Structural properties of the stable core

S.-D. Friedman, V. Gitman, and S. Müller, “Some properties of the Stable Core,” Manuscript.

The first author introduced the inner model stable core while investigating under what circumstances the universe $V$ is a class forcing extension of the inner model ${\rm HOD}$, the collection of all hereditarily ordinal definable sets (missing reference). He showed in (missing reference) that there is a robust $\Delta_2$-definable class $S$ contained in ${\rm HOD}$ such that $V$ is a class-forcing extension of the structure $\langle L[S],\in, S\rangle$, which he called the stable core, by an ${\rm Ord}$-cc class partial order $\mathbb P$ definable from $S$. Indeed, for any inner model $M$, $V$ is a $\mathbb P$-forcing extension of $\langle M[S],\in,S\rangle$, so that in particular, since ${\rm HOD}[S]={\rm HOD}$, $V$ is a $\mathbb P$-forcing extension of $\langle {\rm HOD},\in,S\rangle$.

Let's explain the result in more detail for the stable core $L[S]$, noting that exactly the same analysis applies to ${\rm HOD}$. The partial order $\mathbb P$ is definable in $\langle L[S],\in,S\rangle$ and there is a generic filter $G$, meeting all dense sub-classes of $\mathbb P$ definable in $\langle L[S],\in,S\rangle$, such that $V=L[S][G]$. All standard forcing theorems hold for $\mathbb P$ since it has the ${\rm Ord}$-cc. Thus, we get that the forcing relation for $\mathbb P$ is definable in $\langle L[S],\in,S\rangle$ and the forcing extension $\langle V,\in,G\rangle\models{\rm ZFC}$. However, this particular generic filter $G$ is not definable in $V$. To obtain $G$, we first force with an auxiliary forcing $\mathbb Q$ to add a particular class $F$, without adding sets, such that $V=L[F]$. We then show that $G$ is definable from $F$ and $F$ is in turn definable in the structure $\langle L[S][G], \in, S,G\rangle$, so that $L[S][G]=V$. This gives a formulation of the result as a ${\rm ZFC}$-theorem because we can say (using the definitions of $\mathbb P$ and $\mathbb Q$) that it is forced by $\mathbb Q$ that $V=L[F]$, where $F$ is $V$-generic for $\mathbb Q$, and (the definition of) $G$ is $\langle L[S],\in,S\rangle$-generic, and finally that $F$ is definable in $\langle L[S][G],\in,S,G\rangle$. Of course, a careful formulation would say that the result holds for all sufficiently large natural numbers $n$, where $n$ bounds the complexity of the formulas used.

Without the niceness requirement on $\mathbb P$ that it has the ${\rm Ord}$-cc, there is a much easier construction of a class forcing notion $\mathbb P$, suggested by Woodin, such that $V$ is a class forcing extension of $\langle {\rm HOD},\in, \mathbb P\rangle$. At the same time, some additional predicate must be added to ${\rm HOD}$ in order to realize all of $V$ as a class-forcing extension because, as Hamkins and Reitz observed in [1], it is consistent that $V$ is not a class-forcing extension of ${\rm HOD}$. To construct such a counterexample, we suppose that $\kappa$ is inaccessible in $L$ and force over the Kelley-Morse model $\mathcal L=\langle L_\kappa,\in, L_{\kappa+1}\rangle$ to code the truth predicate of $L_\kappa$ (which is an element of $L_{\kappa+1}$) into the continuum pattern below $\kappa$. The first-order part $L_\kappa[G]$ of this extension cannot be a forcing extension of ${\rm HOD}^{L_\kappa[G]}=L_\kappa$ (by the weak homogeneity of the coding forcing), because the truth predicate of $L_\kappa$ is definable there and this can be recovered via the forcing relation.

