Unfoldable cardinal
The unfoldable cardinals were introduced by Andres Villaveces in order to generalize the definition of weak compactness. Because weak compactness has many different definitions, the one he chose to extend was specifically the embedding property (see weakly compact for more information). The way he did this was analogous to the generalization of huge cardinals to superhuge cardinals.
Contents
Definition
There are unfoldable cardinals and strongly unfoldable cardinals, as well as superstrongly unfoldable (AKA almost-hugely unfoldable AKA strongly uplifting) cardinals. All of these are generalizations of weak compactness.
Unfoldable
A cardinal $\kappa$ is $\theta$-unfoldable iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$. $\kappa$ is then called unfoldable iff it is $\theta$-unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.
Unfoldability can also be characterized by the order-type of nontrivial elementary end-extensions of $V_\kappa$. Let $\mathcal{M}=(M,R_0^\mathcal{M},R_1^\mathcal{M}...)$ be an aribtrary structure of type $(\alpha,\beta)$ with relations $R_0^\mathcal{M},R_1^\mathcal{M}...$ and $\mathcal{N}=(N,R_0^\mathcal{N},R_1^\mathcal{N}...)$ be another arbitrary structure of the same language with relations $R_0^\mathcal{N},R_1^\mathcal{N}...$. Then, one writes $\mathcal{M}\prec_e\mathcal{N}$ iff all of the following hold:
- $\mathcal{M}$ is an elementary substructure of $\mathcal{N}$
- $\mathcal{M}\neq\mathcal{N}$
- For any $a\in M$, $b\in N$, and $\gamma<\beta$, $b R_\gamma^\mathcal{N} a\rightarrow b\in M$
If such holds, $\mathcal{M}$ is said to be nontrivially end elementary extended by $\mathcal{N}$ and $\mathcal{N}$ is a nontrivial end elementary extension of $\mathcal{M}$, abbreviated $\mathcal{N}$ is an eee of $\mathcal{M}$.
A cardinal $\kappa$ is $\lambda$-unfoldable iff $\kappa$ is inaccessible and for any $S\subset V_\kappa$, there are well-founded models $M$ nontrivially end elementary extending $(V_\kappa;\in,S)$ such that $M\not\in V_\lambda$. $\kappa $ is unfoldable iff $M $ can be made to have arbitrarily large rank. In this case, $(V_\kappa;\in,S)\prec_e (M;\in^M,S^M)$ iff $(V_\kappa;\in,S)\prec (M;\in^M,S^M)$ and $(V_\kappa;\in)\prec_e (M;\in^M)$. [1]
$\kappa$ is also unfoldable iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $(V_\kappa;\in,S)$, $(\mathcal{E};\prec_e)$ has abitrarily long chains. The name "unfoldable" comes from the fact that "unfolding" $(V_\kappa;\in,S)$ yields a larger structure with the same properties and a bit of excess information, and this can be done arbitrarily many times on the iterated results of "unfolding". [1]
Long Unfoldable
$\kappa$ is long unfoldable iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $(V_\kappa;\in,S)$, $(\mathcal{E};\prec_e)$ has chains of length $\text{Ord}$.
Every long unfoldable cardinal is unfoldable. [1]
Strongly Unfoldable
A cardinal $\kappa$ is $\theta$-strongly unfoldable iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$ and $V_\theta\subseteq N$.
$\kappa$ is then called strongly unfoldable iff it is $\theta$-strongly unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.
As defined in [2] in analogy with Mitchell ranks, a strongly unfoldable cardinal $\kappa$ is strongly unfoldable of degree $\alpha$, for an ordinal $\alpha$, if for every ordinal $\theta$ it is $\theta$-strongly unfoldable of degree $\alpha$, meaning that for each $A \in H_{\kappa^+}$ there is a $\kappa$-model $M \models \mathrm{ZFC}$ with $A \in M$ and a transitive set $N$ with $\alpha \in M$ and an elementary embedding $j:M \to N$ having critical point $\kappa$ with $j(\kappa)>\max\{\theta, \alpha\}$ and $V_\theta \subset N$, such that $\kappa$ is strongly unfoldable of every degree $\beta < \alpha$ in $N$.[3]
The strongly unfoldable cardinals are exactly the shrewd cardinals. [1]
Superstrongly Unfoldable
Superstrongly unfoldable and almost-hugely unfoldable cardinals are defined and shown to be equivalent to strongly uplifting (described there) in [3].
