Generalised square inaccessibility

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 \langle C_\alpha\mid\alpha\in\kappa^+\cap\text{Lim}\rangle 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^\P 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 \langle c_\alpha\mid\alpha\in C\rangle 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.