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# On strong forms of extendibility

1. Abstract

The large cardinal program studies the properties of large cardinals; cardinals with large cardinal properties. Many large cardinal properties have been discovered: Inaccessibility; Mahloness, weak compactness, indescribability, partition properties, $$0^\sharp$$, measurability, $$0^\dagger$$, and the hierarchy of huge cardinals. The last collection of properties is particularly interesting. The hierarchy of huge cardinals consists of critical point of non-trivial elementary embeddings $$V$$ into inner models $$M$$, with strong closure properties. As was shown by Sato Kentaro, every single large cardinal between measurability and $$I3$$ can be slotted into the hierarchy of $$n-$$fold variants. At the bottom of the hierarchy we have $$n-$$fold superstrongness, $$n-$$fold strongness, then $$n-$$fold Woodiness, equivalent to the existence of for every $$A$$, some $$\alpha\lt\kappa$$ $$n-$$fold $$\lt\kappa-$$strong cardinal. Then there is $$n-$$fold supercompactness Then there is $$n-$$fold extendibility=$$n+1-$$fold strongness, and $$n-$$fold Vopěnkaness, equivalent to the existence of for every $$A$$, some $$\alpha\lt\kappa$$ $$n-$$fold $$\lt\kappa-$$extendibile for $$A$$. Therefore $$n-$$fold Vopěnkaness=$$n+1-$$Woodiness. Then there is $$n-$$fold Shelah cardinals, and the hierarchy of huge variants.

The study of these huge variants of large cardinals is a very promising field, with the possibility to bring forth dozens of new interesting results. Already some incredible connections between Woodiness and Vopěnkaness have been discovered, and this field brings the possibility of thousands of new results. Unfortunately, in its current state, to some degree in $$n-$$fold variants, and very much so at the level of Choiceless cardinals, runs into the problem that many simple problems are open and seem to have no immediate tools to solve them. We give some examples at the end of this problem. The goal of this projected series of papers is to first give a detailed analysis of cardinals between measurability and $$I3$$, and then analyze the cardinals from $$I3$$ through $$I0$$ and then up through the Reinhardt hierarchy.

The idea behind the formulation of extremely large cardinals (A sort of “Huge cardinal program”) is, as described in Large Cardinals beyond Choice, is in a sense anathema to the Ultimate$$-L$$ program. The Ultimate$$-L$$ program has the ultimate hope of finding some core model, the so called Ultimate$$-L$$, a core model for all large cardinals. On the other, extremely large cardinals are based on the idea that for any core model $$M$$, you can find the largest large cardinal consistent with $$V=M$$, and take “$$+1$$” to get a stronger large cardinal beyond $$V=M$$. This indicates a hierarchy of core models, or Ultimate$$-L$$s. For example, Ultimate$$-L_0=L$$ and Ultimate$$-L_1=L[D]$$, for some normal measure $$D$$ over a measurable $$\kappa$$. In this case, the largest large cardinal compatible with $$V=L$$ would be $$\kappa\rightarrow(\alpha)^{\lt\omega}$$ for all countable $$\alpha$$, and the transcendence principle is $$0^\sharp$$, which is below $$\kappa\rightarrow(\omega_1)^{\lt\omega}$$. Similarly, the largest large cardinal compatible with $$V=L[D]$$ would be measurability, $$\kappa$$ is measurable, and the transcendence principle is $$0^\dagger$$, which is below two measurables. So far, Core models have reached the level of infinitely many Woodin cardinals, but no further.

Theorem (Jensen’s Covering Theorem): Exactly one of the following hold:

$$(i)$$ For every singular cardinal $$\gamma$$, $$\gamma$$ is singular in $$L$$ and $$(\gamma^+)^L=\gamma^+$$.

$$(ii)$$ For every uncountable cardinal $$\gamma$$, $$\gamma$$ is inaccessible in $$L$$.

In the first situation, $$L$$ is “close” to $$V$$, in that it correctly computes singular cardinals and successors; it covers $$V$$. In the second instance, $$L$$ is very “far” from $$V$$, to the point every uncountable cardinal $$\gamma$$ is inaccessible in $$L$$. This is called “$$0^\sharp$$ exists.” The hope for Ultimate$$-L$$ is that, in addition to satisfying certain combinatorical propositions such as $$GCH$$, does not suffer from the same transcendence principle. In some formulations, Ultimate$$-L\subseteq HOD$$. However, $$HOD$$ itself suffers from a similar problem.

Theorem (The $$HOD$$ Dichotomy): Let $$\kappa$$ be extendible. Exactly one of the following hold:

$$(i)$$ For every singular cardinal $$\gamma\gt\kappa$$, $$\gamma$$ is singular in $$HOD$$ and $$(\gamma^+)^{HOD}=\gamma^+$$.

$$(ii)$$ For every cardinal $$\gamma\ge\kappa$$, $$\gamma$$ is measurable in $$HOD$$.

This indicates that there is some general transcendence principle, $$0^‡$$, that is transcendent over $$HOD$$ and possibly Ultimate$$-L$$. The hope is that, given such an extendible, Ultimate$$-L$$ would be a weak extender model for the supercompactness of said cardinal. Unfortunately, this may not be possible. The goal of this series of papers is to investigate what happens when the Ultimate$$-L$$ program fails. The goal is to analyze first large cardinals slightly below $$I1$$ and $$I0$$; namely, the double helix, on then upwards and onwards into large cardinals beyond $$V=$$Ultimate$$-L$$. Hopefully, we can discover the exact location of this transcendence principle $$0^‡$$, and then go forward into larger and larger large cardinals. The idea is that because of how close the double helix is to $$0^‡$$, an analysis of $$0^‡$$ is impossible without a detailed knowledge of the double helix. We will discuss $$0^‡$$ and large cardinals beyond Choice a little bit more at the end of that paper.

The study of huge cardinals has been advanced considerably by Sato Kentaro in Double helix in large large cardinals and iteration of elementary embeddings, who researched the double helix significantly. Since then, new large cardinals have been created. Ultrahuge, and huge* cardinals were discovered as strong forms of extendibility. In fact, many large large cardinals here an be thought of as strong forms of extendibility (Hence the title).

Even Reinhardt cardinals were originally formulated as stronger and large extendibles, and $$I3(\kappa,\lambda)$$ and $$I1(\kappa,\lambda)$$ can be though of as strong form of $$\lambda-$$extendibility and $$\lambda+1-$$extendibility respectively in which the target model $$V_\zeta$$ has $$\zeta=\lambda$$, or in the first case as a weakening of $$\omega-$$fold extendible cardinals. Joan Bagaria also discovered the $$C^{(n)}$$ cardinals, and discovered some incredible connections between extendibility and Vopěnka’s principle. The goal of this paper is to generalize these results from the first helix, just as Kentaro’s original paper did. Part 1 is the abstract.

In part 2, we introduce the main notions that will be relevant through out the paper: Ultrahugeness, hyperhugeness, and hugeness*. We establish that every hyperhuge cardinal is ultrahuge, and furthermore that every superhuge* cardinal is hyperhuge. We also establish hyper $$n-$$hugeness is equivalent to $$n+1-$$fold supercompactness. Then we give an analysis of equivalent variants of superhugeness*. Finally, we bound the the consistency strength of super $$n-$$hugeness* below almost $$n+1-$$hugeness.

In part 3, we show that a cardinal is superhuge* if and only if it satisfies a higher analogue of the embedding characterization, $$j: V\rightarrow M$$, of extendibility. We then introduce strong variants of superhugeness, ultrahugeness, hyperhugeness, and superhugeness*: Stationarily superhuge, stationarily ultrahuge, stationarily hyperhuge, and stationarily superhuge*. We bound the the consistency strength of stationary super $$n-$$hugeness* below almost $$n+1-$$hugeness. We also show they obey a linear hierarchy in terms of implication strength, so that each stationarily ultrahuge is stationarily superhuge; each stationarily hyperhuge is stationarily ultrahuge, and each stationarily superhuge* cardinal is stationarily hyperhuge. Finally, we show that stationary superhugeness strongly implies superhugeness.

In part 4, we reference $$C^{(n)}-$$cardinals, and show a cardinal is $$C^{(m)}-$$superhuge* if and only it is $$C^{(m)}-2-$$fold extendible. We also show a cardinal is $$C^{(n)}-$$superhuge* if and only if it it satisfies a higher analogue of the embedding characterization, $$j: V\rightarrow M$$, of extendibility, with target in $$C^{(n)}$$. We then give a characterization of $$C^{(n)}-$$variants of certain types of embedding properties. We show that, if $$C$$ is club, there is some proper class of models $$\mathfrak M_\alpha$$, such that for any non-trivial elementary embedding $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\kappa$$, $$\kappa\in C$$. We then give a characterization of hyper $$n-$$huge cardinals in terms of a higher analogue of Magidor’s characterization. We then show some higher analogues of Joan Bagaria’s theorems on $$C^{(n)}-$$cardinals and Vopěnka’s principle. First we show, an $$n-$$fold variant of Vopěnka’s principle restricted to $$\Pi_1$$ formulas with parameters of rank $$\le\alpha$$ implies the existence of $$n-$$fold supercompact cardinal $$\gt\alpha$$. We also show the converse: The existence of $$n-$$fold supercompact cardinal $$\ge\alpha$$ implies an $$n-$$fold variant of Vopěnka’s principle restricted to $$\Sigma_2$$ formulas with parameters of rank $$\lt\alpha$$. Then we show the least cardinal $$\kappa$$ such that $$VP_n(\kappa,\Pi_1)$$ is $$n-$$fold supercompact, and $$VP_n(\kappa,\Pi_1)$$ if and only if $$\kappa$$ is $$n-$$fold supercompact or a limit of such cardinals.

Then we show, for $$m\gt 0$$, an $$n-$$fold variant of Vopěnka’s principle restricted to $$\Pi_{m+1}$$ formulas with parameters of rank $$\le\alpha$$ implies the existence of $$C^{(m)}-n-$$fold extendible cardinal $$\gt\alpha$$. We also show the converse: The existence of $$C^{(m)}-n-$$fold extendible cardinal $$\ge\alpha$$ implies an $$n-$$fold variant of Vopěnka’s principle restricted to $$\Sigma_{m+2}$$ formulas with parameters of rank $$\lt\alpha$$. Then we show the least cardinal $$\kappa$$ such that $$VP_n(\kappa,\Pi_{m+1})$$ is $$C^{(m)}-n-$$fold extendible , and $$VP_n(\kappa,\underset{\sim}{\Pi}_1)$$ if and only if $$\kappa$$ is $$C^{(m)}-n-$$fold extendible or a limit of such cardinals.

We continue by showing a collection of three equivalent characterizations of Vopěnka’s principle, and three equivalent characterizations of Vopěnka cardinals. We then show that a cardinal is $$n+1-$$fold Woodin for supercompactness if and only if it is $$n+1-$$fold Vopěnka. We then show that a $$C^{(n+1)}-P$$ cardinal is typically much larger than a $$C^{(n)}-P$$ cardinal. Finally, we establish the transcendence of $$I3$$ over $$C^{(n)}-$$variants.

In part 5, we reference that famous theorems that if $$\kappa$$ is supercompact, then $$\kappa\in C^{(2)}$$ and if the $$GCH$$ holds below $$\kappa$$, then it holds everywhere. We give general forms of this for hyper $$n-$$huge, and show almost ultra $$n-$$huge cardinals upward reflect cardinals $$\kappa$$ such that $$o(\kappa)=(2^\kappa)^+$$.

In part 6, we discuss the limit of the double helix; the rank-into-rank axioms. First, we discuss the hierarchy of $$C^{(n)}$$ variants of $$I3$$ and establish the transcendence of $$C^{(n+1)}$$ variants over $$C^{(n)}$$ variants with arbitrarily large target. Then we strengthen the known theorems on the size and shape of the iterability hierarchy above $$I3$$. We show $$IE^{\delta+\omega}$$ implies the consistency of, for every a $$C^{(n)}$$ variant of $$IE^\delta$$ in a strong way. We then discuss discuss the hierarchy of $$C^{(n)}$$ variants of $$I2$$ and establish the transcendence of $$C^{(n+1)}$$ variants over $$C^{(n)}$$ variants with arbitrarily large target. We continue, discussing the hierarchy of $$C^{(n)}$$ variants of Laver’s $$E$$ series and establish the transcendence of $$C^{(n+1)}$$ variants over $$C^{(n)}$$ variants with arbitrarily large targets. We also establish the transcendence of $$\omega-$$fold strong cardinals over $$I2$$, and of $$E_1^2$$ over $$\omega-$$fold strong cardinals. We then establish the transcendence of $$I1$$=$$\omega-$$huge* over $$I2$$. We then discuss discuss the hierarchy of $$C^{(n)}$$ variants of $$I1$$ and establish the transcendence of $$C^{(n+1)}$$ variants over $$C^{(n)}$$ variants with arbitrarily large target.

