Generic Vopěnka's Principle, remarkable cardinals, and the weak Proper Forcing Axiom
J. Bagaria, V. Gitman, and R. Schindler, “Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom,” Arch. Math. Logic, vol. 56, no. 1-2, pp. 1–20, 2017.
Vopěnka's Principle is a large cardinal principle which states that for every proper class $\mathcal C$ of structures of the same type there are $B\neq A$, both in $\mathcal C$, such that $B$ elementarily embeds into $A$. It can be formalized in first-order set theory as a schema, where for each natural number $n$ in the meta-theory there is a formula expressing that Vopěnka's Principle holds for all $\Sigma_n$-definable (with parameters) classes. Following [1], we call ${\rm VP}(\mathbf\Sigma_n)$ the fragment of Vopěnka's Principle for $\Sigma_n$-definable classes and let ${\rm VP}(\Sigma_n)$ be the weaker principle, where parameters are not allowed in the definition of the class (with analogous definitions for $\Pi_n$). Bagaria introduced in [1] a family of Vopěnka-like principles ${\rm VP}(\kappa,\mathbf\Sigma_n)$, where $\kappa$ is a cardinal, which state that for every proper class $\mathcal C$ of structures of the same type that is $\Sigma_n$-definable with parameters in $H_\kappa$ (the collection of all sets of hereditary size less than $\kappa$), $\mathcal C$ reflects below $\kappa$, namely for every $A\in\mathcal C$ there is $B\in H_\kappa\cap \mathcal C$ that elementarily embeds into $A$. Bagaria established a relationship between Vopěnka's Principle fragments and his family of principles ${\rm VP}(\kappa,\mathbf\Sigma_n)$ and provided a complete characterization of Vopěnka's Principle fragments ${\rm VP}(\mathbf\Pi_n)$, as well as the weaker principles ${\rm VP}(\Pi_n)$, in terms of the existence of supercompact and $C^{(n)}$-extendible cardinals [1].
Recall that $C^{(n)}$ denotes the class club of ordinals $\delta$ such that $V_\delta\prec_{\Sigma_n} V$. A cardinal $\kappa$ is called $C^{(n)}$-extendible if for every $\alpha>\kappa$, there is an elementary embedding $j:V_\alpha\to V_\beta$ with critical point $\kappa$ and with $j(\kappa)\in C^{(n)}$. Note that every extendible cardinal is $1$-extendible. Bagaria [1] showed that the weaker principle ${\rm VP}(\Pi_1)$ holds if and only if for some $\kappa$, ${\rm VP}(\kappa,\mathbf\Sigma_2)$ holds, and if and only if there is a supercompact cardinal. Also, for $n\geq 1$, ${\rm VP}(\Pi_{n+1})$ holds if and only if for some $\kappa$, ${\rm VP}(\kappa,\mathbf\Sigma_{n+2})$ holds, and if and only if there is a $C^{(n)}$-extendible cardinal. The results generalize to show that the Vopěnka's Principle fragment ${\rm VP}(\mathbf\Pi_1)$ holds if and only if ${\rm VP}(\kappa,\mathbf\Sigma_2)$ holds for a proper class of $\kappa$, and if and only if there is a proper class of supercompact cardinals. Also, for $n\geq 1$, ${\rm VP}(\mathbf\Pi_{n+1})$ holds if and only if ${\rm VP}(\kappa,\mathbf\Sigma_{n+2})$ holds for a proper class of $\kappa$, and if and only if there is a proper class of $C^{(n)}$-extendible cardinals. Thus, Vopěnka's Principle holds precisely when, for every $n\in\omega$, there is a proper class of $C^{(n)}$-extendible cardinals.
In this article, we introduce and study generic versions of Vopěnka's Principle and its variants. The Generic Vopěnka's Principle states that for every proper class $\mathcal C$ of structures of the same type there are $B\neq A$, both in $\mathcal C$, such that $B$ elementarily embeds into $A$ in some set-forcing extension. We call ${\rm gVP}(\mathbf\Sigma_n)$ the Generic Vopěnka's Principle fragment for $\Sigma_n$-definable (with parameters) classes and we let ${\rm gVP}(\Sigma_n)$ be the weaker principle where parameters are not allowed in the definition of the class (with analogous definitions for $\Pi_n$). We also call ${\rm gVP}(\kappa,\mathbf\Sigma_n)$ the analogous generic version of ${\rm VP}(\kappa,\mathbf\Sigma_n)$.
