View source for N-fold variants

{{DISPLAYTITLE: $n$-fold Variants of Large Cardinals}}
[[Category: Large cardinal axioms]]
[[Category: Critical points]]
''This page is a WIP.''
The $n$-fold variants of large cardinal axioms were created by Sato Kentaro in <cite>Kentaro2007:DoubleHelix</cite> in order to study and investigate the double helix phenomena. The double helix phenomena is the strange pattern in consistency strength between such cardinals, which can be seen below.

[[file:DoubleHelix.png | 600px | center]]

This diagram was created by Kentaro. The arrows denote consistency strength, and the double lines denote equivalency. The large cardinals in this diagram will be detailed on this page (unless found elsewhere on this website).

This page will only use facts from <cite>Kentaro2007:DoubleHelix</cite> unless otherwise stated.

== $n$-fold Variants ==

The $n$-fold variants of large cardinals were given in a very large paper by Sato Kentaro. Most of the definitions involve giving large closure properties to the $M$ used in the original large cardinal in an [[elementary embedding]] $j:V\rightarrow M$. They are very large, but [[rank into rank | rank-into-rank]] cardinals are stronger than most $n$-fold variants of large cardinals. 

Generally, the $n$-fold variant of a large cardinal axiom is the similar to the generalization of [[superstrong]] cardinals to [[superstrong|$n$-superstrong]] cardinals, [[huge]] cardinals to [[huge|$n$-huge]] cardinals, etc. More specifically, if the definition of the original axiom is that $j:V\prec M$ has critical point $\kappa$ and $M$ has some closure property which uses $\kappa$, then the definition of the $n$-fold variant of the axiom is that $M$ has that closure property on $j^n{\kappa}$.

=== $n$-fold Variants Which Are Simply the Original Large Cardinal ===

There were many $n$-fold variants which were simply different names of the original large cardinal. This was due to the fact that some n-fold variants, if only named n-variants instead, would be confusing to the reader (for example the $n$-fold extendibles rather than the [[extendible | $n$-extendibles]]). Here are a list of such cardinals:

*The '''$n$-fold superstrong''' cardinals are precisely the [[superstrong | $n$-superstrong]] cardinals
*The '''$n$-fold almost huge''' cardinals are precisely the [[huge | almost $n$-huge]] cardinals
*The '''$n$-fold huge''' cardinals are precisely the [[huge | $n$-huge]] cardinals
*The '''$n$-fold superhuge''' cardinals are precisely the [[huge | $n$-superhuge]] cardinals
*The '''$\omega$-fold superstrong''' and '''$\omega$-fold Shelah''' cardinals are precisely the [[rank-into-rank|I2]] cardinals

=== $n$-fold supercompact cardinals ===

A cardinal $\kappa$ is '''$n$-fold $\lambda$-supercompact''' iff it is the critical point of some nontrivial elementary embedding $j:V\rightarrow M$ such that $\lambda<j(\kappa)$ and $M^{j^{n-1}(\lambda)}\subset M$ (i.e. $M$ is closed under all of its sequences of length $j^{n-1}(\lambda)$). This definition is very similar to that of the [[huge | $n$-huge]] cardinals.

A cardinal $\kappa$ is '''$n$-fold supercompact''' iff it is $n$-fold $\lambda$-supercompact for every $\lambda$. Consistency-wise, the $n$-fold supercompact cardinals are stronger than the [[superstrong | $n$-superstrong]] cardinals and weaker than the $(n+1)$-fold strong cardinals. In fact, if an $n$-fold supercompact cardinal exists, then it is consistent for there to be a proper class of $n$-superstrong cardinals.

It is clear that the $n+1$-fold $0$-supercompact cardinals are precisely the [[huge|$n$-huge]] cardinals. The $1$-fold supercompact cardinals are precisely the [[supercompact]] cardinals. The $0$-fold supercompact cardinals are precisely the [[measurable]] cardinals.

=== $n$-fold strong cardinals ===

A cardinal $\kappa$ is '''$n$-fold $\lambda$-strong''' iff it is the critical point of some nontrivial elementary embedding $j:V\rightarrow M$ such that $\kappa+\lambda<j(\kappa)$ and $V_{j^{n-1}(\kappa+\lambda)}\subset M$.