In part 7, first we show, an $$\omega-$$fold variant of Vopěnka’s principle restricted to $$\Pi_1$$ formulas with parameters of rank $$\le\alpha$$ implies the existence of $$\omega-$$fold supercompact cardinal $$\gt\alpha$$. We also show the converse: The existence of $$\omega-$$fold supercompact cardinal $$\ge\alpha$$ implies an $$\omega-$$fold variant of Vopěnka’s principle restricted to $$\Sigma_2$$ formulas with parameters of rank $$\lt\alpha$$. Then we show the least cardinal $$\kappa$$ such that $$VP_\omega(\kappa,\Pi_1)$$ is $$n-$$fold supercompact, and $$VP_n(\kappa,\Pi_{m+1})$$ if and only if $$\kappa$$ is $$\omega-$$fold supercompact or a limit of such cardinals.

Then we show, for $$m\gt 0$$, an $$\omega-$$fold variant of Vopěnka’s principle restricted to $$\Pi_{m+1}$$ formulas with parameters of rank $$\le\alpha$$ implies the existence of $$C^{(m)}-\omega-$$fold extendible cardinal $$\gt\alpha$$. We also show the converse: The existence of $$C^{(m)}-\omega-$$fold extendible cardinal $$\ge\alpha$$ implies an $$\omega-$$fold variant of Vopěnka’s principle restricted to $$\Sigma_{m+2}$$ formulas with parameters of rank $$\lt\alpha$$. Then we show the least cardinal $$\kappa$$ such that $$VP_\omega(\kappa,\Pi_{m+1})$$ is $$C^{(m)}-n-$$fold extendible, and $$VP_\omega(\kappa,\underset{\sim}{\Pi}_1)$$ if and only if $$\kappa$$ is $$C^{(m)}-\omega-$$fold extendible or a limit of such cardinals. We also show that the $$\omega-$$fold Woodin for supercompactness cardinals at the same as the $$\omega-$$fold Vopěnka cardinals, and an alternative characterization of $$\omega-$$fold Woodin cardinals. Then we discuss $$I1$$ tower cardinals, and give a characterization of them similar to that of Vopěnka’s principle.

In a sense, Choice can be thought of as a principle similar to $$V=L$$, but less extreme, in that it implies a strong degree of structure to the universe, that doesn’t seem as natural as the other axioms. Most of the other axioms are either fundamental parts of are conception of a set, like exstensionaility, so that they seem to really be part of the definition of a set; or as Axioms of Construction, asserting that certain sets can be constructed; or the Axiom of Infinity, which can be thought of as large cardinal axiom. So what happens when we drop choice.

In part 8, we discuss the Reinhardt hierarchy. We show weakly Reinhardt cardinals are below in consistency strength, both strongly Reinhardt and $$I0$$ cardinals. Then we show that the consistency of the existence of super Reinhardt, or even superstrongly Reinhardt cardinals that are $$\Sigma_3-$$reflecting, implies the consistency of strongly Reinhardt cardinals in a strong way. We also show that if $$\kappa$$ is super Reinhardt, $$V_\kappa\prec V$$, and then we show that the consistency of the existence of super Reinhardt cardinal implies the consistency of Reinhardt cardinals being stationary in the universe. Part 9 are open questions, and part 10 is the references.

An important theorem is that if $$\kappa=\beth_\kappa$$, then $$\kappa\in C^{(1)}$$, because $$V_\kappa=H_\kappa$$. For any embedding property $$P$$, we can define $$o_P(X)$$ as the set of $$\kappa\in X$$ such that there is a normal measure $$D$$ generated by a $$P$$ embedding, such that $$X\cap\lambda\in D$$. In all instances throughout this paper, when we assert the existence of a normal measure concentrating on some large cardinal $$P$$, we can assert the existence of a filter closed under the $$P$$ operation. Similarly, if $$\kappa$$ is $$n-$$fold Vopěnka, then for each $$A\subseteq V_\kappa$$, $$\{\lambda\lt\kappa|\lambda\text{ is }C^{(\lt\omega)}-n-\text{fold extendible for }A\}\in F^{(n)}_{\text{Vop},\kappa}$$ and $$\{\lambda\lt\kappa|\lambda\text{ is }C^{(\lt\omega)}-\omega-\text{fold extendible for }A\}\in F^{(\omega)}_{\text{Vop},\kappa}$$, and the $$n+1-$$fold Woodin for supercompactnes filter is the same as $$n+1-$$fold Vopěnka filter, and $$\omega-$$fold Woodin for supercompactness filter is the same as the $$\omega-$$fold Vopěnka filter.

2. Ultrahuge, hyperhuge, and huge* cardinals.

It is known that a cardinal $$\kappa$$ is extendible if and only if there for every ordinal $$\lambda$$, $$\rho$$, $$\kappa$$ is jointly $$\lambda-$$supercompact and $$\rho-$$superstrong, i.e. there is a non-trivial elementary embedding $$j: V\rightarrow M$$ such that $$j(\kappa)\gt\lambda$$, $$M^\lambda\subseteq M$$, and $$V_\rho\subseteq M$$.

Theorem: The following are equivalent$$^1$$:

$$(i)$$ $$\kappa$$ is jointly $$\lambda-$$supercompact and $$\rho-$$superstrong for every $$\lambda$$,$$\rho$$.

$$(ii)$$ $$\kappa$$ is jointly $$\lambda-$$supercompact and $$\kappa-$$superstrong for every $$\lambda$$.

$$(iii)$$ $$\kappa$$ is extendible.

As a generalization to this, Konstantinos Tsaprounis introduced $$\lambda-$$ultrahugness$$^2$$, which is a generalization of (Super)hugeness and extendibility. $$\kappa$$ is $$\lambda-$$ultrahuge if there is a non-trivial elementary embedding $$j: V\rightarrow M$$ such that $$M^{j(\kappa)}\subseteq M$$ and $$V_{j(\lambda)}\subseteq M$$.

Definition: $$\kappa$$ is $$\lambda-$$ultra $$n-$$huge if and only if there exists a non-trivial elementary embedding $$j: V\rightarrow M$$, with critical point $$\kappa$$, $$M^{j^n(\kappa)}\subseteq M$$, and $$V_{j^n(\lambda)}\subseteq M$$.

Furthermore, $$\kappa$$ is $$n-$$huge*$$^3$$ if and only if there is some $$\alpha$$, $$\beta$$, and a non-trivial elementary embedding $$j:V_\alpha\rightarrow V_\beta$$ with critical point $$\kappa$$, such that $$\alpha\gt j^n(\kappa)$$. $$j(\kappa)$$ is called the target. For the purposes of the rest of the paper, $$\kappa$$ is super $$n-$$huge* if and only if it is $$n-$$huge* with arbitrarily large target.

Definition: $$\kappa$$ is $$\lambda-$$hyper $$n-$$huge if and only if there is a non-trivial elementary embedding $$j: V\rightarrow M$$ with critical point $$\kappa$$, $$j(\kappa)\gt\lambda$$, and $$M^{j^n(\lambda)}\subseteq M$$. $$\kappa$$ is hyper $$n-$$huge if and only if $$\kappa$$ is $$\lambda-$$hyper $$n-$$huge for every $$\lambda$$.

Hence, if $$\kappa$$ is $$\lambda-$$hyper $$n-$$huge, then $$\kappa$$ is $$\lambda-$$ultra $$n-$$huge. The $$\lambda-$$hyper $$n-$$huge cardinals are precisely the $$n+1-$$fold $$\lambda-$$supercompact cardinals, and so are beyond the standard double helix. $$[\lambda]^{\kappa}=\{S\subseteq\lambda||\lambda|\le\kappa\}$$.

Theorem (Sato Kentaro$$^4$$): $$\kappa$$ is $$\lambda-$$hyper $$n+1-$$huge if and only if there exists an increasing sequence $$\{\lambda_i|i\le n\}$$ such that $$\lambda_0=\lambda$$, and there exists a normal, fine measure on $$D$$ on $$[\lambda_n]^{\kappa’}$$, such that $$\lambda_n\gt\kappa’$$ and:

$$\{x\in [\lambda_n]^{\kappa’}|ot(x\cap\lambda_{i+1})=\lambda_i\}\in D$$, for every $$i\lt n$$.

$$\{x\in [\lambda_n]^{\kappa’}|ot(\lambda\cap x)\lt\kappa\}\in D$$.

Theorem: Let $$\lambda\ge\kappa$$ be regular. If $$\kappa$$ is $$\lambda-$$hyper $$n-$$huge, and $$\alpha$$ is $$\lt\kappa-$$hyper $$n-$$huge, then $$\alpha$$ is $$\lambda-$$hyper $$n-$$huge.

$$Proof.$$ Let $$j: V\rightarrow M$$ be a $$\lambda-$$hyper $$n-$$hugeness embedding. Then, as $$j(\alpha)=\alpha$$, $$Ult_D\vDash(\alpha\text{ is }\lt j(\kappa)-\text{hyper }n-\text{huge})$$. Therefore, there is a $$\lambda-$$hyper $$n-$$hugeness measure $$j(D)$$ on $$[j^n(\lambda)]^{\kappa’}$$. But $$|[j^n(\lambda)]^{\kappa’}|=j^n(\lambda)$$ and so $$D$$ is an actual $$\lambda-$$hyper $$n-$$hugeness measure.◼

Theorem: If $$\kappa$$ is super $$n-$$huge*, then $$\kappa$$ is hyper $$n-$$huge and there is a normal measure $$D$$ such that $$\{\lambda\lt\kappa|\lambda\text{ is hyper }n-\text{huge}\}\in D$$.

$$Proof.$$ To get that $$\kappa$$ is hyper $$n-$$huge is easy, in the context of the next part. To get a measure concentrating on such cardinals, let $$\alpha=\beth_\alpha$$, and $$j: V_\alpha\rightarrow V_\beta$$ an $$n-$$hugeness* embeddding. Then $$\alpha$$ is $$\Sigma_1-$$correct and so $$V_\alpha\vDash\kappa\text{ is supercompact}$$, and also $$V_\beta\vDash\kappa\text{ is }j(\kappa)-\text{hyper }n-\text{huge}$$ and so $$V_\beta\vDash V_{j(\kappa)}\vDash\kappa\text{ is hyper }n-\text{huge}$$.

Let $$U=\{\lambda\lt\kappa|\lambda\text{ is }\lt\kappa-\text{hyper }n-\text{huge}\}$$ and $$D$$ the measure generated by $$j$$. As $$\kappa$$ is hyper $$n-$$huge, each such $$\lambda\in U$$ is hyper $$n-$$huge. Furthermore, by the above $$U\in D$$.◼

It seems natural to ask then, whether making small alteration to the definition of super $$n-$$hugeness* would make a difference; what if we drop that $$j(\kappa)\gt\lambda$$, simply that $$\alpha\gt\lambda$$? What if we add $$\alpha\lt j^{n+1}(\kappa)$$?

Theorem: The following are equivalent for $$n\gt 0$$:

$$(i)$$ $$\kappa$$ is super $$n-$$huge*.

$$(ii)$$ For every $$\lambda$$, there is an $$\alpha\gt\lambda$$ and an $$n-$$hugeness* embedding $$j:V_\alpha\rightarrow V_\beta$$ with critical point $$\kappa$$.

$$(iii)$$ For every $$\lambda$$, there is an $$\alpha\gt\lambda$$ and an $$n-$$hugeness* embedding $$j:V_\alpha\rightarrow V_\beta$$ with critical point $$\kappa$$, and $$j^{n+1}(\kappa)\gt\alpha$$.

$$(iiii)$$ For every $$\lambda$$, there is an $$\alpha\gt\lambda$$ and an $$n-$$hugeness* embedding $$j:V_\alpha\rightarrow V_\beta$$ with critical point $$\kappa$$, $$j^{n+1}(\kappa)\gt\alpha$$, and $$j(\kappa)\gt\lambda$$.

$$Proof.$$ $$(iiii)\rightarrow (iii)\rightarrow (ii)$$ is trivial.

$$(ii)\rightarrow (i)$$. Let $$\alpha\gt\lambda$$, and let $$\phi(i)$$ be the assertion that there is an $$n-$$hugeness* embedding $$k: V_\eta\rightarrow V_\zeta$$ such that $$k(\kappa)=j^{i+1}(\kappa)$$. We establish that $$\phi(n)$$ holds by induction, the base case being trivial.

