$ for which the ordering on $\langle g_\alpha: \alpha<\alpha_0\rangle$ is particularly nice: there is an unbounded set $A$ in $\alpha$ and some $i_0$ so that $\langle g_\alpha: \alpha\in A\rangle$ is pointwise increasing at each coordinate above $i_0$. A good scale is just a scale with club many good points. If you have some experience with the definitions, it's not too hard to show that a weak square sequence at $\lambda$ implies that every scale at $\lambda$ is good.

John went over the proof of the following theorem:

Theorem: If Martin's Maximum holds, then there are no good scales on $\lambda$ for any singular $\lambda$ with $\mathrm{cf}(\lambda)=\omega$. In particular, weak square fails at $\lambda$.

The proof, of course, went by defining a poset $\mathbb{P}$, proving that it is stationary preserving, and then applying MM. Suppose there is a good scale on $\langle \lambda_n:n<\omega\rangle$. The poset was a version of Namba forcing: its conditions are trees $T$ whose nodes are sequences of ordinals of countable cofinality with $n$th coordinate less than $\lambda_n$, and which (after some finite stem) split into stationary sets, i.e., for every $t\in T$ above the stem, there are stationary (in $\lambda_n$ for $n$ equal to the length of $t$) many $i$ so that $i$ appended to $t$ is in $T$.

The heart of the argument is the stationary preserving property, which was proven by appealing to the Gale-Stewart theorem for closed games. It was a very clever argument that involved showing that for any name for a club $\dot{C}\subset \lambda^+$, any ground model stationary set $S$, and any condition $T$ in the poset, you could find $\delta\in S$ so that:

$T$ can be thinned to a stationarily splitting subtree $T'$ so that the nodes at the $i$th level of $T'$ forces points in $(\delta_i, \delta)$ into $C$ below $\delta$, where $\delta_i$ is a fixed cofinal sequence in $\delta$.

This ensures that in the generic extension obtained by forcing with $T'$, $\delta\in S\cap C$. This is shown with a "Namba combinatorics" argument. A family of two-player games of length $\omega$ were defined, with a parameter for $\delta<\lambda^+$. In the game with parameter $\delta$, player II will build a branch through $T'$ after $\omega$ moves. Player I's $i$th move consists of blocking a nonstationary set of points below $\lambda_i$, and player II responds by naming the next coordinate of his branch, with the condition that he must play outside of the set that player I just blocked. Player II winning if at each stage $i$, taking $T'$ with his branch as the stem forces that a certain member of $C$ into the interval $(\delta_i,\delta)$. It was shown that player II wins this game for almost every $\delta$.

Now, in the generic extension by $\mathbb{Q}=\mathbb{P}\ast \mathrm{Col}(\omega_1, \lambda^+)$ (which is a stationary preserving poset), it is not hard to see by a density argument that the generic element $h$ of $\prod \lambda_i$ added by $\mathbb{Q}$ is an upper bound of every ground model member of $\prod \lambda_i$, under the eventual domination ordering. In fact it is an exact upper bound in the sense that any element of $\prod \lambda_i$ which is below $h$ is eventually dominated by a ground model function. Furthermore, every coordinate of $h$ has countable cofinality.

Back to the ground model. For any club $C\subset \lambda^+$, we can use MM to find a pseudogeneric $h$ which (VERY roughly) has some of the properties of the actually generic $h$ of the previous paragraph, but just working with the scale functions indexed below some cofinality $\omega_1$ ordinal $\gamma\in C$. Through some arguments which I omit, you can show that $\gamma$ is not good. So the set of nongood points is stationary!

Some vague parting thoughts: Are there any other constructions of such nongood points $\gamma$ for a scale, without MM? (Part of this question involves understanding the essential properties of the point $\gamma$, which I did not talk about). Are they similar to the nongood points you get from a supercompact cardinal? (I think the answer is no.) More generally, can we classify nongood points somehow? On the technical side, is there a general form of the Namba lemma which we can directly apply to see that $\mathbb{P}$ is stationary preserving?