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:

- $\mathcal{M}$ 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
$(V_\kappa;\in)\prec_e (M;\in^M)$.
(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:

- 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. (Villaveces, 1996)
- 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. (Villaveces, 1996)
- 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. (Hamkins & Johnstone, 2014)
- 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. (Hamkins, 2008)
- 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. (Villaveces, 1996)
- 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. (Hamkins, 2008)
- 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.(Cody et al., 2013)
- The existence of a subtle cardinal is consistency-wise stronger than the existence of an unfoldable cardinal. (Villaveces, 1996)
- 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). (Villaveces, 1996)
- 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). (Villaveces, 1996)
- 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 $L$.(Gitman & Welch, 2011)

e.g. GCH, indestructibility, connection to weak forms of PFA

consistency with slim Kurepa trees

- Villaveces, A. (1996). Chains of End Elementary Extensions of Models of Set Theory.
*JSTOR*. - Hamkins, J. D., & Johnstone, T. A. (2010). Indestructible strong un-foldability.
*Notre Dame J. Form. Log.*,*51*(3), 291–321. - Hamkins, J. D., & Johnstone, T. A. (2014).
*Strongly uplifting cardinals and the boldface resurrection axioms*. - Hamkins, J. D. (2008).
*Unfoldable cardinals and the GCH*. - Cody, B., Gitik, M., Hamkins, J. D., & Schanker, J. (2013).
*The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly θ-Supercompact*. - Gitman, V., & Welch, P. (2011). Ramsey-like cardinals II.
*J. Symbolic Logic*,*76*(2), 541–560. http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf