Ground model definability in ${\rm ZF}$
This is a talk at the Joint Mathematics Meeting, special session on 'Choiceless Set Theory and related areas', University of Denver, January 15-16, 2020.
Slides
Laver, and independently Woodin, showed that a ground model is always definable (with a ground model parameter) in its set-forcing extensions [1], [2]. More so, the definition is uniform for all ${\rm ZFC}$-universes: there is a single formula $\varphi(x,y)$ such that whenever $V\models{\rm ZFC}$ and $V[G]$ is a set-forcing extension of $V$ by a poset $\mathbb P\in V$ of size $|\mathbb P|^V=\gamma$, then $V$ is defined in $V[G]$ by $\varphi(x,P^V(\delta))$ with $\delta=(\gamma^+)^V(=(\gamma^+)^{V[G]})$. Let's explain what the formula $\varphi(x,P^V(\delta))$ says. Let ${\rm Z}^*$ be a certain finite fragment of ${\rm ZFC}$. We will say that a transitive model $M\in V[G]$ is good if $M\models {\rm Z}^*$, $P^M(\delta)=P^V(\delta)$ and $(\delta^+)^M=(\delta^+)^{V[G]}$. The formula $\varphi(x,P^V(\delta))$ then holds of a set $x$ if $x$ is an element of a good model $M$ of height $\lambda\gg\delta$ such that $V[G]_\lambda\models {\rm Z}^*$ and the pair $M\subseteq V[G]_\lambda$ has the $\delta$-cover and $\delta$-approximation properties (we will explain what those are in a moment). Here is the reason the formula works. Any forcing extension by a poset of size less than a regular cardinal $\delta$ has the property that there are unboundedly many ordinals $\lambda$ such that both $V_\lambda$ and $V[G]_\lambda$ satisfy ${\rm Z}^*$ and the pair $V_\lambda\subseteq V[G]_\lambda$ has the $\delta$-cover and $\delta$-approximation properties. Also, if there is a good model $M$ of height $\lambda$ with $P^M(\delta)=P^V(\delta)$ and $(\delta^+)^M=(\delta^+)^{V[G]}$ such that that pair $M\subseteq V[G]_\delta\models {\rm Z}^*$ has the the $\delta$-cover and $\delta$-approximation properties, then the model is unique. Hence such a model $M$ of height $\lambda$ can only be $V_\lambda$.
The notions of $\delta$-cover and $\delta$-approximation properties are due to Hamkins [3]. Suppose that $V\subseteq W$ are transitive models of (some fragment of) ${\rm ZFC}$ and $\delta$ is a cardinal in $W$.
- The pair $V\subseteq W$ satisfies the $\delta$-cover property if for every $A\in W$ with $A\subseteq V$ and $|A|^W<\delta$, there is $B\in V$ with $A\subseteq B$ and $|B|^V<\delta$.
- The pair $V\subseteq W$ satisfies the $\delta$-approximation property if whenever $A\in W$ with $A\subseteq V$ and $A\cap a\in V$ for every $a$ of size less than $\delta$ in $V$, then $A\in V$.
Because the existence of cardinalities is fundamental to the definition of the cover and approximation properties, these notions appear to rely essentially on the axiom of choice. Therefore proofs of ground model definability based on these properties cannot be easily modified to work in the absence of choice. It is not known yet whether ground definability holds in ${\rm ZF}$. There are partial positive results and no known counterexamples.
With Tom Johnstone, we adapted the approximation and cover properties arguments to show that for a cardinal $\delta$, uniform ground model definability holds in models of ${\rm ZF}+{\rm DC}_\delta$ for posets admitting a gap at $\delta$.
Theorem: There is a single formula $\varphi(x,y)$ such that whenever $V\models{\rm ZF}+{\rm DC}_\delta$ and $V[G]$ is a set-forcing extension of $V$ by a poset $\mathbb P$ admitting a gap at $\delta$ in $V$, then $V$ is defined in $V[G]$ by $\varphi(x,P^V(\delta))$.
A poset is said to admit a gap at $\delta$ if it has the form $\mathbb R*\dot{\mathbb Q}$ where $\mathbb R$ is a non-trivial forcing of size less than $\delta$ and it is forced by $\mathbb R$ that is $\dot{\mathbb Q}$ is $\leq\delta$-strategically closed. The $\delta$-dependent choice principle ${\rm DC}_\delta$, introduced by Lévy, allows us to make $\delta$-many dependent choices along any relation without terminal nodes. It asserts that for any non-empty set $S$ and any binary relation $R$, if for each sequence $s\in S^{\lt\delta}$ there is a $y\in S$ such that $s$ is $R$ related $y$, then there is a function $f:\delta\to S$ such that $f\upharpoonright\alpha \,R\, f(\alpha)$ for all $\alpha<\delta$. The principle ${\rm DC}_\delta$ is robust for forcing with posets admitting a gap at $\delta$ because they preserve it to the forcing extension (see [4]).
