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 and a cardinal define the forcing axiom as postulating, for every and a -sized collection of dense subsets of , the existence of a filter that meets every . We set .
To mention a few examples:
- Martin’s axiom at , , is , and is for all ;
- Proper forcing axiom, , is ;
- Martin’s maximum, , is ;
- Subcomplete forcing axiom, , is .
Here if we focus on the case for Martin’s axiom the first three axioms come in increasing actual strength, meaning that is implied by which again follows from , and also follows from . One peculiar feature of (and ) is that it implies that fails. More precisely, under . On the contrary, is consistent with . As for a few applications to get an idea of the mathematical usefulness, we’ll mention the following.
- (Bella-Nyikus ’91, ) Every compact Hausdorff space of size strictly less than is sequentially compact;
- (Shelah ’74, ) There exists a non-free Whitehead group;
- (Baumgartner ’73, ) Every two -dense sets of reals are isomorphic.
- (Shelah-Steprans ’88, ) Every automorphism of is trivial; i.e. is induced by a function .
- (Farah ’11, ) Every automorphism of the Calkin algebra is inner.
In (3), a set is -dense if has size for every pair of reals . 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 , or is it wildly far from it? In the case of it’s quite innocent: it’s implied by and thus consistent relative to , and even is consistent relative to .
As for , and , we quickly fly through the roof in terms of (upper bounds of) consistency strength.
Theorem (Foreman-Magidor-Shelah ’88). , and thus also and , 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 , which is in the area of a Woodin limit of Woodins. Sargsyan and Trang has recently shown the lower bound of and up to this point.
Theorem (Sargsyan-Trang ’16). Assume either or . Then there exists a transitive model containing the ordinals and the reals, which satisfies .
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 be a cardinal. Then
- is threadable if fails;
- If then is square inaccessible if fails.
Recall that implies , so every threadable successor cardinal is also square inaccessible. Now the interest in square inaccessible and threadable cardinals originates from the following theorem of Todorčević, and recently Fuchs has shown that the same result holds assuming as well, improving on a result of Jensen (’14) that implies that every successor cardinal is square inaccessible.
Theorem (Todorčević ’84). implies that every cardinal is threadable.
Theorem (Fuchs ’16). implies that every cardinal 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.
- Singular strong limit;
- Weakly compact;
- Inaccessible Jónsson;
- 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 and , even stronger versions of known as and , 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).
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 is square inaccessible and 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 .
Theorem (Jensen-Schimmerling-Schindler-Steel ’09). Let be regular and countably closed and suppose that both is square inaccessible and 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 is equiconsistent with the theory .
Here is a certain iterability hypothesis called Strategic Branches Hypothesis (at ). 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.