# Anti-large cardinal principles

I'm interested in axioms that prevent the existence of large cardinals. More precisely:

(Informal definition) $\Phi$ is an anti-large-cardinal axiom iff $V \models \Phi \Rightarrow V \not \models \Psi$ for some garden-variety large cardinal axiom $\Psi$.

Probably the most famous are:

Variety 1 Axioms of constructibility (e.g. $V=L$, $V=L[x]$).

$V=L$ for example implies that there are no measurable cardinals in $V$. Often we don't need the full power of $L$ to get anti-large cardinal features though. For instance:

Variety 2 $\square$-principles

often prevent the existence of large cardinals via their anti-reflection features. For example $\square_\kappa$ prevents the existence of cardinals that reflect stationary sets to $\alpha < \kappa$ (and hence rule out cardinals like supercompacts).

Trivially, we might look at axioms of the following form:

Variety 3 $\neg \Phi$ or $\neg Con(\Phi)$ for some large cardinal axiom $\Phi$.

These obviously and boringly prevents the existence of a large cardinal in the model. Another possible candidate (formalisable in $NBG + \Sigma^1_1$ comprehension) is:

Variety 4 Inner model hypotheses.

For example, the vanilla inner model hypothesis that any parameter-free first-order sentence true in an inner model of an outer model of $V$ is already true in an inner model of $V$. This prevents the existence of inaccessibles in $V$. Other variants rule out cardinals not consistent with $L$.

Final example (if we're being super liberal about what we allow to be revised):

Variety 5 Choice axioms.

For instance $AC$ prevents the existence of Reinhardt (and super-Reinhardt, Berkeley etc.) cardinals in $V$ (assuming that these are, in fact, consistent with $ZF$, which is pretty non-trivial).

My question:

Are there other kinds of axiom with anti-large-cardinal features? Especially non-trivial ones (i.e. not like Variety 3).

• $PA$ is equivalent to $ZF$ with the axiom of infinity replaced by its negation. That fact sort of fits in here. – Oscar Cunningham Apr 26 '17 at 17:20
• @OscarCunningham. Indeed. I regard that as an example of 3 (after all, the axiom of infinity seems like a large cardinal axiom to me). – Neil Barton Apr 26 '17 at 17:50
• In fact, if we look at the space below the axiom of infinity, the principle superexponentiation is not total'' (equivalent to the negation of the Finite Ramsey Theorem over $EFA$) would be a similar principle. In general assertions about fast growing functions behave a bit like large cardinals in the arithmetic realm (Peter Koellner made this point to me, and there's some similar remarks in his SEP article: plato.stanford.edu/entries/large-cardinals-determinacy). – Neil Barton Apr 26 '17 at 17:54
• @NeilBarton Wouldn't it make more sense to look at principles which imply the inconsistency of small large cardinals? This is because these principles actually act like the large cardinals themselves; the weaker the large cardinal axiom is that these principles deny, the closer the principles are to inconsistency with ZFC (viewing ZFC itself as the weakest large cardinal axiom). Just a preference though, not necessary of course. – Zetapology Nov 19 '17 at 18:31

Foreman's maximality principle (the statement: any non-trivial forcing either adds a new real or collapses some cardinals) implies there are no inaccessible cardinal.

Also the consistency of Magidor's question (tree property at all regular cardinals above $\aleph_1$) implies the non-existence of inaccessible cardinals.

Another example is the following maximality principle introduced at A new maximality principle and its consequences:

$(MP_*):$ For all uncountable regular cardinals $\kappa, 2^{<\kappa}=\kappa^{+}$ and all trees of height and size $\kappa$ are specialized.

• Thanks for this. In what sense is Foreman's Maximality Principle a maximality'' principle? It strikes me as more of a `dichotomy' principle (or similar), but I may be missing something here. – Neil Barton Apr 25 '17 at 9:37
• @NeilBarton It is called a maximality principle since it implies the existence of many sets, for example it implies that the GCH fails everywhere, or there are no $\kappa$-Souslin trees for all uncountable regular cardinals $\kappa$ (so all trees are in some sense fat). – Mohammad Golshani Apr 26 '17 at 3:38
• @NeilBarton There's also a more subtle fact. It is a consequence of the proper forcing axiom (actually, $\mathsf{BPFA}$ suffices) that any forcing that adds a subset of $\omega_1$ must either add a real or collapse $\omega_2$. The principle is an attempt at an ultimate generalization of this result. In their 1986 paper on the maximality principle, Foreman, Magidor and Shelah show that $0^\sharp$ implies that any nontrivial forcing that belongs to $L$ adds a real when used to force over $V$. The principle claims a similar consequence without restricting where the poset used to force comes from. – Andrés E. Caicedo Apr 26 '17 at 15:44
• @NeilBarton Finally, Martin's maximum (which appeared shortly thereafter), seems to have been at least partially motivated by an attempt to prove the consistency of the principle (see the statement of the second theorem in page 40 of the 1986 paper). – Andrés E. Caicedo Apr 26 '17 at 15:46