Let $$k: V_\eta\rightarrow V_\zeta$$, and as $$i\lt m$$, we can apply $$j(k)$$ to get some $$k’: V_{j(\eta)}\rightarrow V_{j(\zeta)}$$ with $$k'(j(\kappa))=j^{i+2}(\kappa)$$. But then, $$j'(x)=k'(j(x))$$ gives us the necessary form of embedding.

Then if $$\alpha\lt j^{n+1}(\kappa)$$, the proof is complete. Else $$j(\alpha)=\alpha$$, in which case a rank-into-rank argument gives the rest.

$$(i)\rightarrow (iiii)$$. Let $$j(\kappa)\gt\lambda$$. Then $$j’=j\restriction V_{j^n(\kappa)+1}$$ is an $$n-$$hugeness* embedding $$j’: V_{j^n(\kappa)+1}\rightarrow V_{j^{n+1}(\kappa)+1}$$. As $$j^{n+1}(\kappa)\gt j^n(\kappa)$$, this is the requite type of embedding.◼

Theorem: If $$\kappa$$ is almost $$n+1-$$huge, for $$n\gt 0$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\lambda\lt\kappa|V_\kappa\vDash\lambda\text{ is super }n-\text{huge*}\}\in D$$.

$$Proof.$$ Temporarily denote by $$\alpha\rightarrow(\beta)$$ the assertion $$\alpha$$ is $$n-$$huge* with target $$\beta$$. Let $$j: V\rightarrow M$$ be an almost $$n+1-$$hugeness embedding, and let $$k=j\restriction V_{j^n(\kappa)+1}$$, and so $$\kappa$$ is $$n-$$huge*.

Let $$D$$ be the measure generated by $$j$$. By closure $$k\in M$$, and so $$M\vDash \kappa\rightarrow j(\kappa)$$. Therefore $$\{\lambda\lt\kappa|\lambda\rightarrow(\kappa)\}\in D$$. By a similar argument, for each such $$\lambda$$ $$M\vDash\lambda\rightarrow\kappa$$ and so $$\{\alpha\lt\kappa|\lambda\rightarrow(\alpha)\}\in D$$, and so is unbounded.◼

3. Strong forms of extendibility and stationarily superhuge cardinals

Definition: $$\kappa$$ is jointly $$\lambda-$$hyper $$n-$$huge and $$\rho-$$super $$n-$$strong if and only if there is a non-trivial elementary embedding $$j: V\rightarrow M$$ with critical point $$\kappa$$, $$M^{j^n(\lambda)}\subseteq M$$, $$V_{j^{n+1}(\rho)}\subseteq M$$.

$$\kappa$$ is jointly hyper $$n-$$huge and $$n+1-$$strong if and only if it is jointly $$\lambda-$$hyper $$n-$$huge and $$\rho-$$super $$n-$$strong for every $$\lambda$$,$$\rho$$. $$\kappa$$ is jointly hyper $$n-$$huge and super $$n-$$strong if and only if it is $$\kappa$$ is jointly $$\lambda-$$hyper $$n-$$huge and $$\kappa-$$super $$n-$$strong.

Theorem: The following are equivalent:

$$(i)$$ $$\kappa$$ is jointly hyper $$n-$$huge and $$n+1-$$strong.

$$(ii)$$ $$\kappa$$ is jointly hyper $$n-$$huge and super $$n-$$strong.

$$(iii)$$ $$\kappa$$ is super $$n-$$huge*.

$$Proof.$$ $$(i)\rightarrow (ii)$$. This is trivial.

$$(ii)\rightarrow (iii)$$. Let $$\lambda\gt\kappa$$ be in $$C^{(3)}$$. Let $$j: V\rightarrow M$$ witness that $$\kappa$$ is jointly hyper $$n-$$huge and $$\lambda-$$super $$n-$$strong. Then $$j\restriction V_{j^n(\kappa)}: V_{j^n(\lambda)}\rightarrow V_{j^{n+1}(\lambda)}^M$$ is a non-trivial elementary embedding, and $$j^n(\lambda)\gt j^n(\kappa)$$, so that $$M\vDash\kappa\text{ is }n-\text{huge* with target above }\lambda$$. Then, as $$j(\kappa)$$ is inaccessible $$V_{j(\kappa)}\vDash\lambda\in C^{(2)}$$ and so $$M\vDash V_{j(\kappa)}\vDash\lambda\in C^{(2)}$$, and as $$M\vDash j(\kappa)\in C^{(2)}$$, $$M\vDash\lambda\in C^{(2)}$$ and so $$M\vDash V_\lambda\vDash\kappa\text{ is }n-\text{huge* with target above }\lambda$$ and so $$V_\lambda\vDash\kappa\text{ is }n-\text{huge* with target above }\lambda$$, and as $$\lambda\in C^{(3)}$$, the same holds in $$V$$.

$$(iii)\rightarrow (i)$$. Let $$j: V_\lambda\rightarrow V_{\lambda’}$$ be an $$n-$$hugeness* embedding with $$\lambda=\beth_\lambda$$, and $$\lambda$$ regular. Let $$E$$ be the extender derived from $$j$$. Now let $$j_E: V\rightarrow M_E$$ be the embedding derived for $$E$$. While we can’t yet get some $$k_E$$ cummuting with $$j$$ and $$j_E$$, we can get a restricted version of it, by letting $$k_E^*([a,f])=j^n(f)(a)$$, for all $$[a,f]\in V_{j_E^{n+1}(\lambda)}$$, where $$a\in [j(\lambda)]^{\lt\omega}$$ and $$f: [\lambda]^{|a|}\rightarrow V_\lambda$$. Then $$k_E^*$$, $$j_E^n\restriction V_\lambda$$, and $$j^n\restriction V_\lambda$$ are commutative. Then $$k_E^*$$ is surjective and so the identity so that in particular, $$V_{j_E^n(\lambda)}^{M_E}=V_{j^n(\lambda)}$$. The rest is then as in proposition 2.18 in $$^1$$.◼

The natural generalization of asserting that $$j(\kappa)$$ can be made arbitrarily large, is asserting that for every club $$C$$, $$j^n(\kappa)\in C$$$$^5$$. Given any embedding property $$P$$, this is called being stationarily $$P$$. In accordance with our previous theorem, we give the following.

Theorem: If $$\kappa$$ is almost $$n+1-$$huge, for $$n\gt 0$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\lambda\lt\kappa|V_\kappa\vDash\lambda\text{ is stationarily super }n-\text{huge*}\}\in D$$.

$$Proof.$$ Note that in our previous theorem, we showed $$\{\alpha\lt\kappa|\lambda\rightarrow(\alpha)\}\in D$$, and so unbounded. As $$D$$ contains every club set, it is also stationary.◼

Theorem: If $$\kappa$$ is stationarily super $$n-$$huge*, then $$\kappa$$ is stationarily hyper $$n-$$huge.

$$Proof.$$ Let $$j: V_\alpha\rightarrow V_\beta$$ be an $$n-$$hugeness* embedding. The previous arguments show that if $$\kappa$$ is $$n-$$huge* with target $$j(\kappa)$$, then $$\kappa$$ is $$\lt j(\kappa)-$$hyper $$n-$$huge with the same target, and so if the targets are stationary for $$n-$$hugeness*, the same is true for hyper$$n-$$hugeness.◼

Theorem: If $$\kappa$$ is stationarily super $$n-$$huge, then there is a normal measure $$D$$ on $$\kappa$$, such that $$\{\lambda\lt\kappa|\lambda\text{ is super }n-\text{huge}\}\in D$$.

$$Proof.$$ Let $$C_\phi=\{\alpha\lt\kappa|V_\alpha\vDash\phi\leftrightarrow\phi\}$$. Then, for each $$\phi$$ there is a non-trivial elementary embedding $$j(\kappa)\in C_\phi$$, and as $$V_\kappa\prec V_{j(\kappa)}$$, $$\kappa\in C_\phi$$. Therefore $$V_{j(\kappa)}\vDash (\kappa\text{ is super }n-\text{huge})$$. Furthermore, $$M\vDash(V_{j(\kappa)}\vDash (\kappa\text{ is super }n-\text{huge}))$$ and as $$\kappa\in C_\phi$$, $$j(\kappa)\in C_{\phi^M}$$ and so $$M\vDash (\kappa\text{ is super }n-\text{huge})$$.◼

4. $$C^{(n)}-$$superhuge cardinals.

Definition: $$\kappa$$ is $$n-$$fold $$\eta-$$extendible for $$A$$ if and only if there is a sequence $$\{\zeta_i|i\le n\}$$ with $$\zeta_0=\kappa+\eta$$, and an increasing sequence $$\{\kappa_i|i\le n\}$$ with $$\kappa_0=\kappa$$ and $$\kappa_{i+1}\gt\zeta_i$$, and embeddings $$e_{i,j}: V_{\zeta_i}\rightarrow V_{\zeta_j}$$ with critical point $$\kappa_i$$, $$e_{i,j}(\kappa_i)=\kappa_j$$, $$e_{i+2,i+1}(e_{i,i+1}(x))=e_{i,i+1}(x)$$, and $$e_{i,j}^+(A\cap V_{\zeta_i})=A\cap V_{\zeta_j}$$.

Definition: If $$P$$ is some embedding property (Witnessed by a family of embeddings $$j$$), then $$\kappa$$ is $$C^{(n)}-P$$ if and only if, in addition, $$V_{j(\kappa)}\prec_{\Sigma_n} V$$.

In the case when $$P$$ is some $$n-$$fold variant of a property, we replace $$j$$ with $$j^n$$. Note that this typically implies $$j(\kappa)\in C^{(n)}$$, as typically $$V_{j(\kappa)}\prec V_{j^n(\kappa)}$$.

Theorem: $$\kappa$$ is super $$C^{(m)}-n-$$huge* if and only if $$\kappa$$ is $$C^{(m)}-n+1-$$extendible.

$$Proof.$$ $$(i)\rightarrow (ii)$$. Let $$j(\kappa)\gt\kappa+\eta$$, and let $$j: V_\alpha\prec V_\beta$$ be an $$n-$$hugeness* embedding, with $$j^k(\kappa)\in C^{(m)}$$. Let $$e^{(i)}=j^{i+}(j)$$. We can assume $$\alpha=e^{(i)}(\kappa+\eta)$$. Then, if $$\lambda$$ is the critical point of $$e^{(i)}$$, $$\lambda=j^i(\kappa)$$. Let $$e_{i,j+1}(x)=e^{(j+1)}\circ e_{i,j}$$ and $$e_{i,i}(x)=x$$. Then $$e_{i+1,i+2}(e_{i,i+1}(x))=e^{(i+2)}\circ e^{(i+1)}(x)$$ and $$e^{(i+2)}\circ e^{(i+1)}(x)=e_{i,i+2}(x)$$. Furthermore, $$V_{e^{(i)}(\kappa+\eta)}\vDash\phi(x)\leftrightarrow V_{e_{i,j}(e^{(i)}(\kappa+\eta))}\vDash\phi(e_{i,j}(x))$$. Finally, $$e_{i,j}(e^{(i)}(\kappa+\eta))=e^{(j)}(\kappa+\eta)$$. We can iterate this process up to $$n+1$$ iterates, because $$j^{n+1}(\kappa)\lt\beta$$.

$$(i)\rightarrow (ii)$$. Let $$\alpha$$ be any ordinal, and $$V_\kappa\prec V_{\kappa+\eta}$$, with $$\eta\gt\alpha$$. For each $$x\in\zeta_n$$, let $$i$$ be the least ordinal such that $$x\in\zeta_i$$, and let $$j(x)=e^{(i)}(x)$$. Then each $$V_{\zeta_i}\prec V_{\zeta_j}$$, whenever $$i\lt j$$. Therefore $$V_{\zeta_n}\vDash\phi(x)$$ if and only if $$V_{\zeta_i}\vDash\phi(x)$$ if and only if $$V_{\zeta_{i+1}}\vDash\phi(j(x))$$ if and only if $$V_{\zeta_{n+1}}\vDash\phi(j(x))$$. Furthermore, $$\zeta_n\gt j^n(\kappa)$$.◼

In such a situation we say $$j(\kappa)\in C^{(n)}$$. $$\kappa$$ is $$C^{(m)}-n-$$fold $$\eta-$$extendible if each $$\kappa_i\in C^{(m)}$$. $$\kappa$$ is $$C^{(m)+}-n-$$fold $$\eta-$$extendible if for some $$\eta’\ge\eta$$, there is a $$C^{(m)+}-n-$$fold sequence of embeddings with $$\zeta_0=\zeta$$, each $$\kappa_i\in C^{(m)}$$, and each $$\zeta_i\in C^{(m)}$$.