A very different recent partial result on ${\rm ZF}$-ground model definability is due to Usuba [5]. Usuba showed that if a ${\rm ZF}$-universe has a proper class of Löwenheim-Skolem cardinals, a notion which he introduces, then uniform ground model definability holds for it.
Theorem: There is a single formula $\varphi(x,y)$ such that whenever $V$ is a model of ${\rm ZF}$ with a proper class of Löwenheim-Skolem cardinals and $V[G]$ is a set-forcing extension of $V$ by a poset $\mathbb P\in V_\kappa$ for a Lowenheim-Skolem cardinal $\kappa$, then $V$ is defined in the extension $V[G]$ by $\varphi(x,V_\kappa)$.
Usuba's proof is another modification of the arguments using cover and approximation, where he replaces the notion of cardinality by a rough measure on sets, defining for a set $x$ that the measure $|\!|x|\!|$ is the least ordinal $\alpha$ such that there is a surjection from $V_\alpha$ onto $x$.
An uncountable cardinal $\kappa$ is a Löwenheim-Skolem (${\rm LS}$) cardinal if for every $\gamma<\kappa\leq\alpha$ and $x\in V_\alpha$, there is $\beta>\alpha$ and an elementary $X\prec V_\beta$ such that
- $V_\gamma\subseteq X$,
- $x\in X$,
- the transitive collapse of $X$ belongs to $V_\kappa$,
- $(X\cap V_\alpha)^{V_\gamma}\subseteq X$.
The existence of ${\rm LS}$ cardinals is absolute throughout the generic multiverse [5]. It follows that if we can force choice over a ${\rm ZF}$-universe, then since its forcing extension has a proper class of ${\rm LS}$ cardinals, it must have a proper class of ${\rm LS}$ cardinals as well. Thus, in particular, ground model definability holds for any ${\rm ZF}$-universe, such as $L(\mathbb R)$, over which we can force the axiom of choice.
At the same time, it is known that there are ${\rm ZF}$ universes without any ${\rm LS}$-cardinals. Let's explain how. Usuba showed that if $\kappa$ is an ${\rm LS}$-cardinal, then the club filter on $\kappa^+$ must be $\kappa$-complete [5]. Asaf Karagila showed that every model $V$ of ${\rm ZFC}$ satisfying ${\rm GCH}$ has an extension to a model of ${\rm ZF}$ with the same cofinalities in which the club filter is not $\sigma$-complete on every regular cardinal $\kappa$ [6]. This model cannot have any ${\rm LS}$ cardinals because if $\kappa$ in it was an ${\rm LS}$-cardinal, then $\kappa^+$ would be a regular cardinal (since cofinalities from $V$ are preserved) with a $\kappa$-complete club filter.
One important consequence of uniform ground model definability is that it makes it possible to define in a first-order way the collection of all grounds of a ${\rm ZFC}$-universe. A priori this is a collection of classes, a second-order notion, but using ground model definability we can write down a first-order formula $\psi(x,y)$ such that for every set $a$, the class $W_a=\{x\mid \psi(x,a)\}$ is a ground of $V$ and for every $W$ ground of $V$, there is a set $a$ such that $W=W_a$. The first-order definability of the collection of grounds makes it possible to study their structure in ${\rm ZFC}$. This has led to the very fruitful subject of set-theoretic geology, initiated by Fuchs, Hamkins, and Reitz, which explores the structure of grounds of a set-theoretic universe. Ground model definability for ${\rm ZF}$ would allow us to extend the same analysis to ${\rm ZF}$-universes, and Usuba's theorem has now made such an analysis possible for a large collection of models, namely those with a proper class of ${\rm LS}$ cardinals. We will have to wait to find out whether ground definability holds fully in ${\rm ZF}$.
References
- R. Laver, “Certain very large cardinals are not created in small forcing extensions,” Ann. Pure Appl. Logic, vol. 149, no. 1-3, pp. 1–6, 2007. Available at: http://dx.doi.org/10.1016/j.apal.2007.07.002
- H. Woodin, “Recent developments on Cantor’s Continuum Hypothesis,” in Proceedings of the continuum in Philosophy and Mathematics, 2004.
- J. D. Hamkins, “Extensions with the approximation and cover properties have no new large cardinals,” Fund. Math., vol. 180, no. 3, pp. 257–277, 2003. Available at: http://dx.doi.org/10.4064/fm180-3-4
- V. Gitman and T. A. Johnstone, “On ground model definability,” in Infinity, Computability, and Metamathematics: Festschrift in honour of the 60th birthdays of Peter Koepke and Philip Welch, London, GB: College publications, 2014.
- T. Usuba, “Choiceless Löwenheim-Skolem property and uniform definability of grounds,” Manuscript.
- A. Karagila, “Fodor’s lemma can fail everywhere,” Acta Math. Hungar., vol. 154, no. 1, pp. 231–242, 2018. Available at: https://doi.org/10.1007/s10474-017-0768-5