A cardinal $\kappa$ is '''$n$-fold strong''' iff it is $n$-fold $\lambda$-strong for every $\lambda$. Consistency-wise, the $(n+1)$-fold strong cardinals are stronger than the $n$-fold supercompact cardinals, equivalent to the $n$-fold extendible cardinals, and weaker than the $(n+1)$-fold Woodin cardinals. More specifically, in the rank of an (n+1)-fold Woodin cardinal there is an $(n+1)$-fold strong cardinal.

It is clear that the $(n+1)$-fold $0$-strong cardinals are precisely the [[superstrong|$n$-superstrong]] cardinals. The $1$-fold strong cardinals are precisely the [[strong]] cardinals. The $0$-fold strong cardinals are precisely the [[measurable]] cardinals.

=== $n$-fold extendible cardinals ===

For ordinal $η$, class $F$, positive natural $n$ and $κ+η<κ_1<···<κ_n$:
* Cardinal $κ$ is '''$n$-fold $η$-extendible for $F$''' with targets $κ_1,...,κ_n$ iff there are $κ+η=ζ_0<ζ_1<···<ζ_n$ and an iteration sequence $\vec e$ through $〈(V_{ζ_i},F∩V_{ζ_i})|i≤n〉$ with $\mathrm{crit}(\vec e)=κ$, and $e_{0,i}(κ)=κ_i$.
* Cardinal $κ$ is '''$n$-fold extendible for $F$''' iff, for every $η$, $κ$ is $n$-fold $η$-extendible for $F$.
* Cardinal $κ$ is '''$n$-fold extendible''' iff it is $n$-fold extendible for $\varnothing$.

$n$-fold extendible cardinals are precisely $n+1$ strong cardinals.

$n$-fold $1$-extendibility is implied by $(n+1)$-fold $1$-strongness and implies $n$-fold superstrongness.

=== $n$-fold Woodin cardinals ===

A cardinal $\kappa$ is '''$n$-fold Woodin''' iff for every function $f:\kappa\rightarrow\kappa$ there is some ordinal $\alpha<\kappa$ such that $\{f(\beta):\beta<\alpha\}\subseteq\alpha$ and $V_{j^{n}(f)(j^{n-1}(\alpha))}\subset M$. Consistency-wise, the $(n+1)$-fold Woodin cardinals are stronger than the $(n+1)$-fold strong cardinals, and weaker than the $(n+1)$-fold Shelah cardinals. Specifically, in the rank of an $(n+1)$-fold Shelah cardinal there is an $(n+1)$-fold Woodin cardinal, and every $(n+1)$-fold Shelah cardinal is also an $(n+1)$-fold Woodin cardinal.

The $2$-fold Woodin cardinals are precisely the [[Vopenka|Vopěnka]] cardinals (therefore precisely the [[Woodin#Shelah|Woodin for supercompactness]] cardinals). In fact, the $n+1$-fold Woodin cardinals are precisely the $n$-fold Vopěnka cardinals. The $1$-fold Woodin cardinals are precisely the [[Woodin]] cardinals.

''(More to be added)''

== $\omega$-fold variants ==

The $\omega$-fold variant is a very strong version of the $n$-fold variant, to the point where they even beat some of the [[rank-into-rank]] axioms in consistency strength. Interestingly, they follow a somewhat backwards pattern of consistency strength relative to the original double helix. For example, $n$-fold strong is much weaker than $n$-fold Vopěnka (the jump is similar to the jump between a [[strong]] cardinal and a [[Vopenka|Vopěnka]] cardinal), but $\omega$-fold strong is much, much stronger than $\omega$-fold Vopěnka.