Theorem: The following are equivalent:

$$(i)$$ $$\kappa$$ is $$C^{(m)}-$$jointly hyper $$n-$$huge and $$n+1-$$strong.

$$(ii)$$ $$\kappa$$ is $$C^{(m)}-$$jointly hyper $$n-$$huge and super $$n-$$strong.

$$(iii)$$ $$\kappa$$ is $$C^{(m)}-$$super $$n-$$huge*.

$$Proof.$$ $$(i)\rightarrow (ii)$$. This is trivial.

$$(ii)\rightarrow (iii)$$. Let $$\lambda\gt\kappa$$ be in $$C^{(m+2)}$$. Let $$j: V\rightarrow M$$ witness that $$\kappa$$ is jointly hyper $$n-$$huge and $$\lambda-$$super $$n-$$strong. Then $$j\restriction V_{j^m(\kappa)}: V_{j^m(\lambda)}\rightarrow V_{j^{m+1}(\lambda)}^M$$ is a non-trivial elementary embedding, and $$j^n(\lambda)\gt j^n(\kappa)$$, so that $$M\vDash\kappa\text{ is }n-\text{huge* with target above }\lambda$$. Then, as $$j(\kappa)\in C^{(m)}$$, $$V_{j(\kappa)}\vDash\lambda\in C^{(m+1)}$$ and so $$M\vDash V_{j(\kappa)}\vDash\lambda\in C^{(m+1)}$$, and as $$M\vDash j(\kappa)\in C^{(m+1)}$$, $$M\vDash\lambda\in C^{(m+1)}$$ and so $$M\vDash V_\lambda\vDash\kappa\text{ is }n-\text{huge* with target above }\lambda$$ and so $$V_\lambda\vDash\kappa\text{ is }C^{(m,k)}-n-\text{huge* with target above }\lambda$$, and as $$\lambda\in C^{(m+2)}$$, the same holds in $$V$$.

$$(iii)\rightarrow (i)$$. Let $$j: V_\lambda\rightarrow V_{\lambda’}$$ be an $$n-$$hugeness* embedding with $$\lambda=\beth_\lambda$$, and $$\lambda$$ regular. Let $$E$$ be the extender derived from $$j$$. Now let $$j_E: V\rightarrow M_E$$ be the embedding derived for $$E$$. While we can’t yet get some $$k_E$$ cummuting with $$j$$ and $$j_E$$, we can get a restricted version of it, by letting $$k_E^*([a,f])=j^n(f)(a)$$, for all $$[a,f]\in V_{j_E^{n+1}(\lambda)}$$, where $$a\in [j(\lambda)]^{\lt\omega}$$ and $$f: [\lambda]^{|a|}\rightarrow V_\lambda$$. Then $$k_E^*$$, $$j_E^n\restriction V_\lambda$$, and $$j^n\restriction V_\lambda$$ are commutative. Then $$k_E^*$$ is surjective and so the identity so that in particular, $$V_{j_E^n(\lambda)}^{M_E}=V_{j^n(\lambda)}$$. The rest is then as in proposition 2.18 in $$^1$$.◼

Theorem: Let $$P$$ be some $$\Sigma_m-$$embedding property. For $$n\gt 0$$, $$\kappa$$ is $$C^{(n)}-P$$ if and only if for every $$\Pi_{n+1}-$$definable club $$C$$, there is a $$P$$ embedding $$j: V\rightarrow M$$ with critical point $$\kappa$$, and $$j(\kappa)\in C$$.

$$Proof.$$ Let $$C$$ be $$\Pi_{n+1}-$$definable. Then $$C$$ is club below $$j(\kappa)$$, as $$\Pi_{n+1}$$ formulas are downward absolute, and so $$j(\kappa)\in C$$. Conversely, $$C^{(n)}$$ is $$\Pi_n+1-$$definable.◼

Theorem: If $$C$$ is club, then there exists a proper class of natural models $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},R_\alpha)$$ definable from $$C$$ such that, for any $$j: \mathfrak M_\alpha\rightarrow\mathfrak M_\beta$$ with critical point $$\kappa$$, then $$\kappa\in C$$.

$$Proof.$$ Let $$\gamma_\alpha$$ be the least limit point of $$C$$ above $$\alpha$$. Let $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\gamma_\alpha,C\cap\alpha+1)$$. Let $$j: \mathfrak M_\alpha\rightarrow\mathfrak M_\beta$$ be a non-trivial elementary embedding with critical point $$\kappa$$, and assume $$\kappa\notin C$$, so that $$\gamma\lt\kappa$$, where $$\gamma=\text{sup} C\cap\kappa$$. Let $$\delta$$ be the least ordinal in $$C$$ greater than $$\kappa$$ such that $$\delta\lt\gamma_\alpha$$. Since $$\delta$$ is definable from $$\gamma$$ in $$\mathfrak M_\alpha$$, and since $$j(\gamma)=\gamma$$, then $$j(\delta)=\delta$$, and so $$j\restriction V_{\delta+2}$$ is a non-trivial elementary embedding from $$V_{\delta+2}$$ into itself, contradicting Kunen’s theorem.◼

Definition: $$\kappa$$ is $$n-$$fold $$\lambda-$$supercompact for $$A$$ if and only if there is a sequence $$\{\zeta_i|i\le n\}$$ with $$\zeta_1=\kappa+\lambda$$, and an increasing sequence $$\{\kappa_i|i\le n\}$$ with $$\kappa_1=\kappa$$, and embeddings $$e_{i,j}: V_{\zeta_i}\rightarrow V_{\zeta_j}$$ with critical point $$\kappa_i$$, $$e_{i,j}(\kappa_i)=\kappa_j$$, $$e_{i+2,i+1}(e_{i,i+1}(x))=e_{i,i+1}(x)$$, and $$e_{i,j}^+(A\cap V_{\zeta_i})=A\cap V_{\zeta_j}$$.

Theorem: $$\kappa$$ is $$n+1-$$fold supercompact (For $$\emptyset$$) if and only if $$\kappa$$ is hyper $$n-$$huge.

$$Proof.$$ For the forward direction, let $$\lambda\gt\kappa$$. Then, let $$e_{i,k}=j^k\restriction V_{j^i(\kappa)}$$. By closure for $$M$$, we can easily get that $$M\vDash(j(\kappa)\text{ is }n-\text{fold }\lambda-\text{supercompact})$$, and the rest follows by elementarity.

For the converse, let $$\zeta_1=\lambda+\omega$$ and let $$\zeta_0=\beta+\omega$$. Let $$\lambda_i=e_{1,i}(\lambda)$$, so that $$\lambda_0=\lambda$$. Let $$\kappa’=\kappa_n$$. Then let $$D=\{X\subseteq [\lambda_n]^{\kappa’}|e_{0,n}”(\lambda_n)\in e_{0,n}(X)\}$$. Note that $$ot(e_{0,i}”(\lambda)\cap\lambda_{i+1})=\lambda_i$$, and $$ot(e_{0,i}”(\lambda)\cap\lambda)\lt\beta$$. Then take $$e(D)$$.◼

Definition: We write $$VP_n(\kappa,\underset{\sim}{\Gamma })$$ if and only if for every proper class $$C$$ of natural models $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},R_\alpha)$$ $$\Gamma-$$definable with parameters in $$H_\kappa$$, for every $$A\in C$$, there is an increasing sequence $$\{\kappa_i|i\le n\}$$, and embeddings $$e_{i,j}: \mathfrak M_{\zeta_i}\rightarrow \mathfrak M_{\zeta_j}$$ with critical point $$\kappa_i$$ and $$\kappa_{i+1}\gt\zeta_i$$, $$e_{i,j}(\kappa_i)=\kappa_j$$, and $$e_{i+2,i+1}(e_{i,i+1}(x))=e_{i,i+1}(x)$$, such that $$\mathfrak M_{\zeta_0}\in H_\kappa$$ and $$\mathfrak M_{\zeta_1}=A$$. $$VP_n(\underset{\sim}{\Gamma })$$ if and only if $$C$$ of natural models $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},R_\alpha)$$ $$\Gamma-$$definable, there is an increasing sequence $$\{\kappa_i|i\le n\}$$, and embeddings $$e_{i,j}: \mathfrak M_{\zeta_i}\rightarrow \mathfrak M_{\zeta_j}$$ with critical point $$\kappa_i$$ and $$\kappa_{i+1}\gt\zeta_i$$, $$e_{i,j}(\kappa_i)=\kappa_j$$, and $$e_{i+2,i+1}(e_{i,i+1}(x))=e_{i,i+1}(x)$$. $$VP_n(\Gamma)$$ is the weakening of $$VP_n(\underset{\sim}{\Gamma })$$ by removing parameters.

Theorem: If $$\kappa$$ is $$n-$$fold supercompact, then $$VP_n(\kappa,\underset{\sim}{\Sigma}_2)$$.

$$Proof.$$ First, $$V_\kappa=H_\kappa$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be some $$\Sigma_n$$ definable sequence $$C$$, with parameter $$p$$. Let $$V_{\kappa+\eta}\ni p$$ be $$\Sigma_2$$ elementary in $$V$$, $$\zeta_0=\kappa+\eta$$, and let $$\zeta_i$$ be a sequence witnessing (Partial) $$n-$$fold supercompactness. $$\alpha\lt\kappa_1$$ and $$\alpha\ge\kappa_0$$. As $$\kappa_i\in C^{(1)}$$, then $$V_{\zeta_i}\prec_{\Sigma_{n+1}} V_{\zeta_j}$$ and each $$\zeta_i\in C^{(1)}$$, for each $$i\le n$$. Then $$e_{i,j}$$ restricts to an embedding $$k_{i,j}: \mathfrak M_\alpha^{V_{\zeta_i}}\rightarrow \mathfrak M_{k_{i,j}(\alpha)}^{V_{\zeta_j}}$$. But $$\mathfrak M_\alpha^{V_{\zeta_i}}=\mathfrak M_\alpha$$ for each $$i\lt n$$. In particular, $$V_{\zeta_n}$$ think there is a sequence of the necessary type, and so the same holds in $$V$$.◼

Theorem: $$VP_n(\Pi_1)$$ with parameters of rank $$\le\alpha$$ implies the existence of a $$n-$$fold supercompact cardinal $$\gt\alpha$$. Therefore $$VP_n(\underset{\sim}{\Pi}_1 )$$ implies the existence of a proper class of $$n-$$fold supercompact cardinals.

$$Proof.$$ Assume to the contrary. Let $$C$$ be the class of $$\mathfrak M_\xi=(V_{\gamma_\xi+2},\in,\{\xi\},\gamma_\xi,\{\gamma\})_{\gamma\le\alpha}$$, where $$\gamma_\xi$$ is the least limit ordinal above $$\xi$$ such that there is no $$\kappa\gt\alpha$$ $$n-$$fold $$\lt\gamma_\xi-$$supercompact. This is $$\Pi_1$$ definable with parameter $$\alpha$$. Let $$e{i,j}: \mathfrak M_{\zeta_i}\rightarrow\mathfrak M_{\zeta_j}$$. By Kunen’s theorem $$\zeta_i\lt\zeta_j$$. Then let $$\kappa_1\lt\zeta_1$$ be standard. By an argument similar to the theorem on hyper $$n-$$huge, $$\kappa_1$$ is then $$n-$$fold $$\lt\lambda-$$supercompact. Contradiction. The second statement follows immediately from the first.◼

Theorem: The following are equivalent for $$n\gt 0$$:

$$(i)$$ $$VP_n(\Pi_1)$$.

$$(ii)$$ $$VP_n(\kappa,\underset{\sim}{\Sigma}_2 )$$ for some $$\kappa$$.

$$(iii)$$ There exists a $$n-$$fold supercompact cardinal.

