# Superextendibles defined analogously to superstrong cardinals: Where are they consistency strength-wise?

A common trend in large cardinal axioms dealing with critical points of elementary embeddings from $$V$$ into a transitive class $$M$$ is to make some large cardinal axiom with an ordinal parameter, similar to "$$M$$ is closed under $$\theta$$-sequences" or "$$V_\theta$$ is a subset of $$M$$". The resulting cardinals from these definitions are the $$\theta$$-supercompacts and the $$\theta$$-strongs.

After this, expansions could be made to make the cardinal even larger; specifically, by taking that parameter of $$M$$ and making it $$j^n(\kappa)$$. When doing this to $$\theta$$-strong cardinals, one achieves $$n$$-superstrong cardinals. When doing this to supercompact cardinals, one achieves $$n$$-huge cardinals.

Of course, a natural question at this point is to see what happens when one does it to $$\eta$$-extendible cardinals. Here are the results:

Let $$\kappa$$ be $$n$$-superextendible if there is some ordinal $$\eta$$ such that $$\kappa$$ is the critical point of some nontrivial elementary embedding $$j:V_\theta\prec V_\eta$$ such that $$\theta>j^n(\kappa)$$.

So far, here are the results:

• The $$0$$-superextendible cardinals are precisely the $$0$$-extendible cardinals
• I3 cardinals are $$n$$-superextendible for every $$n$$ (this property could be called $$\omega$$-superextendible)
• The $$1$$-superextendible cardinals, if one lets an arbitrary such be $$\kappa$$, are $$\kappa$$-extendible. Of course, every $$1$$-superextendible cardinal is a supercompact cardinal and is, in fact, a stationary limit of such due to this property.
• If a cardinal is $$n+1$$-superextendible it is $$n$$-superextendible

These properties are, yes, very obvious. Are there any other possibly less obvious properties that I am missing? Where do these cardinals fall under consistency strength?

• The strength of these $n$-fold large cardinal axioms have been calibrated in the paper ac.els-cdn.com/S0168007207000127/… Oct 21, 2017 at 23:51
• @Joseph_Van_Name That definition is slightly stronger than this one I believe. Correct me if I'm wrong though Oct 24, 2017 at 1:46

Let $$\kappa$$ be $$n$$-Superextendible if there is some ordinal $$\eta$$ such that $$\kappa$$ is the critical point of some nontrivial elementary embedding $$j:V_\theta\prec V_\eta$$ such that $$\theta>j^n(\kappa)$$.

Without loss of generality $$\theta=j^n(\kappa)+1$$ and $$\eta=j^{n+1}(\kappa)+1$$ (otherwise take $$j^n\upharpoonright V_{\kappa+1}$$). Thus this is the same property that Sato 2007 (Joseph Van Name's link doesn't work) would call $$n+1$$-fold 1-extendible. As Master's answer points out, $$n+1$$-fold 1-extendible cardinals are also called n-huge*. By this Mathoverflow answer by Joel David Hamkins, $$n$$-fold 1-extendible cardinals are $$n$$-superstrong and by proposition 8.5 of Sato's paper, if $$\kappa$$ is $$n$$-fold $$2^\kappa$$-supercompact, it is a limit of $$n$$-fold 1-extendible cardinals.

When doing this to $$\theta$$-Strong cardinals, one achieves n-Superstrong cardinals. When doing this to Supercompact cardinals, one achieves n-Huge cardinals.

A weaker large cardinal property analogous to those has $$\theta \ge j^n(\kappa)$$ or, without loss of generality, $$\theta=j^n(\kappa)$$. This property can be called $$n+1$$-fold 0-extendible. Solovay, Reinhardt and Kanamori 1978 calls the assertion that there exists a 2-fold 0-extedibility embedding $$A_2$$. By theorems 8.1, 8.2 and 8.3 of that paper, 2-fold 0-extedible cardinals are between almost huge and huge cardinals in the large cardinal hierarchy.

These large cardinals actually already have a name. They are called $$n$$-huge* cardinals.

Theorem: If $$\kappa$$ is $$n$$-huge*, then $$\kappa$$ is $$n$$-huge and a limit of $$n$$-huge cardinals.

Proof. Let $$j: V_\alpha\rightarrow V_\beta$$ be an $$n$$-hugeness* embedding, with $$\lambda=j^n(\kappa)$$. Furthermore, let $$Y=\{j(\alpha)|\alpha\lt\lambda\}$$ and $$D=\{X\subseteq\lambda|Y\in j(X)\}$$. It is immediately a normal measure. Now note that $$ot(Y\cap j^{i+1}(\kappa))=j^i(\kappa)$$. For the rest, note that $$D\in V_\beta$$.■

Theorem: If $$\kappa$$ is $$n+1$$-huge, then $$\kappa$$ is $$n$$-huge* and a limit of $$n$$-huge* cardinals.

Proof. Let $$j: V\rightarrow M$$ be an $$n+1$$-hugeness embedding, with $$\lambda=j^n(\kappa)$$, and let $$k=j\restriction V_{\lambda+1}$$. Then $$k\in M$$ and so $$M\vDash\kappa\text{ is }n\text{-huge*}$$ and so $$M\vDash V_{j(\lambda)}\vDash(\kappa\text{ is }n\text{-huge*})$$, but $$V_{j(\lambda)}\subseteq M$$ and so $$\kappa$$ is actually $$n$$-huge*. A similar argument goes for the rest .■

Refrences: Virtual large cardinals, Victoria Gitman and Ralf Schindler.