Universal indestructibility is another example. In our joint paper,

Arthur Apter and I proved that it is relatively consistent that there is a supercompact cardinal and every supercompact and partially supercompact cardinal is fully Laver indestructible. The proof used the "trial-by-fire" forcing, where one proceeds in a grand forcing iteration, killing at every stage $\delta$ the most supercompactness of $\delta$ that could be killed by $<\delta$-directed closed forcing. The difficult part is to show that with a suitable large cardinal assumption, something does indeed survive this trial by fire.

Meanwhile, relating to your question, we also proved that under universal indestructibility, there can be at most one supercompact cardinal. Indeed, if universal indestructibility holds, then no cardinal is supercompact beyond a measurable cardinal.

• Great! Strange that it's just one (does the other one interfere by being in the range of an embedding? via reflection maybe). I'll check out the paper! – Neil Barton Apr 25 '17 at 9:32

Here is a different kind of example, not a full instance of what you request, but halfway. Namely, a natural principle that implies either that there are no large cardinals of a certain kind or else a proper class of them.

Consider the maximality principle, which is the scheme asserting of any set-theoretic assertion $\varphi$ that if there is some forcing extension in which $\varphi$ becomes true and remains true in all further forcing extensions, then it is already true.

This assertion can be expressed in modal terms by $$\diamondsuit\Box\varphi\to\varphi$$ using the forcing modalities. For example, see my paper, Hamkins, Joel David, A simple maximality principle., J. Symb. Log. 68, No. 2, 527-550 (2003). ZBL1056.03028. (my blog post) A version of the principle was introduced by Jouko Väänänen.

One of the first observations to make about the maximality principle is that it implies that there is no medium ground when it comes to certain kinds of larger cardinals.

• If MP holds, then either there are no inaccessible cardinals, or a proper class of them. The reason is that if there are just a few, then one can force so as to collapse them and thereby come to have none, in a way that persists to all further extensions. So there must have been none to begin with.

• If MP holds, then either there are no Mahlo cardinals or a proper class of them. Similarly, if there were only a bounded number of them, then you could force to kill them all and this would remain true in all further extensions; so it must have been true already.

• It should be called large cardinal property with 0-1 law! – Rahman. M Apr 25 '17 at 7:57
• Ah, this is nice. It's an especially nice case where the property of being a button is in full effect. – Neil Barton Apr 25 '17 at 9:30

The following is part of unpublished work by Toshimichi Usuba:

Definition. A cardinal $\kappa$ is hyper-huge if for every $\lambda > \kappa$ there is an elementary embedding $$j \colon V \to M$$ into some inner model $M$ such that $\operatorname{crit}(j) = \kappa$, $\lambda < j(\kappa)$ and $^{j(\lambda)} M \subseteq M$.

Definition. An inner model $M$ of $V$ is a ground if there is some set forcing $\mathbb P \in M$ and some $\mathbb P$-generic filter $G$ over $M$ such that $M[G] = V$.

Theorem (Usuba). If there is a hyper-huge cardinal, then there are only set many grounds.

He also proved that $V$ has a $\subseteq$-minimal ground (which is known as the bedrock) iff there are only set many grounds. Thus the nonexistence of the bedrock is an anti-large cardinal axiom.

On the other hand, in Inner Model Theoretic Geology, Fuchs and Schindler provided an example for which the bedrock does not exist, namely: If $L[E]$ is a pure extender model without a strong cardinal satisfying some 'nice inner model theoretic properties' (*) then the bedrock of $L[E]$ does not exist. (**)

(*) $L[E]$ is assumed to be tame, fully iterable in $V$, internally not fully iterable as guided by $\mathcal{P}$-constructions and the extender sequence $E$ and its canonical extensions by $\operatorname{Col}(\omega, \theta)$, for cutpoints $\theta$ of $L[E]$, are ordinal definable in their respective models.

(**) Please note that they provide a much more detailed analysis of the class of grounds for this model.