Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.
View the Project on GitHub neugierde/cantors-attic
Quick navigation
The upper attic
The middle attic
The lower attic
The parlour
The playroom
The library
The cellar
Sources
Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
Latest working snapshot at the wayback machine
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.
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.
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:
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 $(V_\kappa;\in)\prec_e (M;\in^M)$. (Villaveces, 1996)
$\kappa$ 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”. (Villaveces, 1996)
$\kappa$ 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. (Villaveces, 1996)
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 (Hamkins & Johnstone, 2010) 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 $N$.(Hamkins & Johnstone, 2014)
Superstrongly unfoldable and almost-hugely unfoldable cardinals are defined and shown to be equivalent to strongly uplifting (described there) in (Hamkins & Johnstone, 2014).
Here is a list of relations between unfoldability and other large cardinal axioms:
e.g. GCH, indestructibility, connection to weak forms of PFA
consistency with slim Kurepa trees