Consistency strength of forcing axioms

Previously I’ve only been talking about large cardinals and determinacy theories as if they were the only consistency hierarchies around. There is another important class of axioms, which has the added benefit of being, dare I say it, more useful to mathematicians not working in set theory. The reason for this is probably that these forcing axioms have a handful of consequences of a non-set theoretic nature, making them easier to apply in (mathematical) practice. When it comes to the consistency strength of these axioms though, things get a lot more hazy: we know very little about the strength of (almost all of) these axioms. I’ll introduce these axioms here and state what is known to date.

What is a forcing axiom first of all? The axioms that definitely fit this label are particular instances of the following schema.

Definition. For a class of forcing posets \mathcal C and a cardinal \lambda define the forcing axiom \textsf{FA}_\lambda(\mathcal C) as postulating, for every \mathbb P\in\mathcal C and \mathcal D\subseteq\mathbb P a {\leq\lambda}-sized collection of dense subsets of \mathbb P, the existence of a filter G\subseteq\mathbb P that meets every D\in\mathcal D. We set \textsf{FA}(\mathcal C):=\textsf{FA}_{\aleph_1}(\mathcal C).

To mention a few examples:

  • Martin’s axiom at \lambda<\mathfrak c, \textsf{MA}_\lambda, is \textsf{FA}_\lambda(\text{ccc}), and \textsf{MA} is \textsf{MA}_\lambda for all \lambda<\mathfrak c;
  • Proper forcing axiom, \textsf{PFA}, is \textsf{FA}(\text{proper});
  • Martin’s maximum, \textsf{MM}, is \textsf{FA}(\text{preserves stationary sets of }\omega_1);
  • Subcomplete forcing axiom, \textsf{SCFA}, is \textsf{FA}(\text{subcomplete}).

Here if we focus on the \aleph_1 case for Martin’s axiom the first three axioms come in increasing actual strength, meaning that \textsf{MA}_{\aleph_1} is implied by \textsf{PFA} which again follows from \textsf{MM}, and \textsf{SCFA} also follows from \textsf{MM}. One peculiar feature of \textsf{PFA} (and \textsf{MM}) is that it implies that \textsf{CH} fails. More precisely, \mathfrak c=\aleph_2 under \textsf{PFA}. On the contrary, \textsf{SCFA} is consistent with \textsf{CH}. As for a few applications to get an idea of the mathematical usefulness, we’ll mention the following.

Theorem(s).

  1. (Bella-Nyikus ’91, \textsf{MA}_{\aleph_1}) Every compact Hausdorff space of size strictly less than 2^{\aleph_1} is sequentially compact;
  2. (Shelah ’74, \textsf{MA}+\lnot\textsf{CH}) There exists a non-free Whitehead group;
  3. (Baumgartner ’73, \textsf{PFA}) Every two \aleph_1-dense sets of reals are isomorphic.
  4. (Shelah-Steprans ’88, \textsf{PFA}) Every automorphism of \mathcal P(\mathbb N)/\text{Fin} is trivial; i.e. is induced by a function f:\mathbb N\to\mathbb N.
  5. (Farah ’11, \textsf{PFA}) Every automorphism of the Calkin algebra is inner.

In (3), a set X\subseteq\mathbb R is \aleph_1-dense if (a,b)\cap X has size \aleph_1 for every pair of reals a<b. But okay, say we agree that these forcing-type axioms are indeed useful. Then how strong of a hypothesis are we really assuming? Is it just innocently consistent with \textsf{ZFC}, or is it wildly far from it? In the case of \textsf{MA} it’s quite innocent: it’s implied by \textsf{CH} and thus consistent relative to \textsf{ZFC}, and even \textsf{MA}+\lnot\textsf{CH} is consistent relative to \textsf{ZFC}.

As for \textsf{PFA}, \textsf{SCFA} and \textsf{MM}, we quickly fly through the roof in terms of (upper bounds of) consistency strength.

Theorem (Foreman-Magidor-Shelah ’88). \textsf{MM}, and thus also \textsf{PFA} and \textsf{SCFA}, are consistent relative to a supercompact cardinal.

How about the lower bound? This is a slow process, as the main (probably only) tool we got for showing lower consistency bounds is via inner model theory, so it ultimately depends on how far the inner model theory programme has come. As it’s incredibly far from a supercompact right now, we simply don’t have the tools yet to find an equiconsistency. As I mentioned in my previous post, inner models have been constructed up to \textsf{LSA}, which is in the area of a Woodin limit of Woodins. Sargsyan and Trang has recently shown the lower bound of \textsf{PFA} and \textsf{SCFA} up to this point.

Theorem (Sargsyan-Trang ’16). Assume either \textsf{PFA} or \textsf{SCFA}. Then there exists a transitive model containing the ordinals and the reals, which satisfies \textsf{LSA}.

A strategy we could also take, which is interesting and useful in its own right, is try “chopping the axiom into smaller parts” and looking at the consistency strength of these parts. One of the parts we’re particularly interested in are failures of square principles – I’ll use the following terminology in the following.