Relations to Other Cardinals
Here is a list of relations between unfoldability and other large cardinal axioms:
- For every finite $m$ and $n$, unfoldability implies the consistency of the existence of a $\Pi_m^n$-indescribable cardinal (specifically, such cardinals exist in $V_\kappa\cap L$ for some $\kappa$). Furthermore, if $V=L$, then the least $\Pi_m^n$-indescribable cardinal is less than the least unfoldable cardinal, and every unfoldable cardinal is totally indescribable. [1]
- Any strongly unfoldable cardinal is totally indescribable and a limit of totally indescribable cardinals. Therefore the consistency strength of unfoldability is stronger than total indescribability. [1]
- Every superstrongly unfoldable cardinal (i.e. strongly uplifting cardinal) is strongly unfoldable of every ordinal degree \(\alpha\), and a stationary limit of cardinals that are strongly unfoldable of every ordinal degree and so on. [3]
- The $\kappa$ which are $\kappa$-unfoldable are precisely those which are $\kappa$-strongly unfoldable, which are precisely those that are weakly compact. Furthermore, if $V=L$, then any $\theta$-unfoldable cardinal is $\theta$-strongly unfoldable. Therefore, unfoldability and strong unfoldability are equiconsistent. [4]
- The assertion that a Ramsey cardinal and a strongly unfoldable cardinal both exists is stronger than the assertion that there exists both an unfoldable cardinal and a strongly unfoldable cardinal. [1]
- Although unfoldable cardinals are consistency-wise stronger than weakly compact cardinals, if there is a strongly unfoldable cardinal, then in a forcing extension, the least weakly compact cardinal is also the least unfoldable cardinal.[5]
- The existence of a subtle cardinal is consistency-wise stronger than the existence of an unfoldable cardinal. [1]
- If a subtle cardinal and an unfoldable cardinal exist and $V=L$, then the least unfoldable cardinal is larger than the least subtle cardinal (and therefore much larger than the least weakly compact). [1]
- Any Ramsey cardinal is unfoldable. If there is a weakly compact cardinal above an $\omega_1$-Erdos cardinal, then the least unfoldable is less than that (therefore less than the least Ramsey). [1]
- Even though it may seem odd at first, if both exist then the least I3 cardinal is less than the least strongly unfoldable cardinal.
- $ω_1$-iterable cardinals are strongly unfoldable in $L$.[6]
Relation to forcing
From [4]:
- If $κ$ is unfoldable, then in a forcing extension by forcing of size $κ$ the unfoldability of $κ$ is indestructible by $\mathrm{Add}(κ, θ)$ for any $θ$. (Main Theorem 2)
- If $κ$ is strongly unfoldable, then in a forcing extension by forcing of size $κ$ the strong unfoldability of $κ$ is indestructible by $\mathrm{Add}(κ, 1)$ but not by $\mathrm{Add}(κ, θ)$ for any cardinal $θ > κ$. (Theorem 7)
- Any unfoldable cardinal can be made, by forcing of size $κ^+$, $κ$ unfoldable but not strongly unfoldable. (Corollary 8)
- In a forcing extension preserving all strongly unfoldable cardinals, GCH fails at every inaccessible cardinal. (Theorem 9)
- Interestingly, if there is an unfoldable cardinal then there is a forcing extension in which all unfoldable cardinals in $V$ are unfoldable in the forcing extension and GCH fails at every inaccessible cardinal. Therefore, GCH can fail at every unfoldable cardinal.
TODO: connection to weak forms of PFA
TODO: consistency with slim Kurepa trees
References
- Villaveces, Andrés. Chains of End Elementary Extensions of Models of Set Theory. JSTOR , 1996. arχiv bibtex
- Hamkins, Joel David and Johnstone, Thomas A. Indestructible strong un-foldability. Notre Dame J Form Log 51(3):291--321, 2010. bibtex
- Hamkins, Joel David and Johnstone, Thomas A. Strongly uplifting cardinals and the boldface resurrection axioms. , 2014. arχiv bibtex
- Hamkins, Joel David. Unfoldable cardinals and the GCH. , 2008. arχiv bibtex
- Cody, Brent, Gitik, Moti, Hamkins, Joel David, and Schanker, Jason. The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly θ-Supercompact. , 2013. arχiv bibtex
- Gitman, Victoria and Welch, Philip. Ramsey-like cardinals II. J Symbolic Logic 76(2):541--560, 2011. www arχiv MR bibtex