Furthermore, the following are equivalent for $$n\gt 0$$:

$$(i)$$ $$VP_n(\underset{\sim}{\Pi}_1)$$.

$$(ii)$$ $$VP_n(\kappa,\underset{\sim}{\Sigma}_2 )$$ for proper class of $$\kappa$$.

$$(iii)$$ There exists a proper class of $$n-$$fold supercompact cardinals.

$$Proof.$$ All of this is a corollary to the above. For the first part, $$(iiii)\rightarrow (iii)$$ and $$(iii)\rightarrow (ii)$$ is by the previous theorems. $$(ii)\rightarrow (i)$$ and $$(i)\rightarrow (iiii)$$ is by the previous theorem. For the second part, $$(ii)\rightarrow (i)$$ and $$(iiii)\rightarrow (iii)$$ are obvious, and $$(iii)\rightarrow (ii)$$ follows from the previous theorems. $$(i)\rightarrow (iiii)$$ follow from the previous theorem.◼

Theorem: The following are equivalent for $$n\gt 0$$:

$$(i)$$ $$\kappa$$ is the least $$n-$$fold supercompact cardinal.

$$(ii)$$ $$\kappa$$ is the least cardinal such that $$VP_n(\kappa,\underset{\sim}{\Sigma}_2 )$$.

$$(iii)$$ $$\kappa$$ is the least cardinal such that $$VP_n(\kappa,\underset{\sim}{\Pi}_1)$$.

$$Proof.$$ The only tricky part is $$(iii)\rightarrow (i)$$. Assume to the contrary. Let $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\lambda$$, where $$\gamma_\alpha$$ is a limit ordinal with uncountable cofinality, and there is no $$\kappa’\le\alpha$$ that is $$n-$$fold $$\lambda-$$supercompact. This is $$\Pi_1-$$definable. Then, if $$\alpha$$ is the critical point of $$e_{0,1}: \mathfrak M_{\zeta_0}\rightarrow\mathfrak M_{\zeta_1}$$. Then each $$e_{i,i+1}(\kappa’)\in C^{(m)}$$, and so $$\kappa’$$ is $$n-$$fold $$\lambda-$$supercompact.◼

Theorem: For $$n\gt 0$$, $$VP_n(\kappa,\underset{\sim}{\Pi}_1)$$ if and only if $$\kappa$$ is $$n-$$fold supercompact or a limit of such cardinals.

$$Proof.$$ The converse direction is clear. For the forward, assume to the contrary that $$\kappa$$ is not $$n-$$fold supercompact or a limit of such cardinals. Let $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\lambda\{\gamma\})_{\gamma\le\zeta}$$, where $$\gamma_\alpha$$ is a limit ordinal with uncountable cofinality, and there is no $$\kappa’\le\alpha$$ that is $$n-$$fold $$\lambda-$$supercompact. This is $$\Pi_{m+1}-$$definable. But, if $$\kappa$$ is the critical point $$e_{1,2}$$ it is $$n-$$fold $$\lambda-$$supercompact.◼

Theorem: If $$\kappa$$ is $$C^{(m)}-n-$$fold extendible, then $$VP_n(\kappa,\underset{\sim}{\Sigma}_{m+2} )$$.

$$Proof.$$ First, $$V_\kappa=H_\kappa$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be some $$\Sigma_n$$ definable sequence $$C$$, with parameter $$p$$. Let $$V_{\kappa+\eta}\ni p$$ be $$\Sigma_{m+2}$$ elementary in $$V$$, $$\zeta_0=\kappa+\eta$$, and let $$\zeta_i$$ be a sequence witnessing $$n-$$fold $$\eta-$$extendibility. $$\alpha\lt\kappa_1$$ and $$\alpha\ge\kappa_0$$. As $$\kappa_i\in C^{(n)}$$, then $$V_{\zeta_i}\prec_{\Sigma_{n+1}} V_{\zeta_j}$$ and each $$\zeta_i\in C^{(n)}$$, for each $$i\lt n$$. Then $$e_{i,j}$$ restricts to an embedding $$k_{i,j}: \mathfrak M_\alpha^{V_{\zeta_i}}\rightarrow \mathfrak M_{k_{i,j}(\alpha)}^{V_{\zeta_j}}$$. But $$\mathfrak M_\alpha^{V_{\zeta_i}}=\mathfrak M_\alpha$$ for each $$i\lt n$$. In particular, $$V_{\zeta_n}$$ think there is a sequence of the necessary type, and so the same holds in $$V$$.◼

Theorem: $$m\gt 1$$ and $$VP_n(\Pi_{m+1})$$ with parameters of rank $$\le\alpha$$ implies the existence of a $$C^{(m)+}-n-$$fold extendible cardinal $$\gt\alpha$$. Therefore $$VP_n(\underset{\sim}{\Pi}_{m+1} )$$ implies the existence of a proper class of $$n-$$fold $$C^{(m)+}-n-$$extendible cardinals.

$$Proof.$$ Let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$C^{(m)+}-n-$$fold $$\eta-$$extendible, and $$\kappa$$ otherwise. Let $$C=\{\lambda\in C^{(m)}|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$\mathfrak N_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha,\{\gamma\})_{\gamma\le\alpha}$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$. This is $$\Pi_{m+1}$$. We show that if $$\kappa$$ is the critical point of $$e_{0,1}: \mathfrak N_{\zeta_0}\rightarrow \mathfrak N_{\zeta_1}$$, then $$\kappa$$ is $$C^{(m)+}-n-$$fold extendible. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Since $$\kappa_i\lt\gamma_{\zeta_i}$$ and $$\gamma_{\zeta_i}\in C$$, $$g(\kappa_i)\lt\gamma_{\zeta_i}$$. It follows that $$e_{i,j}\restriction V_{g(\kappa_i)}\rightarrow V_{e_{i,j}(g(\kappa_i))}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt e_{i,j}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But then, these properties show that $$\kappa$$ is $$C^{(m)+}-n-$$fold $$\eta-$$extendible. The second statement follows immediately from the first.◼

Theorem: The following are equivalent for $$n$$,$$m\gt 0$$:

$$(i)$$ $$VP_n(\Pi_{m+1})$$.

$$(ii)$$ $$VP_n(\kappa,\underset{\sim}{\Sigma}_{m+2} )$$ for some $$\kappa$$.

$$(iii)$$ There exists a $$C^{(m)}-n-$$fold extendible cardinal.

$$(iiii)$$ There exists a $$C^{(m)+}-n-$$fold extendible cardinal.

Furthermore, the following are equivalent for $$n$$,$$m\gt 0$$:

$$(i)$$ $$VP_n(\underset{\sim}{\Pi}_{m+1})$$.

$$(ii)$$ $$VP_n(\kappa,\underset{\sim}{\Sigma}_{m+2} )$$ for proper class of $$\kappa$$.

$$(iii)$$ There exists a proper class of $$C^{(m)}-n-$$fold extendible cardinals.

$$(iiii)$$ There exists a proper class of $$C^{(m)+}-n-$$fold extendible cardinals.

$$Proof.$$ All of this is a corollary to the above. For the first part, $$(iiii)\rightarrow (iii)$$ and $$(iii)\rightarrow (ii)$$ is by the previous theorems. $$(ii)\rightarrow (i)$$ and $$(i)\rightarrow (iiii)$$ is by the previous theorem. For the second part, $$(ii)\rightarrow (i)$$ and $$(iiii)\rightarrow (iii)$$ are obvious, and $$(iii)\rightarrow (ii)$$ follows from the previous theorems. $$(i)\rightarrow (iiii)$$ follow from the previous theorem.◼

Theorem: The following are equivalent for $$n$$,$$m\gt 0$$:

$$(i)$$ $$\kappa$$ is the least $$C^{(m)+}-n-$$fold extendible cardinal.

$$(ii)$$ $$\kappa$$ is the least cardinal such that $$VP_n(\kappa,\underset{\sim}{\Sigma}_{m+2} )$$.

$$(iii)$$ $$\kappa$$ is the least cardinal such that $$VP_n(\kappa,\underset{\sim}{\Pi}_{m+1})$$.

$$Proof.$$ The only tricky part is $$(iii)\rightarrow (i)$$. Assume to the contrary. Let $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\lambda, C^{(n)}\cap\gamma_\alpha)$$, where $$\gamma_\alpha$$ is a limit point of $$C^{(n)}$$ with uncountable cofinality, $$\lambda\in C^{(n)}$$, and there is no $$\kappa’\le\alpha$$ that is $$C^{(m)}-n-$$fold $$\lambda-$$extendible. This is $$\Pi_{m+1}-$$definable. Then, if $$\alpha$$ is the critical point of $$e_{0,1}: \mathfrak M_{\zeta_0}\rightarrow\mathfrak M_{\zeta_1}$$. Then each $$e_{i,i+1}(\kappa’)\in C^{(m)}$$, and so $$\kappa’$$ is $$C^{(m)}-n-$$fold $$\lambda-$$extendible.◼

Theorem: For $$n$$,$$m\gt 0$$, $$VP_n(\kappa,\underset{\sim}{\Pi}_{m+1})$$ if and only if $$\kappa$$ is $$C^{(m)}-n-$$fold extendible or a limit of such cardinals.

$$Proof.$$ The converse direction is clear. For the forward, assume to the contrary that $$\kappa$$ is not $$C^{(m)}-n-$$fold extendible or a limit of such cardinals. Let $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\lambda, C^{(n)}\cap\alpha+1,\{\gamma\})_{\gamma\le\zeta}$$, where $$\gamma_\alpha$$ is a limit point of $$C^{(m)}$$ with uncountable cofinality, $$\lambda\in C^{(m)}$$, and there is no $$\kappa’\le\alpha$$ that is $$C^{(m)}-n-$$fold $$\lambda-$$extendible. This is $$\Pi_{m+1}-$$definable. Then, if $$\alpha$$ is the critical point of $$e_{0,1}: \mathfrak M_{\zeta_0}\rightarrow\mathfrak M_{\zeta_1}$$. Then each $$e_{i,i+1}(\kappa’)\in C^{(m)}$$, and so $$\kappa’$$ is $$C^{(m)}-n-$$fold $$\lambda-$$extendible.◼

Theorem: The following are conservative for $$n\gt 0$$:

$$(i)$$ For every $$\Gamma$$, $$VP_n(\Gamma)$$.

$$(ii)$$ For every $$m$$, there is an $$C^{(m)}-n-$$fold extendible cardinal.

$$(iii)$$ For every $$A$$, there is $$n-$$fold extendible for $$A$$ cardinal.

$$Proof.$$ Let $$(M,\in,N)$$ be a model of $$NBG$$. Then $$(M,\in,Def(M))$$ is a model of $$NBG$$ with the same first order theory, in which every class is definable. We show that if $$M_0=(M,\in,Def(M))$$ is such a model of $$(ii)$$ or $$(iii)$$, then it is a model of $$(i)$$.

$$(ii)$$ extends $$(i)$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be some $$\Sigma_n$$ definable class, with parameter $$p$$. Let $$V_{\kappa+\eta}\ni p$$ be $$\Sigma_n$$ elementary in $$V$$, $$\zeta_0=\kappa+\eta$$, and let $$\zeta_i$$ be a sequence witnessing $$n-$$fold $$\eta-$$extendibility. $$\alpha_i\lt\kappa_{i+1}$$ and $$\alpha\gt\kappa_i$$. As $$\kappa_i\in C^{(n)}$$, then $$V_{\zeta_i}\prec V_{\zeta_j}$$. Then $$e_{i,j}$$ restricts to an embedding $$k_{i,j}: \mathfrak M_\alpha^{V_{\zeta_i}}\rightarrow \mathfrak M_{j(\alpha)}^{V_{\zeta_j}}$$. But $$\mathfrak M_\alpha^{V_{\zeta_i}}=\mathfrak M_\alpha$$.

$$(iii)$$ extends $$(i)$$. Let $$A=\{\mathfrak M_\alpha|\alpha\in Ord\}$$, and $$F(\beta)=\text{sup}\{\alpha\lt\beta|rank(\mathfrak M_\alpha)\lt\beta\}$$. Then we have non-trivial elementary embeddings from some $$(V_{\zeta_i},\in,\mathfrak M_\alpha)_{\alpha\lt F(\zeta_i)}$$ into $$(V_{\zeta_j},\in,\mathfrak M_\alpha)_{\alpha\lt F(\zeta_j)}$$.