While the definition of the partial order $\mathbb P$ is fairly involved, the stability predicate $S$ simply codes the elementarity relations between sufficiently nice initial segments $H_\alpha$ (the collection of all sets with transitive closure of size less than $\alpha$) of $V$. Given a natural number $n\geq 1$, call a cardinal $\alpha$ $n$-good if it is a strong limit cardinal and $H_\alpha$ satisfies $\Sigma_n$-collection. The predicate $S$ consists of triples $(n,\alpha,\beta)$ such that $n\geq 1$, $\alpha$ and $\beta$ are $n$-good cardinals and $H_\alpha\prec_{\Sigma_n}H_\beta$. We will denote by $S_n$ the $n$-th slice of the stability predicate $S$, namely $S_n=\{(\alpha,\beta)\mid (n,\alpha,\beta)\in S\}$.

Clearly the stable core $L[S]\subseteq{\rm HOD}$, and the first author showed in (missing reference) that it is consistent that $L[S]$ is smaller than ${\rm HOD}$. The stable core has several nice properties which fail for ${\rm HOD}$ such as that it is partially forcing absolute and, assuming ${\rm GCH}$, is preserved by forcing to code the universe into a real (missing reference)

In order to motivate the many questions which arise about the stable core let us briefly discuss the set-theoretic goals of studying inner models.

The study of canonical inner models has proved to be one of the most fruitful directions of modern set-theoretic research. The canonical inner models, of which Gödel's constructible universe $L$ was the first example, are built bottom-up by a canonical procedure. The resulting fine structure of the models leads to regularity properties, such as the ${\rm GCH}$ and $\square$, and sometimes even absoluteness properties. But all known canonical inner models are incompatible with sufficiently large large cardinals, and indeed each such inner model is very far from the universe in the presence of sufficiently large large cardinals in the sense, for example, that covering fails and the large cardinals are not downward absolute.

The inner model ${\rm HOD}$ was introduced by Gödel, who showed that in a universe of ${\rm ZF}$ it is always a model of ${\rm ZFC}$. But unlike the constructible universe which also shares this property, ${\rm HOD}$ has turned out to be highly non-canonical. While $L$ cannot be modified by forcing, ${\rm HOD}$ can be easily changed by forcing because we can use forcing to code information into ${\rm HOD}$. For instance, any subset of the ordinals from $V$ can be made ordinal definable in a set-forcing extension by coding its characteristic function into the continuum pattern, so that it becomes an element of the ${\rm HOD}$ of the extension. Indeed, by coding all of $V$ into the continuum pattern of a class-forcing extension, Roguski showed that every universe $V$ is the ${\rm HOD}$ of one of its class-forcing extensions [2]. Thus, any consistent set-theoretic property, including all known large cardinals, consistently holds in ${\rm HOD}$. At the same time, the ${\rm HOD}$ of a given universe can be very far from it. It is consistent that every measurable cardinal is not even weakly compact in ${\rm HOD}$ and that a universe can have a supercompact cardinal which is not even weakly compact in ${\rm HOD}$ [3]. It is also consistent that ${\rm HOD}$ is wrong about all successor cardinals [4].

Does the stable core behave more like the canonical inner models or more like ${\rm HOD}$? Is there a fine structural version of the stable core, does it satisfy regularity properties such as the ${\rm GCH}$? Is there a bound on the large cardinals that are compatible with the stable core? Or, on the other hand, are the large cardinals downward absolute to the stable core? Can we code information into the stable core using forcing?

In this article, we show the following results about the structure of the stable core, which answer some of the aforementioned questions as well as motivate further questions about the structure of the stable core in the presence of sufficiently large large cardinals.

Measurable cardinals are consistent with the stable core.

Theorem:

1. The stable core of $L[\mu]$, the canonical model for one measurable cardinal, is $L[\mu]$. In particular, the stable core can have a measurable cardinal.
2. If $\vec U=\langle U_\alpha\mid\alpha\in{\rm Ord}\rangle$ is a discrete collection of normal measures, then the stable core of $L[\vec U]$ is $L[\vec U]$. In particular, the stable core can have a discrete proper class of measurable cardinals.

