Which $L$-like principles are known to be relatively consistent with large cardinals?

Question

For which of the standard large cardinal axioms $\varphi_{LC}$ and which $L$-like principles $\psi$ (e.g. GCH, $\mathrm{V}=\mathrm{HOD},$ the ground axiom, and various diamond and square principles) is it known that $\mathrm{ZFC} + \varphi_{LC} + \psi$ is equiconsistent with $\mathrm{ZFC} + \varphi_{LC}?$

A quick motivation: if one wishes to use forcing to prove the consistency of some theory $T$ relative to a large cardinal axiom below $0^{\sharp},$ say an inaccessible, one might as well start from a model of that large cardinal plus $V=L$ for a clearer picture of our ground universe. For large cardinals above $0^{\sharp}$ but still within the grasp of current inner model theory (roughly, up to a Woodin limit of Woodins), we can replace $V=L$ with a more general inner model axiom and still get most of its nice properties. For higher larger cardinals, we can't simply declare the universe is a canonical inner model, but I have heard that many of the nice features of inner models are known to be replicable via class forcing, so I'm interested to learn the state of knowledge regarding such equiconsistencies.

Answer 1

Indeed, many of the structural features that hold in the constructible universe $L$ are obtainable by forcing in a way that accommodates large cardinals.

GCH. The standard forcing of the GCH is an Easton-support iteration that simply iteratively forces the next instance directly, and this forcing has very nice structural features that will enable to preserve most all of the usual large cardinals by the usual lifting arguments. The usual master condition arguments, the same arguments used in the Laver preparation, show that this forcing will preserve all supercompact cardinals, and one can use the same methods to preserve larger large cardinals. For example, every measurable cardinal, every strong cardinal, every supercompact cardinal, almost every partially supercompact cardinal, every rank-to-rank cardinal, and so forth, is preserved by the canonical forcing of the GCH. So the GCH is relatively consistent with most all of the usually considered large cardinals.

The basic methods are a central method of my paper on the lottery preparation:

• Hamkins, Joel David, The lottery preparation, Ann. Pure Appl. Logic 101, No. 2-3, 103-146 (2000). ZBL0949.03045.

One generally gets equiconsistency for each large cardinal notion separately, since the lifting arguments generally don't need more than the LC notion in the ground model in order to realize the same LC notion in the forcing extension. (There are a few exceptions to this, such as partial strongness at up to a singular limit cardinal and other cases like that.)

V=HOD. The situation is exactly similar with V=HOD. One of the standard ways to force V=HOD is to force the continuum coding axiom CCA, which is the assertion that every set of ordinals is coded into the GCH pattern on an interval of regular cardinals.

If we simply force with the Ord-length Easton-support iteration that performs the lottery sum of GCH forcing or not-GCH forcing at each regular cardinal, that is, the forcing that chooses generically at each stage whether to have the GCH for that cardinal or not, then we will get the CCA and hence V=HOD in the forcing extension.

This forcing also exhibits the same closure conditions that enable all the usual master condition arguments. Every supercompact cardinal is preserved by the forcing, as well as much stronger notions, including rank-to-rank. So this again provides the desired models of large cardinals with V=HOD.

Note also that V=HOD is generally a consequence of indestructibility, since if a large cardinal $\kappa$ is indestructible by ${<}\kappa$-directed closed forcing, then since every set of ordinals below $\kappa$ can be coded by such forcing into the GCH pattern above $\kappa$, this must already be true below $\kappa$ by reflection. And so $V_\kappa$ will have already satisfied the continuum coding axiom. In particular, a proper class of indestructible supercompact cardinals (or much less) already implies V=HOD outright.

For a reference, you can see my paper with Arthur Apter

• Apter, Arthur W.; Hamkins, Joel David, Universal indestructibility, Kobe J. Math. 16, No. 2, 119-130 (1999). ZBL0953.03060. arxiv:9808004

This notion of universal indestructibility is much stronger than necessary, since we are making every supercompact, every partially supercompact, every strong, every measurable, and so forth indestructible. But the lifting argument is there and this method is completely standard and flexible.

See also the forcing of V=HOD with rank-to-rank cardinals in my paper on the Wholeness axiom:

• Hamkins, Joel David, The wholeness axioms and V=HOD, Arch. Math. Logic 40, No. 1, 1-8 (2001). ZBL0969.03063 arxiv:9902079.

Ground Axiom. The ground axiom, introduced by myself and Jonas Reitz, is similarly consistent with large cardinals. This follows by one of the main dissertation results of Jonas Reitz, who was my student at the time.

• Reitz, Jonas, The Ground Axiom, J. Symb. Log. 72, No. 4, 1299-1317 (2007). ZBL1135.03018.

Extending this work, we proved together with Woodin that that the ground axiom is also consistent with V$\neq$HOD, in a manner that accommodates large cardinals.

• Hamkins, Joel David; Reitz, Jonas; Woodin, W. Hugh, The ground axiom is consistent with $V \neq \text{HOD}$, Proc. Am. Math. Soc. 136, No. 8, 2943-2949 (2008). ZBL1145.03029.

Basically, the forcing is an Easton-support iteration of adding a Cohen subset to each regular cardinal. This forces both the GCH and the ground axiom, and the closure features accommodate the master condition arguments that allow supercompact and higher large cardinals to be preserved, including rank-to-rank.

Diamond. As for the diamond principles $\Diamond_\kappa$, these are generally outright consequences of $\kappa$ being a large cardinal. Indeed, the Laver function idea holds generally for almost all the usual large cardinal notions, and serves as a generalization of the usual diamond. See my paper:

• Joel David Hamkins, A class of strong diamond principles, arxiv:0211419.

Meanwhile, one can easily force diamond at other cardinals between the large cardinals, and the existence of diamonds will be a consequence of sufficient indestructibility, since the forcing is very tame. I believe that the canonical forcing of the GCH will also have the effect of forcing diamonds at all the cardinals as well.

Obtaining the properties in inner models. Surprisingly, it also often possible to obtain the desired features not just by forcing to an outer model, but also by going to an inner model. These results are the central feature of my paper with Arthur Apter:

• Apter, Arthur W.; Gitman, Victoria; Hamkins, Joel David, Inner models with large cardinal features usually obtained by forcing, Arch. Math. Logic 51, No. 3-4, 257-283 (2012). ZBL1250.03104.

For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal, which therefore has V=HOD up to that point. And one can also arrange GCH at $\kappa$ and in an interval above it, and so forth.

Lifting arguments. About 25 years ago, I started a book project detailing these standard lifting arguments that enable all these kinds of forcing iterations to preserve large cardinals, covering the standard Silver iterations, GCH forcing, V=HOD forcing, indestructibility forcing, the lottery preparation, and more.

Alas, the book project was never completed, but I have a draft copy available for people working in this area. Perhaps someday I shall make a suitable version available generally.