$$(i)$$ extends to $$(ii)$$. Finally, it remains to verify that every model $$M_0$$ of $$(i)$$ is a model of $$(ii)$$ and $$(iii)$$. First, for $$(ii)$$, let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$C^{(m)}-n-$$fold $$\eta-$$extendible, and $$\kappa$$ otherwise. Let $$C=\{\lambda\in C^{(m)}|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$\mathfrak N_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha)$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$. We show that if $$\kappa$$ is the critical point of $$e_{0,1}: \mathfrak N_{\zeta_0}\rightarrow \mathfrak N_{\zeta_1}$$, then $$\kappa$$ is $$C^{(m)}-n-$$fold extendible. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Since $$\kappa_i\lt\gamma_{\zeta_i}$$ and $$\gamma_{\zeta_i}\in C$$, $$g(\kappa_i)\lt\gamma_{\zeta_i}$$. It follows that $$e_{i,j}\restriction V_{g(\kappa_i)}\rightarrow V_{e_{i,j}(g(\kappa_i))}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt e_{i,j}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But then, these properties show that $$\kappa$$ is $$C^{(m)}-n-$$fold $$\eta-$$extendible.

$$(i)$$ extends to $$(iii)$$. Let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$n-$$fold $$\eta-$$extendible for $$A$$, and $$\kappa$$ otherwise. Let $$C=\{\lambda\in C^{(m)}|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$\mathfrak N_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha, A\cap V_{\gamma_\alpha})$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$. We show that if $$\kappa$$ is the critical point of $$e_{0,1}: \mathfrak N_{\zeta_0}\rightarrow \mathfrak N_{\zeta_1}$$, then $$\kappa$$ is $$n-$$fold extendible for $$A$$. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Since $$\kappa_i\lt\gamma_{\zeta_i}$$ and $$\gamma_{\zeta_i}\in C$$, $$g(\kappa_i)\lt\gamma_{\zeta_i}$$. It follows that $$e_{i,j}\restriction V_{g(\kappa_i)}\rightarrow V_{e_{i,j}(g(\kappa_i))}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt e_{i,j}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But then, these properties show that $$\kappa$$ is $$n-$$fold $$\eta-$$extendible for $$A$$.◼

Theorem: The following are equivalent:

$$(i)$$ $$\kappa$$ is $$n+1-$$fold Vopěnka.

$$(ii)$$ For every $$A\subseteq V_\kappa$$, there is some $$\kappa’\lt\kappa$$ $$n-$$fold $$\lt\kappa-$$extendible for $$A$$.

$$(iii)$$ For every $$A\subseteq V_\kappa$$, there is some $$\kappa’\lt\kappa$$ $$n-$$fold $$\lt\kappa-$$extendible for $$A$$.

$$Proof.$$ $$(i)\rightarrow (ii)$$. Let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$n-$$fold $$\eta-$$extendible for $$A$$, and $$\kappa$$ otherwise. Let $$C=\{\lambda|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$\mathfrak N_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha, A\cap V_{\gamma_\alpha})$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$. We show that if $$\kappa$$ is the critical point of $$e_{0,1}: \mathfrak N_{\zeta_0}\rightarrow \mathfrak N_{\zeta_1}$$, then $$\kappa$$ is $$n-$$fold extendible for $$A$$. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Since $$\kappa_i\lt\gamma_{\zeta_i}$$ and $$\gamma_{\zeta_i}\in C$$, $$g(\kappa_i)\lt\gamma_{\zeta_i}$$. It follows that $$e_{i,j}\restriction V_{g(\kappa_i)}\rightarrow V_{e_{i,j}(g(\kappa_i))}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt e_{i,j}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But then, these properties show that $$\kappa$$ is $$n-$$fold $$\eta-$$extendible for $$A$$.

$$(ii)\rightarrow (iii)$$. Let $$\lambda\gt\kappa’$$. Then, let $$e_{i,k}=j^k\restriction V_{j^i(\kappa’)}$$. By closure for $$M$$, we can easily get that $$M\vDash(j(\kappa’)\text{ is }n-\text{fold }\lambda-\text{supercompact for }A)$$, and the rest follows by elementarity.

$$(iii)\rightarrow (i)$$. Let $$A=\{\mathfrak M_\alpha|\alpha\in Ord\}$$, and $$F(\beta)=\text{sup}\{\alpha\lt\beta|rank(\mathfrak M_\alpha)\lt\beta\}$$. Then we have non-trivial elementary embeddings from some $$(V_{\zeta_i},\in,\mathfrak M_\alpha)_{\alpha\lt F(\zeta_i)}$$ into $$(V_{\zeta_j},\in,\mathfrak M_\alpha)_{\alpha\lt F(\zeta_j)}$$.◼

Theorem: $$\kappa$$ is $$n+1-$$fold Woodin for supercompactness if and only if it is $$n+1-$$fold Vopěnka.

$$Proof.$$ By a simple variation of the standard proof, we can get that $$\kappa$$ is $$n+1-$$fold Woodin for supercompactness if and only if, for every $$A\subseteq V_\kappa$$, there is some $$\alpha\lt\kappa$$ such that $$\alpha$$ is $$\lt\kappa-$$supercompact for $$A$$. Therefore, $$\kappa$$ is $$n+1-$$fold Woodin for supercompactness if and only if it is $$n+1-$$fold Vopěnka.◼

Theorem: If $$P$$ (With target $$\lambda’$$) is $$\Pi_n$$ and $$\kappa$$ is $$C^{(n)}-P$$ and $$C^{(n+1)}-$$superstrong, for $$n\gt 1$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\lambda\lt\kappa|\lambda\text{ is }C^{(n)}-\text{superstrong}\land C^{(n)}-P\}\in D$$.

$$Proof.$$ Let $$j: V\rightarrow M$$ witness $$C^{(n+2)}-$$superstrongness. As $$j(\kappa)\in C^{(n+1)}$$, then $$V_{j(\kappa)}\vDash(\kappa\text{ is }C^{(n)}-\text{superstrong}\land C^{(n)}-P)$$. Then $$M\vDash V_{j(\kappa)}\vDash(\kappa\text{ is }C^{(n)}-\text{superstrong}\land C^{(n)}-P)$$ and so $$\{\lambda\lt\kappa|V_{j(\kappa)}\vDash(\kappa\text{ is }C^{(n)}-\text{superstrong}\land C^{(n)}-P)\}\in D$$. But the assertion $$\lambda$$ is $$C^{(n)}-P$$ is $$\Sigma_{n+1}$$.◼

Theorem: If $$I3(\kappa,\lambda)$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\lambda’\lt\kappa|\forall n\lt\omega(\lambda’\text{ is super }C^{(n)}-\lt\omega-\text{huge*}\}\in D$$.

$$Proof.$$ Let $$j: V_\lambda\rightarrow V_\lambda$$. It is clear $$V_{j(\kappa)}\prec V_\lambda$$. Then $$V_\lambda\vDash(\forall n\lt\omega(\kappa\text{ is super }C^{(n)}-\lt\omega-\text{huge*}))$$. Then let $$D$$ be the measure generated by $$j$$.◼

We will discuss rank-into-rank cardinals more later.

5. The universe above ultrahuge, hyperhuge cardinals.

The following motivating example is typically associated with supercompactness, but actually follows from strongness.

Theorem: Let $$\kappa$$ be $$\lambda-$$supercompact ($$\lambda-$$strong) as witnessed by, $$j$$,$$M$$. Then, if $$F=F^M$$ be such that $$2^\alpha=F(\alpha)$$ for every $$\alpha\lt\kappa$$. Then $$2^\alpha=F(\alpha)$$ for every $$\alpha\le\lambda$$. Furthermore, let $$\lambda’\gt j(\kappa)$$ be such that $$\kappa$$ is $$\lambda’-$$supercompact ($$\lambda’-$$strong), and let $$F$$ be $$\Sigma_2$$. Then, if $$2^\alpha=F(\alpha)$$ for every $$\alpha\lt\kappa$$, then $$2^\alpha=F(\alpha)$$ for every $$\alpha\lt j(\kappa)$$.

$$Proof.$$ By elementarity, $$M\vDash (\forall\alpha\lt j(\kappa)(2^\alpha=F(\alpha))$$ and so $$\forall\alpha\lt j(\kappa)(2^\alpha=F^M(\alpha))$$. For the second part, assume to the contrary that there is some $$\alpha\lt j(\kappa)$$ such that $$2^\alpha\neq F(\alpha)$$. Let $$j’: V\rightarrow M’$$ witness $$\lambda’-$$supercompactness ($$\lambda’-$$strongness). Then $$\alpha\in V_{j(\kappa)}$$ is a witness to this. Therefore $$M’\vDash V_{j(\kappa)}\vDash\exists\alpha(2^\alpha\neq F(\alpha))$$ and so $$M’\vDash\exists\alpha(2^\alpha\neq F(\alpha))$$. But $$j(\kappa)\lt j'(\kappa)$$ and $$M’\vDash (\forall\alpha\lt j'(\kappa)(2^\alpha=F(\alpha))$$.◼

Theorem: Let $$\kappa$$ be $$\lambda-$$hyper $$n-$$huge as witnessed by, $$j$$,$$M$$. Then, if $$F=F^M$$ be such that $$2^\alpha=F(\alpha)$$ for every $$\alpha\lt\kappa$$. Then $$2^\alpha=F(\alpha)$$ for every $$\alpha\le j^n(\lambda)$$.

$$Proof.$$ By the same argument as before, $$(2^\alpha)^M=F^M(\alpha)$$. But, as every $$X\subseteq\alpha$$ has $$|X|\le j^n(\lambda)$$, $$(2^\alpha)^M=2^\alpha$$.◼

Theorem: If $$\kappa$$ is almost ultra $$n+1-$$huge, there is some $$\lambda-$$ultra $$n-$$hugeness embedding $$j$$, such that $$o(j^{n+1}(\kappa))=(2^{j^{n+1}(\kappa)})^+$$.

$$Proof.$$ Let $$\lambda\gt\kappa+\kappa$$, and $$j: V\rightarrow M$$. Then, for each $$\alpha\lt j^n(\kappa)$$, $$V_{j^{n+1}(\kappa)+\alpha}\subseteq M$$. In particular, if $$M\vDash o(j^i(\kappa))=(2^{j^i(\kappa)})^+$$, then $$o(j^i(\kappa))=(2^{j^i(\kappa)})^+$$.◼

6. Rank-into-Rank axioms.

Definition: $$I3(\kappa,\lambda)_n$$ if and only if there exists an $$I3$$ embedding $$j: V_\lambda\rightarrow V_\lambda$$ with critical point $$\kappa$$, and $$j^k(\kappa)\in C^{(n)}$$ for every $$k$$.

Theorem: If $$I3(\kappa,\lambda)_{n+1}$$ and $$n\gt 0$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha( I3(\beta,\gamma)_n))\}\in D$$.

$$Proof.$$ Let $$j: V_\lambda\rightarrow V_\lambda$$. Note that, if $$j(\kappa)\in C^{(n+1)}$$, then $$V_{j(\kappa)}\vDash I3(\kappa,\lambda)_n$$, and so, if $$D$$ is the measure generated by $$j$$, $$\{\beta\lt\kappa|V_{j(\kappa)}\vDash\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(I3(\beta,\gamma)_n))\}\in D$$. But $$j(\kappa)\in C^{(n)}$$, and so $$\{\beta\lt\kappa|V_{j(\kappa)}\vDash\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(I3(\beta,\gamma)_n))\}=\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha (I3(\beta,\gamma)_n))\}$$.◼

Definition: $$IE^\alpha(\kappa,\lambda)_n$$ if and only if there exists an $$I3$$ embedding $$j: V_\lambda\rightarrow V_\lambda$$ with critical point $$\kappa$$, and $$j^k(\kappa)\in C^{(n)}$$ for every $$k$$, such $$j$$ is $$\alpha-$$iterable. Furthermore $$IE^\alpha(\kappa,\lambda)\leftrightarrow IE^\alpha(\kappa,\lambda)_0$$.

$$IE(\kappa,\lambda)\leftrightarrow IE^{\omega_1}(\kappa,\lambda)$$. From here we use the same definitions as $$^7$$.