It turns out that an elementary embedding $j:B\to A$ between first-order structures exists in some set-forcing extension if and only if it already exists in $V^{\rm{Coll}(\omega,B)}$. We show that to every pair of structures $B$ and $A$ of the same type, we can associate a closed game $G(B,A)$ such that $B$ elementarily embeds into $A$ in $V^{\rm {Coll}(\omega,B)}$ precisely when a particular player has a winning strategy in that game. The game $G(B,A)$ is a variant of an Ehrenfeucht-Fraïssé game of length $\omega$, where player I starts out by playing some $b_0\in B$ and player II responds by playing $a_0\in A$. Players I and II continue to alternate, choosing elements $b_n$ and $a_n$ from their respective structures at stage $n$ of the game. Player II wins if for every formula $\varphi(x_0,\ldots,x_n)$, $$B\models \varphi(b_0,\ldots,b_n)\leftrightarrow A\models\varphi(a_0,\ldots,a_n),$$ and otherwise player I wins. Since if player II loses she must do so at some finite stage of the game, the game $G(B,A)$ is closed and hence determined by the Gale-Stewart theorem [2]. Thus, either player I or player II has a winning strategy. We show that player II has a winning strategy precisely when $B$ elementarily embeds into $A$ in $V^{\rm {Coll}(\omega,B)}$. It follows that each first-order fragment of Generic Vopěnka's Principle is characterized by the existence of certain winning strategies in its associated class of closed games.
The consistency strength of Generic Vopěnka's Principle fragments is measured by a hierarchy of cardinals, the $n$-remarkable cardinals we introduce here, which generalize Schindler's remarkable cardinals analogously to how $C^{(n)}$-extendible cardinals generalize extendible cardinals. A remarkable cardinal (which is $1$-remarkable by our definition) is a type of generic supercompact cardinal and, correspondingly, an $n$-remarkable cardinal (for $n>1$) is a type of generic $C^{(n)}$-extendible cardinal. The $n$-remarkable cardinals sit relatively low in the large cardinal hierarchy. Call a large cardinal completely remarkable if it is $n$-remarkable for every $n\in\omega$. Completely remarkable cardinals can exist in $L$ and the consistency of a completely remarkable cardinal follows from a $2$-iterable cardinal. We show that the Generic Vopěnka's Principle fragment ${\rm gVP}(\Pi_n)$ is equiconsistent with an $n$-remarkable cardinal.
Theorem: The following are equiconsistent.
- ${\rm gVP}(\Pi_n)$.
- ${\rm gVP}(\kappa,\mathbf\Sigma_{n+1})$ for some $\kappa$.
- There is an $n$-remarkable cardinal.
The result generalizes to the bold-face ${\rm gVP}(\mathbf\Pi_n)$ principles.
Theorem: The following are equiconsistent.
- ${\rm gVP}(\mathbf\Pi_n)$.
- ${\rm gVP}(\kappa,\mathbf\Sigma_{n+1})$ for a proper class of $\kappa$.
- There is a proper class of $n$-remarkable cardinals.
The notion of a generic embedding existing in some forcing extension leads naturally to a weak version of the Proper Forcing Axiom ${\rm PFA}$, which we introduce and study here. Schindler and Claverie showed in [3] that ${\rm PFA}$ has the following equivalent formulation.
Theorem: The following are equivalent.
- ${\rm PFA}$
- If ${\mathcal M} = (M;\in,(R_i \mid i<\omega_1))$ is a transitive model, $\varphi(x)$ is a $\Sigma_1$-formula, and ${\mathbb Q}$ is a proper forcing such that $$\Vdash_{\mathbb Q} \varphi({\mathcal M}),$$ then there is in $V$ some transitive ${\bar{\mathcal M}}=({\bar M};\in,({\bar R}_i \mid i<\omega_1))$ together with some elementary embedding $$j : {\bar {\mathcal M}} \rightarrow {\mathcal M}$$ such that $\varphi({\bar {\mathcal M}})$ holds.
By weakening this formulation of ${\rm PFA}$ to say that the embedding $j$ exists in $V^{\rm {Coll}(\omega,\bar M)}$, we obtain the weak Proper Forcing Axiom ${\rm wPFA}$.
We show that ${\rm wPFA}$ is equiconsistent with a remarkable cardinal.
Theorem:
- If $\kappa$ is remarkable, then there is a forcing extension in which ${\rm wPFA}$ holds.
- If ${\rm wPFA}$ holds, then $\omega_2^V$ is remarkable in $L$.
The principle ${\rm wPFA}$ implies ${\rm PFA}_{\aleph_2}$, the Proper Forcing Axiom for meeting antichains of size $\leq\aleph_2$, but it does not imply ${\rm PFA}_{\aleph_3}$.
References
- J. Bagaria, “$C^{(n)}$-cardinals,” Arch. Math. Logic, vol. 51, no. 3-4, pp. 213–240, 2012.
- D. Gale and F. M. Stewart, “Infinite games with perfect information,” in Contributions to the theory of games, vol. 2, Princeton University Press, Princeton, N. J., 1953, pp. 245–266.
- B. Claverie and R. Schindler, “Woodin’s axiom $(∗)$, bounded forcing axioms, and precipitous ideals on $\omega_1$,” J. Symbolic Logic, vol. 77, no. 2, pp. 475–498, 2012.