=== $\omega$-fold extendible ===

For ordinal $η$ and class $F$:
* Cardinal $κ$ is '''$ω$-fold $η$-extendible for $F$''' iff there are $κ+η=ζ_0<ζ_1<ζ_2<...$ and an iteration sequence $\vec e$ through $〈(V_{ζ_i},F∩V_{ζ_i})|i∈ω〉$ with $\mathrm{crit}(\vec e)=κ$, and $e^{(1)}(κ)>κ+η$.
* Cardinal $κ$ is '''$ω$-fold extendible for $F$''' iff, for every $η$, $κ$ is $ω$-fold $η$-extendible for $F$.
* Cardinal $κ$ is '''$ω$-fold extendible''' iff it is $ω$-fold extendible for $\varnothing$.

Results:
* An $ω$-fold extendible cardinal $κ$ is the $κ$-th element of the class of the critical points of all witnesses of I3.
* If $κ$ is a regular cardinal and $F⊂V_κ$, we have$\{α<κ|(V_κ,F)\models \text{“$α$ is $ω$-fold extendible for $F$”}\}∈F^{(ω)}_{Vop,κ}$.
* If there is an $ω$-fold Vopěnka cardinal, then the existence of a proper class of $ω$-fold extendible cardinals is consistent.

=== $\omega$-fold Vopěnka ===
Definition:
* A '''set $X$''' is '''$ω$-fold Vopěnka''' for a cardinal $κ$ iff, for every $κ$-natural sequence $〈\mathcal{M}_α|α<κ〉$, there are an increasing sequence $〈α_n|n∈ω〉$ with $α_n<κ$ and an iteration sequence $\vec e$ through $〈\mathcal{M}_{α_n}|n∈ω〉$ such that $\mathrm{crit}(\vec e)∈X$ .
* A '''cardinal''' $κ$ is '''$ω$-fold Vopěnka''' iff $κ$ is regular and $κ$ is $ω$-fold Vopěnka for $κ$.
* $F^{(ω)}_{Vop,κ}=\{X∈\mathcal{P}(κ)|\text{κ \\ X is not $ω$-fold Vopěnka for $κ$}\}$.

Results:
* An $ω$-fold superstrong cardinal is the $κ$-th $ω$-fold Vopěnka cardinal.
* The critical point $κ$ of a witness of $IE^ω$ is the $κ$-th $ω$-fold Vopěnka cardinal.

=== $\omega$-fold Woodin ===

A cardinal $\kappa$ is '''$\omega$-fold Woodin''' iff for every function $f:\kappa\rightarrow\kappa$ there is some ordinal $\alpha<\kappa$ such that $\{f(\beta):\beta<\alpha\}\subseteq\alpha$ and $V_{j^{\omega}(f)(\alpha))}\subset M$.

Consistency-wise, the existence of an $\omega$-fold Woodin cardinal is stronger than the [[rank-into-rank|I2]] axiom, but weaker than the existence of an $\omega$-fold strong cardinal. In particular, if there is an $\omega$-fold strong cardinal $\kappa$ then $\kappa$ is $\omega$-fold Woodin and has $\kappa$-many $\omega$-fold Woodin cardinals below it, and $V_\kappa$ satisfies the existence of a proper class of $\omega$-fold Woodin cardinals.

=== $\omega$-fold strong ===

A cardinal $\kappa$ is '''$\omega$-fold $\lambda$-strong''' iff it is the critical point of some nontrivial elementary embedding $j:V\rightarrow M$ such that $\kappa+\lambda<j(\kappa)$ and $V_{j^\omega(\kappa+\lambda)}\subset M$.

$\kappa$ is '''$\omega$-fold strong''' iff it is $\omega$-fold $\lambda$-strong for every $\lambda$.

Consistency-wise, the existence of an $\omega$-fold strong cardinal is stronger than the existence of an $\omega$-fold Woodin cardinal and weaker than the assertion that there is a $\Sigma_4^1$-elementary embedding $j:V_\lambda\prec V_\lambda$ with an uncountable critical point $\kappa<\lambda$ (this is a weakening of the [[rank-into-rank|I1]] axiom known as $E_2$). In particular, if there is a cardinal $\kappa$ which is the critical point of some elementary embedding witnessing the $E_2$ axiom, then there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]] over $\kappa$ which contains the set of all cardinals which are $\omega$-fold strong in $V_\kappa$ and therefore $V_\kappa$ satisfies the existence of a proper class of $\omega$-fold strong cardinals.

{{References}}