We can code information into the stable core over $L$ or $L[\mu]$ using forcing.

Theorem: Suppose $\mathbb P\in L$ is a forcing notion and $G\subseteq \mathbb P$ is $L$-generic. Then there is a further forcing extension $L[G][H]$ such that $G\in L[S^{L[G][H]}]$ (the universe of the stable core). An analogous result holds for $L[\mu]$.

An extension of the coding results shows that the ${\rm GCH}$ can fail badly in the stable core.

Theorem:

1. There is a class-forcing extension of $L$ such that in its stable core the ${\rm GCH}$ fails at every regular cardinal.
2. There is a class-forcing extension of $L[\mu]$ such that in its stable core there is a measurable cardinal and the ${\rm GCH}$ fails on a tail of regular cardinals.

Measurable cardinals need not be downward absolute to the stable core.

Theorem: There is a forcing extension of $L[\mu]$ in which the measurable cardinal $\kappa$ of $L[\mu]$ remains measurable, but it is not even weakly compact in the stable core.

Although we don't know whether the stable core can have a measurable limit of measurables, the stable core has inner models with measurable limits of measurables, and much more. Say that a cardinal $\kappa$ is $1$-measurable if it is measurable, and, for $n < \omega$, $(n+1)$-measurable if it is measurable and a limit of $n$-measurable cardinals. Write $m_0^\#$ for $0^\#$ and $m_n^\#$ for the minimal mouse which is a sharp for a proper class of $n$-measurable cardinals, namely, an active mouse $\mathcal M$ such that the critical point of the top extender is a limit of $n$-measurable cardinals in $\mathcal M$. Here we mean mouse in the sense of [5] (Sections 1 and 2), i.e., a mouse has only total measures on its sequence. The mouse $m_n^{\#}$ can also be construed as a fine structural mouse with both total and partial extenders (see [6], Section 4).

Theorem: For all $n<\omega$, if $m_{n+1}^\#$ exists, then $m_n^\#$ is in the stable core.

Moreover, we obtain the following characterization of natural inner models of the stable core.

Theorem: Let $n < \omega$ and suppose that $m_n^\#$ exists. Then whenever $$C_1\supseteq C_2\supseteq\ldots\supseteq C_n$$ are class clubs of uncountable cardinals such that for every $1 < i\leq n$ and every $\gamma \in C_i$, $\langle H_\gamma,\in, C_1,\ldots,C_{i-1}\rangle\prec_{\Sigma_1} \langle V,\in, C_1,\ldots,C_{i-1}\rangle,$ then $L[C_1,\ldots,C_n]$ is a hyperclass-forcing extension of a (truncated) iterate of $m_n^\#$.

References

1. J. D. Hamkins and J. Reitz, “The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD,” ArXiv e-prints, Sep. 2017. Available at: http://jdh.hamkins.org/the-universe-need-not-be-a-class-forcing-extension-of-hod
2. S. Roguski, “Cardinals and iterations of ${\rm HOD}$,” Colloq. Math., vol. 58, no. 2, pp. 159–161, 1990. Available at: https://doi.org/10.4064/cm-58-2-159-161
3. Y. Cheng, S.-D. Friedman, and J. D. Hamkins, “Large cardinals need not be large in HOD,” Ann. Pure Appl. Logic, vol. 166, no. 11, pp. 1186–1198, 2015. Available at: https://doi.org/10.1016/j.apal.2015.07.004
4. J. Cummings, S. D. Friedman, and M. Golshani, “Collapsing the cardinals of HOD,” J. Math. Log., vol. 15, no. 2, pp. 1550007, 32, 2015. Available at: https://doi.org/10.1142/S0219061315500075
5. W. J. Mitchell, “Beginning inner model theory,” in Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 1449–1495. Available at: https://doi.org/10.1007/978-1-4020-5764-9_18
6. M. Zeman, Inner Models and Large Cardinals. Walter de Gruyter, 2002. Available at: https://books.google.at/books?id=V0bLReBbRDEC