Theorem: If $$IE^\delta(\kappa,\lambda)$$ and $$\delta\lt\omega_1$$ is a limit ordinal, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\kappa’\lt\kappa|\forall\alpha\lt\kappa'(\exists\beta\gt\alpha(\forall\alpha’\lt\kappa'(\exists\gamma’\gt\alpha(\forall\delta’\lt\delta(IE^{\delta’}(\beta,\gamma’)\land V_{\gamma’}\prec V_{\kappa’}))))\}\in D$$.

$$Proof.$$ Our proof is similar to $$^7$$. Let $$j: V_\lambda\rightarrow V_\lambda$$ be a non-trivial elementary embedding. Let $$\phi(\alpha,\beta)\leftrightarrow\exists\gamma’\gt\alpha(\forall\delta’\lt\delta(IE^{\delta’}(\beta,\gamma’)\land V_{\gamma’}\prec V))$$. Amend the proof of Claim 4.3 in $$^7$$ by requiring each $$V_{\gamma_n}\prec V_\lambda$$. By the same proof as in $$^7$$, for each $$\alpha$$ the class of such cardinals is unbounded in $$V_\lambda$$. Then, as $$\phi(\alpha,\beta)$$ is describable by a formula with constant symbols, $$\forall\zeta\lt\kappa(\exists\beta\gt\gamma(\forall\alpha\lt\kappa(V_\lambda\vDash\phi(\alpha,\beta))))$$ if and only if $$\forall\zeta\lt\kappa(\exists\beta\gt\zeta(\forall\alpha\lt\kappa(V_\kappa\vDash\phi(\alpha,\beta))))$$ if and only if $$V_\kappa\vDash\forall\zeta(\exists\beta\gt\zeta(\forall\alpha(\phi(\alpha,\beta))))$$. Then, if $$D$$ is the measure generated by $$j$$, $$\{\kappa’\lt\kappa|\forall\alpha\lt\kappa'(\exists\beta\gt\zeta(\forall\alpha’\lt\kappa'(\exists\gamma’\gt\alpha(\forall\delta’\lt\delta(IE^{\delta’}(\beta,\gamma’)\land V_{\gamma’}\prec V_{\kappa’}))))\}\in D$$.◼

Theorem: If $$IE^\alpha(\kappa,\lambda)_{n+1}$$ and $$n\gt 0$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha( IE^\alpha(\beta,\gamma)_n))\}\in D$$.

$$Proof.$$ Let $$j: V_\lambda\rightarrow V_\lambda$$. Note that, if $$j(\kappa)\in C^{(n+1)}$$, then $$V_{j(\kappa)}\vDash IE^\alpha(\kappa,\lambda)_n$$, and so, if $$D$$ is the measure generated by $$j$$, $$\{\beta\lt\kappa|V_{j(\kappa)}\vDash\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(IE^\alpha(\beta,\gamma)_n))\}\in D$$. But $$j(\kappa)\in C^{(n)}$$, and so $$\{\beta\lt\kappa|V_{j(\kappa)}\vDash\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(IE^\alpha(\beta,\gamma)_n))\}=\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha (IE^\alpha(\beta,\gamma)_n))\}$$.◼

Definition: $$\kappa$$ is $$\omega-$$fold $$\lambda-$$strong if and only if there is a non-trivial elementary embedding $$j: V\rightarrow M$$ such that there is $$\delta\gt\lambda$$ with $$j(\delta)=\delta$$, $$V_\delta\subseteq M$$, and $$j^+(A)=A$$. $$\kappa$$ is $$\omega-$$fold strong $$A$$ if and only if it is $$\omega-$$fold $$\lambda-$$strong for every $$\lambda$$.

Theorem: $$\kappa$$ is $$\omega-$$fold $$\lambda-$$strong if and only if there is some $$\delta\gt\lambda$$ such that $$I2(\kappa,\delta)$$.

$$Proof.$$ Let $$j: V\rightarrow M$$ be an $$\omega-$$fold $$\lambda-$$strongness embedding. Then $$j\restriction V_\delta: V_\delta\rightarrow V_{j(\delta)}^M$$. But $$j(\delta)=\delta$$ and so $$V_{j(\delta)}^M=V_\delta$$. For the other direction, it is standard$$^6$$ to get a $$j: V\rightarrow M$$ with $$j(\delta)=\delta$$ and $$V_\delta\subseteq M$$.◼

Theorem: If $$\kappa$$ is $$\omega-$$fold strong, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\beta\lt\kappa|\exists\gamma(I2(\beta,\gamma))\}$$.

$$Proof.$$ Let $$I2(\kappa,\lambda)$$, and $$j: V\rightarrow M$$ be a non-trivial elementary embedding with critical point $$\kappa$$, such that $$\delta\gt\lambda$$, $$j(\delta)=\delta$$, and $$V_\delta\subseteq M$$. Then, if $$j’: V_\lambda\rightarrow V_\lambda$$ witness $$I2(\kappa,\lambda)$$, $$j’\in V_\delta$$ and so $$M\vDash\exists\gamma(I2(\kappa,\gamma))$$.◼

Definition: $$E_n^m(\kappa,\lambda)$$ if and only if there is a non-trivial $$\Sigma_{2m}^1-$$elementary embedding $$j: V_\lambda\rightarrow V_\lambda$$ with critical point $$\kappa$$ and $$j^k(\kappa)\in C^{(n)}$$ for every $$k$$.

Theorem: If $$E_n^{m+1}(\kappa,\lambda)$$ and $$m\gt 0$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(E_n^m(\beta,\gamma)))\}\in D$$. In particular, the consistency strength of $$E_1^2$$ is above that of an $$\omega-$$fold strong cardinal.

$$Proof.$$ Let $$j: V_\lambda\rightarrow V_\lambda$$. Note that, if $$j(\kappa)\in C^{(n+2)}$$, then $$V_{j(\kappa)}\vDash E_n^m(\kappa,\lambda)$$, and so, if $$D$$ is the measure generated by $$j$$, $$\{\beta\lt\kappa|V_{j(\kappa)}\vDash\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(E_n^m(\beta,\gamma)))\}\in D$$. But $$j(\kappa)\in C^{(n)}$$, and so $$\{\beta\lt\kappa|V_{j(\kappa)}\vDash\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(E_n^m(\beta,\gamma)))\}=\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha (E_n^m(\beta,\gamma)))\}$$.

For the second part, let $$\phi(\beta,\alpha)$$ be the statement $$\beta$$ is $$\omega-$$fold $$\alpha-$$strong. Note that for each $$\forall\alpha\lt j(\kappa)(\phi(\kappa,\alpha))$$. Also $$\forall\alpha\lt j(\kappa)(\phi(\kappa,\alpha))$$ if and only if $$\forall\alpha\lt j(\kappa)(\phi^{V_{j(\kappa)}}(\kappa,\alpha))$$ if and only if $$V_{j(\kappa)}\vDash\forall\alpha(\phi(\kappa,\alpha))$$.◼

Denote by $$E_n(\kappa,\lambda)$$ the special case $$E_n^0(\kappa,\lambda)\leftrightarrow E_n^1(\kappa,\lambda)$$.

Theorem: If $$E_{n+1}(\kappa,\lambda)$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(\forall m\lt\omega(E_n(\beta,\gamma)\land V_\gamma\prec V_\kappa)))\}\in D$$.

$$Proof.$$ Use theorem 6.33 from $$^8$$, except modify the argument to use $$\kappa_1$$ instead of $$\kappa_0$$, so that $$V_{\kappa_1}\prec V_{\text{crit}I_0}\prec…\prec V_{\text{crit}I_n}\prec V_{\kappa_1}$$, so that we can find some $$\alpha\lt\lambda$$ and some non-trivial $$\Sigma_n^1-$$elementary embedding $$j_\alpha: V_\alpha\rightarrow V_\alpha$$, with critical point $$\kappa$$, and we can assume $$\alpha\gt\kappa_1$$.◼

Theorem: If $$I1(\kappa,\lambda)$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(\forall n\lt\omega(E_n(\beta,\gamma)\land V_\gamma\prec V_\kappa)))\}\in D$$.

$$Proof.$$ We establish that, for each $$n$$, the measure $$D$$ generated by $$j: V_{\lambda+1}\rightarrow V_{\lambda+1}$$ contains $$X=\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(E_n(\beta,\gamma)\land V_\gamma\prec V_\kappa)))\}$$, and take the intersection. This follows as each such $$X$$ is definable in terms of $$\kappa$$ in $$V_{\lambda+1}$$, and so $$\kappa\in j(X)$$.◼

Definition: $$I1(\kappa,\lambda)_n$$ if and only if there exists an $$I1$$ embedding $$j: V_\lambda\rightarrow V_\lambda$$ with critical point $$\kappa$$, and $$j^k(\kappa)\in C^{(n)}$$ for every $$k$$.

Theorem: If $$I1(\kappa,\lambda)_{n+1}$$ and $$n\gt 0$$, then there is a normal measure $$D$$ on $$\kappa$$ such that $$\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha( I1(\beta,\gamma)_n))\}\in D$$.

$$Proof.$$ Let $$j: V_\lambda\rightarrow V_\lambda$$. Note that, if $$j(\kappa)\in C^{(n+2)}$$, then $$V_{j(\kappa)}\vDash I1(\kappa,\lambda)_n$$, and so, if $$D$$ is the measure generated by $$j$$, $$\{\beta\lt\kappa|V_{j(\kappa)}\vDash\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(I1(\beta,\gamma)_n))\}\in D$$. But $$j(\kappa)\in C^{(n)}$$, and so $$\{\beta\lt\kappa|V_{j(\kappa)}\vDash\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(I1(\beta,\gamma)_n))\}=\{\beta\lt\kappa|\forall\alpha\lt\kappa(\exists\gamma\gt\alpha (I1(\beta,\gamma)_n))\}$$.◼

7. Tower cardinals.

Definition: $$\kappa$$ is $$\omega-$$fold $$\lambda-$$supercompact for $$A$$ if and only if there is a sequence $$\{\zeta_i|i\lt\omega\}$$ with $$\zeta_1=\kappa+\lambda$$, and an increasing sequence $$\{\kappa_i|i\le n\}$$ with $$\kappa_1=\kappa$$, and embeddings $$e_{i,j}: V_{\zeta_i}\rightarrow V_{\zeta_j}$$ with critical point $$\kappa_i$$, $$e_{i,j}(\kappa_i)=\kappa_j$$, $$e_{i+2,i+1}(e_{i,i+1}(x))=e_{i,i+1}(x)$$, and $$e_{i,j}^+(A\cap V_{\zeta_i})=A\cap V_{\zeta_j}$$.

Definition: We write $$VP_\omega(\kappa,\underset{\sim}{\Gamma})$$ if and only if for every proper class $$C$$ of natural models $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},R_\alpha)$$ $$\Gamma-$$definable with parameters in $$H_\kappa$$, for every $$A\in C$$, there is an increasing sequence $$\{\kappa_i|i\lt\omega\}$$, and embeddings $$e_{i,j}: \mathfrak M_{\zeta_i}\rightarrow \mathfrak M_{\zeta_j}$$ with critical point $$\kappa_i$$ and $$\kappa_{i+1}\gt\zeta_i$$, $$e_{i,j}(\kappa_i)=\kappa_j$$, and $$e_{i+2,i+1}(e_{i,i+1}(x))=e_{i,i+1}(x)$$, such that $$\mathfrak M_{\zeta_0}\in H_\kappa$$ and $$\mathfrak M_{\zeta_1}=A$$. $$VP_\omega(\underset{\sim}{\Gamma})$$ if and only if for every proper class $$C$$ of natural models $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},R_\alpha)$$ $$\Gamma-$$definable, there is an increasing sequence $$\{\kappa_i|i\lt\omega\}$$, and embeddings $$e_{i,j}: \mathfrak M_{\zeta_i}\rightarrow \mathfrak M_{\zeta_j}$$ with critical point $$\kappa_i$$ and $$\kappa_{i+1}\gt\zeta_i$$, $$e_{i,j}(\kappa_i)=\kappa_j$$, and $$e_{i+2,i+1}(e_{i,i+1}(x))=e_{i,i+1}(x)$$. $$VP_\omega(\Gamma)$$ is the weakening of $$VP_\omega(\underset{\sim}{\Gamma})$$ by removing parameters.