Definition (Caicedo-Larson-Sargsyan-Schindler-Steel-Zeman ’15). Let \kappa be a cardinal. Then

  • \kappa is threadable if \Box(\kappa) fails;
  • If \kappa=\rho^+ then \kappa is square inaccessible if \Box_\rho fails.

Recall that \Box_\kappa implies \Box(\kappa^+), so every threadable successor cardinal is also square inaccessible. Now the interest in square inaccessible and threadable cardinals originates from the following \textsf{PFA} theorem of Todorčević, and recently Fuchs has shown that the same result holds assuming \textsf{SCFA} as well, improving on a result of Jensen (’14) that \textsf{SCFA} implies that every successor cardinal \kappa\geq\aleph_2 is square inaccessible.

Theorem (Todorčević ’84). \textsf{PFA} implies that every cardinal \kappa\geq\aleph_2 is threadable.

Theorem (Fuchs ’16). \textsf{SCFA} implies that every cardinal \kappa\geq\aleph_2 is threadable.

This has led square-failure principles to be regarded as belonging to the hierarchy of forcing axioms. We can contemplate the consistency strength of specific failures of the square principles, yielding a wide array of new axioms. I’ll here consider the strength of square inaccessibility of successors of the following cardinals.

  • Regular;
  • Singular;
  • Singular strong limit;
  • Weakly compact;
  • Jónsson;
  • Inaccessible Jónsson;
  • Measurable;
  • Weakly compact Woodin.

At this point we have a lot of axioms to consider, and we haven’t even covered variants of the forcing axioms such as bounded variants \textsf{BPFA} and \textsf{BMM}, even stronger versions of \textsf{MM} known as \textsf{MM}^{++} and \textsf{MM}^{+++}, and more. I’ll say a bit more about the square inaccesibility, but first here’s an overview of what is currently known about the consistency strength of the various axioms (see my Diagrams tab for a pdf download).

forcing_axioms_2

First of all, don’t be fooled in thinking that e.g. we’re close to finding an equiconsistency for a square inaccesible successor of a weakly compact Woodin: I’ve cherry-picked certain large cardinals and especially the area between Woodin cardinals and a Woodin limit of Woodins is highly inflated. But okay, let’s justify some of the points in this diagram – I’m here going to focus on the square-failure principles. Firstly, Jensen and Solovay showed that the existence of a square inaccessible successor of a regular cardinal is equiconsistent with the existence of a Mahlo cardinal. As for the upper bounds of the remaining cases we got the following results.

Theorem (Jensen ’98). Successors of subcompact cardinals are square inaccessible.

Theorem (Zeman ’91). Assuming the existence of a measurable subcompact, there exists a generic extension of V in which \aleph_{\omega+1} is square inaccessible and \textsf{GCH} holds.

Jensen’s result gives a measurable subcompact as an upper bound for all the square inaccessibles of the non-singular variant and Zeman’s ensures that this same upper bound also works for singulars and singular strong limits. Note that subcompacts are both weakly compact and Woodin, so we can lower this upper bound slightly in the case of weakly compacts and weakly compact Woodins. As for the lower bounds, we got the following results.

Theorem (Mitchell-Schimmerling-Steel ’94). If there exists a square inaccessible successor of a singular cardinal then there exists an inner model with a Woodin cardinal.

Theorem (Adolf ’17). If there exists a square inaccessible successor of a singular strong limit cardinal then there exists a transitive M containing all the ordinals and reals such that M\models\textsf{ZF}+\textsf{AD}^++\Theta\text{ is regular}.

Theorem (Jensen-Schimmerling-Schindler-Steel ’09). Let \kappa\geq\aleph_3 be regular and countably closed and suppose that both \kappa^+ is square inaccessible and \kappa is threadable. Then there is a proper class model that satisfies “there is a proper class of strong cardinals” and “there is a proper class of Woodin cardinals”.

The first two theorems immediately give lower bounds for the singular cases. The last theorem is useful to us because of the following.

Theorem (Todocevic ’86). Every weakly compact cardinal is threadable.

Theorem (Rinot ’14). Every regular Jónsson cardinal is threadable.

As both weakly compacts and inaccessible Jónssons are countably closed this supplies us with lower bounds in the weakly compact, inaccessible Jónsson and measurable case. The Jónsson lower bound is then just the minimum of the singular lower bound and the inaccessible Jónsson lower bound, which is then an inner model with a Woodin cardinal. When it comes to lower bounds for square inaccessible successors of weakly compact Woodin cardinals we simply got the trivial lower bound of a weakly compact Woodin, which is strictly above a Woodin limit of Woodins. The reason why I included this one is due to the following recent result of Neeman and Steel.

Theorem (Neeman-Steel ’15). The theory \textsf{SBH}_\delta+\delta^+\text{ is a square inaccessible successor of a weakly compact Woodin} is equiconsistent with the theory \textsf{SBH}_\delta+\delta\text{ is subcompact}.

Here \textsf{SBH}_\delta is a certain iterability hypothesis called Strategic Branches Hypothesis (at \delta). So if this hypothesis turns out to be true then we really get an equiconsistency result for square inaccessible successors of weakly compact Woodins.

That was it! Phew, and that was just the square principles! I’ll leave it to the reader to find the consistency upper- and lower bounds for the remaining forcing axioms.