Generalised Square Inaccesibility

Jensen’s square principle $\Box_\kappa$ has proven very useful in measuring the non-compactness of various successor cardinals as well as being an essential tool in finding new lower bounds for forcing axioms like the Proper Forcing Axiom. It should be noted however, that $\Box_\kappa$ is not really about $\kappa$, but about $\kappa^+$. To remedy this confusion, Caicedo et al ('17) came up with the term square inaccessible instead, where $\kappa^+$ is square inaccessible if $\Box_\kappa$ fails. It seems as though we can only talk about successor cardinals being square inaccessible then, but results from Krueger ('13) and Todorčević ('87) allow us generalise this to all uncountable regular cardinals. I’ll introduce this generalisation here and note that the celebrated result of Jensen ('72), stating that there aren’t any successor square inaccessible cardinals in L, does not hold for all cardinals.

Let’s start off with the original square principle.

Definition (Jensen '72). Let $\kappa$ be a cardinal. We say that $\Box_\kappa$ holds if there exists a sequence $\left< C_\alpha\mid\alpha\in\kappa^+\cap\text{Lim}\right<$ such that

1. $C_\alpha\subseteq\alpha$ is club
2. $C_\beta\cap\alpha=C_\alpha$ for every pair of limit ordinals $\alpha\leq\beta<\kappa^+$
3. $\text{ot}(C_\alpha)\leq\kappa$ for each limit ordinal $\alpha<\kappa^+$.

We say that a sequence $\vec C$ satisfying conditions (1)-(2) above is a coherent sequence of clubs. In the same paper, Jensen showed that, in L, $\Box_\kappa$ holds for every (infinite) cardinal $\kappa$. Besides being interesting in its own right, it sparked a lot of interest when Devlin-Jensen ('74) a couple of years later proved the covering lemma for L, in particular stating that if $0^\sharp$ doesn’t exist then $\kappa^+=\kappa^{+L}$ for every V-singular cardinal $\kappa$. This meant that the failure of $\Box_\kappa$ for a singular $\kappa$ implies $0^\sharp$! Since the failure of $\Box_\kappa$ for a regular cardinal $\kappa$ merely had the consistency strength of a Mahlo cardinal, this disparity was quite interesting. How far can it be pushed?

Solovay started by showing that $\Box$ holds inside $L[U]$ for $U$ a measure, Welch ('79) showed it for the core model K below a measurable, Jensen ('94) for K below $0^¶$ and Schimmerling-Zeman ('01) then showed it for every Mitchell-Steel core model. This was accompanied by a proof in Mitchell-Schimmerling-Steel ('97) of the (weak) covering lemma for K below a Woodin, pushing the lower consistency bound of $\lnot\Box_\kappa$ for a singular $\kappa$ up to a Woodin cardinal.

If we take a step back at this point we note that $\Box_\kappa$ is really a statement about $\kappa^+$, which is why the covering lemma was so crucial for the consistency strength application above. But what if we consider the analogous thing for any uncountable regular cardinal? Before this, however, let’s introduce some terminology that turns out to be useful in this context.

Definition (Caicedo et al '17). An uncountable successor cardinal $\kappa^+$ is called square inaccessible if $\Box_\kappa$ fails.

We can then rephrase the above theorems concerning core models to saying that there aren’t any square inaccessible successor cardinals in any Mitchell-Steel core model. Is this still true for limit cardinals as well? What does that even mean? Krueger ('13) managed to come up with the correct definition of square inaccessibility for an arbitrary uncountable regular cardinal (his definition is a bit different than the following, as he defines the analogue of being fully square inaccessible, i.e. the failure of the weak square $\Box_\kappa^*$ instead of the failure of $\Box_\kappa$).

Definition (Krueger '13). Let $\kappa$ be an uncountable regular cardinal. Then $\kappa$ is square inaccessible if whenever $C\subseteq\kappa$ is club and $\left< c_\alpha\mid\alpha\in C\right<$ is a coherent sequence of clubs then there exists $\alpha\in C$ such that $\text{ot}(c_\alpha)=\alpha$.

He then, in the same paper, shows that the two definitions agree whenever $\kappa$ is a successor cardinal, so this is really a natural strengthening of the concept. Here’s the catch however: every Mahlo cardinal is square inaccessible. This is because we can’t have a club of singular cardinals in a Mahlo cardinal, so there has to be many $\alpha\in C$ for which $\text{ot}(c_\alpha)=\alpha$. This puts a damper on the situation, as Mahlo cardinals are downwards absolute to L – we thus get the following corollary.

Corollary. If there exists a Mahlo cardinal then there are square inaccessible cardinals in L.

In fact, he proves, together with a result of Todorčević ('87), that an inaccesible cardinal $\kappa$ is fully square inaccessible (by what I mean his generalised notion of the failure of $\Box_\kappa^*$) if and only if $\kappa$ is Mahlo. This is a great analogy to an inaccessible cardinal $\kappa$ being fully threadable if and only if $\kappa$ is weakly compact, also due to Todorčević ('87). In other words, from a consistency point of view, it’s not really that the generalised concept of square inaccesibility is less interesting, but more that the focus is really about the existence of square inaccessible successor cardinals, or even successor cardinals failing weaker square principles, like the successor (fully) threadable cardinals.