Theorem: If $$\kappa$$ is $$\omega-$$fold supercompact, then $$VP_\omega(\kappa,\underset{\sim}{\Sigma}_2)$$.

$$Proof.$$ First, $$V_\kappa=H_\kappa$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be some $$\Sigma_n$$ definable sequence $$C$$, with parameter $$p$$. Let $$V_{\kappa+\eta}\ni p$$ be $$\Sigma_2$$ elementary in $$V$$, $$\zeta_0=\kappa+\eta$$, and let $$\zeta_i$$ be a sequence witnessing (Partial) $$\omega-$$fold supercompactness, with $$\zeta_\omega=lim_{n\rightarrow\omega}\zeta_n$$. $$\alpha\lt\kappa_1$$ and $$\alpha\ge\kappa_0$$. As $$\kappa_i\in C^{(1)}$$, then $$V_{\zeta_i}\prec_{\Sigma_2} V_{\zeta_j}$$ and each $$\zeta_i\in C^{(1)}$$, for each $$i\lt\omega$$. Then $$e_{i,j}$$ restricts to an embedding $$k_{i,j}: \mathfrak M_\alpha^{V_{\zeta_i}}\rightarrow \mathfrak M_{k_{i,j}(\alpha)}^{V_{\zeta_j}}$$. But $$\mathfrak M_\alpha^{V_{\zeta_i}}=\mathfrak M_\alpha$$ for each $$i\lt n$$. In particular, $$V_{\zeta_\omega}$$ think there is a sequence of the necessary type, and so the same holds in $$V$$.◼

Theorem: $$VP_\omega(\Pi_1)$$ with parameters of rank $$\le\alpha$$ implies the existence of an $$\omega-$$fold supercompact cardinal $$\gt\alpha$$. Therefore $$VP_\omega(\underset{\sim}{\Pi_1})$$ implies the existence of a proper class of $$\omega-$$fold supercompact cardinals.

$$Proof.$$ Assume to the contrary. Let $$C$$ be the class of $$\mathfrak M_\xi=(V_{\gamma_\xi+2},\in,\{\xi\},\gamma_\xi,\{\gamma\})_{\gamma\le\alpha}$$, where $$\gamma_\xi$$ is the least limit ordinal above $$\xi$$ such that there is no $$\kappa\gt\alpha$$ $$\omega-$$fold $$\lt\gamma_\xi-$$supercompact. This is $$\Pi_1$$ definable with parameter $$\alpha$$. Let $$e{i,j}: \mathfrak M_{\zeta_i}\rightarrow\mathfrak M_{\zeta_j}$$. By Kunen’s theorem $$\zeta_i\lt\zeta_j$$. Then let $$\kappa_1\lt\zeta_1$$ be standard. $$\kappa_1$$ is then $$\omega-$$fold $$\lt\lambda-$$supercompact. Contradiction. The second statement follows immediately from the first.◼

Theorem: The following are equivalent for $$n\gt 0$$:

$$(i)$$ $$VP_\omega(\Pi_1)$$.

$$(ii)$$ $$VP_\omega(\kappa,\underset{\sim}{\Sigma}_2 )$$ for some $$\kappa$$.

$$(iii)$$ There exists a $$\omega-$$fold supercompact cardinal.

Furthermore, the following are equivalent for $$\omega\gt 0$$:

$$(i)$$ $$VP_\omega(\underset{\sim}{\Pi}_1)$$.

$$(ii)$$ $$VP_\omega(\kappa,\underset{\sim}{\Sigma}_2 )$$ for proper class of $$\kappa$$.

$$(iii)$$ There exists a proper class of $$\omega-$$fold supercompact cardinals.

$$Proof.$$ All of this is a corollary to the above. For the first part, $$(iiii)\rightarrow (iii)$$ and $$(iii)\rightarrow (ii)$$ is by the previous theorems. $$(ii)\rightarrow (i)$$ and $$(i)\rightarrow (iiii)$$ is by the previous theorem. For the second part, $$(ii)\rightarrow (i)$$ and $$(iiii)\rightarrow (iii)$$ are obvious, and $$(iii)\rightarrow (ii)$$ follows from the previous theorems. $$(i)\rightarrow (iiii)$$ follow from the previous theorem.◼

Theorem: The following are equivalent for:

$$(i)$$ $$\kappa$$ is the least $$\omega-$$fold supercompact cardinal.

$$(ii)$$ $$\kappa$$ is the least cardinal such that $$VP_\omega(\kappa,\underset{\sim}{\Sigma}_2 )$$.

$$(iii)$$ $$\kappa$$ is the least cardinal such that $$VP_\omega(\kappa,\underset{\sim}{\Pi}_1)$$.

$$Proof.$$ The only tricky part is $$(iii)\rightarrow (i)$$. Assume to the contrary. Let $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\lambda)$$, where $$\gamma_\alpha$$ is a limit point of $$C^{(n)}$$ with uncountable cofinality, $$\lambda\in C^{(n)}$$, and there is no $$\kappa’\le\alpha$$ that is $$\omega-$$fold $$\lambda-$$supercompact. This is $$\Pi_1-$$definable. Then, if $$\alpha$$ is the critical point of $$e_{0,1}: \mathfrak M_{\zeta_0}\rightarrow\mathfrak M_{\zeta_1}$$. Then each $$e_{i,i+1}(\kappa’)$$, and so $$\kappa’$$ is $$\omega-$$fold $$\lambda-$$supercompact.◼

Theorem: For $$VP_\omega(\kappa,\underset{\sim}{\Pi}_1)$$ if and only if $$\kappa$$ is $$\omega-$$fold supercompact or a limit of such cardinals.

$$Proof.$$ The converse direction is clear. For the forward, assume to the contrary that $$\kappa$$ is not $$\omega-$$fold supercompact or a limit of such cardinals. Let $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\lambda, \{\gamma\})_{\gamma\le\zeta}$$, where $$\gamma_\alpha$$ is a limit ordinal with uncountable cofinality, and there is no $$\kappa’\le\alpha$$ that is $$\omega-$$fold $$\lambda-$$supercompact. This is $$\Pi_{m+1}-$$definable. But, if $$\kappa$$ is the critical point $$e_{1,2}$$ it is $$\omega-$$fold $$\lambda-$$supercompact.◼

Theorem: If $$\kappa$$ is $$C^{(m)}-\omega-$$fold extendible, then $$VP_\omega(\kappa,\underset{\sim}{\Sigma_{m+2}})$$.

$$Proof.$$ First, $$V_\kappa=H_\kappa$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be some $$\Sigma_n$$ definable sequence $$C$$, with parameter $$p$$. Let $$V_{\kappa+\eta}\ni p$$ be $$\Sigma_{n+2}$$ elementary in $$V$$, $$\zeta_0=\kappa+\eta$$, and let $$\zeta_i$$ be a sequence witnessing $$C^{(n)}-\omega-$$fold $$\eta-$$extendibility. $$\alpha_i\lt\kappa_{i+1}$$ and $$\alpha\gt\kappa_i$$. As $$\kappa_i\in C^{(n)}$$, then $$V_{\zeta_i}\prec_{\Sigma_{n+1}} V_{\zeta_j}$$ and each $$\zeta_i\in C^{(n)}$$, for each $$i\le n$$. Then $$e_{i,j}$$ restricts to an embedding $$k_{i,j}: \mathfrak M_\alpha^{V_{\zeta_i}}\rightarrow \mathfrak M_{k_{i,j}(\alpha)}^{V_{\zeta_j}}$$. But $$\mathfrak M_\alpha^{V_{\zeta_i}}=\mathfrak M_\alpha$$ for each $$i\lt\omega$$.◼

Theorem: $$m\gt 1$$ and $$VP_\omega(\Pi_{m+1})$$ with parameters of rank $$\le\alpha$$ implies the existence of a $$C^{(m)+}-\omega-$$fold extendible cardinal $$\gt\alpha$$. Therefore $$VP_\omega(\underset{\sim}{\Pi}_{m+1})$$ implies the existence of a proper class of $$C^{(m)+}-\omega-$$fold extendible cardinals.

$$Proof.$$ Let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$C^{(m)+}-n-$$fold $$\eta-$$extendible, and $$\kappa$$ otherwise. Let $$C=\{\lambda\in C^{(m)}|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$\mathfrak N_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha,\{\gamma\})_{\gamma\le\alpha}$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$. This is $$\Pi_{m+1}$$. We show that if $$\kappa$$ is the critical point of $$e_{0,1}: \mathfrak N_{\zeta_0}\rightarrow \mathfrak N_{\zeta_1}$$, then $$\kappa$$ is $$C^{(m)+}-\omega-$$fold extendible. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Since $$\kappa_i\lt\gamma_{\zeta_i}$$ and $$\gamma_{\zeta_i}\in C$$, $$g(\kappa_i)\lt\gamma_{\zeta_i}$$. It follows that $$e_{i,j}\restriction V_{g(\kappa_i)}\rightarrow V_{e_{i,j}(g(\kappa_i))}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt e_{i,j}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But then, these properties show that $$\kappa$$ is $$C^{(m)+}-\omega-$$fold $$\eta-$$extendible. The second statement follows immediately from the first.◼

Theorem: The following are equivalent for $$m\gt 0$$:

$$(i)$$ $$VP_\omega(\Pi_{m+1})$$.

$$(ii)$$ $$VP_\omega(\kappa,\underset{\sim}{\Sigma_{m+2}})$$ for some $$\kappa$$.

$$(iii)$$ There exists a $$C^{(m)}-\omega-$$fold extendible cardinals.

$$(iiii)$$ There exists a $$C^{(m)+}-\omega-$$fold extendible cardinals.

Furthermore, the following are equivalent for $$m\gt 0$$:

$$(i)$$ $$VP_\omega(\underset{\sim}{\Pi}_{m+1})$$.

$$(ii)$$ $$VP_\omega(\kappa,\underset{\sim}{\Sigma}_{m+2})$$ for proper class of $$\kappa$$.

$$(iii)$$ There exists a proper class of $$C^{(m)}-\omega-$$fold extendible cardinals.

$$(iiii)$$ There exists a proper class of $$C^{(m)+}-\omega-$$fold extendible cardinals.

$$Proof.$$ All of this is a corollary to the above. For the first part, $$(iiii)\rightarrow (iii)$$ and $$(iii)\rightarrow (ii)$$ is by the previous theorems. $$(ii)\rightarrow (i)$$ and $$(i)\rightarrow (iiii)$$ is by the previous theorem. For the second part, $$(ii)\rightarrow (i)$$ and $$(iiii)\rightarrow (iii)$$ are obvious, and $$(iii)\rightarrow (ii)$$ follows from the previous theorems. $$(i)\rightarrow (iiii)$$ follow from the previous theorem.◼

Theorem: The following are equivalent for $$m\gt 0$$:

$$(i)$$ $$\kappa$$ is the least $$C^{(m)+}-\omega-$$fold extendible cardinal.

$$(ii)$$ $$\kappa$$ is the least cardinal such that $$VP_\omega(\kappa,\underset{\sim}{\Sigma_{m+2}})$$.

$$(iii)$$ $$\kappa$$ is the least cardinal such that $$VP_\omega(\kappa,\underset{\sim}{\Pi}_{m+1})$$.

$$Proof.$$ The only tricky part is $$(iii)\rightarrow (i)$$. Assume to the contrary. Let $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\lambda, C^{(n)}\cap\gamma_\alpha)$$, where $$\gamma_\alpha$$ is a limit point of $$C^{(n)}$$ with uncountable cofinality, $$\lambda\in C^{(n)}$$, and there is no $$\kappa’\le\alpha$$ that is $$C^{(m)}-n-$$fold $$\lambda-$$extendible. This is $$\Pi_{m+1}-$$definable. Then, if $$\alpha$$ is the critical point of $$e_{0,1}: \mathfrak M_{\zeta_0}\rightarrow\mathfrak M_{\zeta_1}$$. Then each $$e_{i,i+1}(\kappa’)\in C^{(m)}$$, and so $$\kappa’$$ is $$C^{(m)}-\omega-$$fold $$\lambda-$$extendible.◼

Theorem: For $$m\gt 0$$, $$VP_\omega(\kappa,\underset{\sim}{\Pi_{m+1}})$$ if and only if $$\kappa$$ is $$C^{(m)}-\omega-$$fold extendible or a limit of such cardinals.

$$Proof.$$ The converse direction is clear. For the forward, assume to the contrary that $$\kappa$$ is not $$C^{(m)}-\omega-$$fold extendible or a limit of such cardinals. Let $$\mathfrak M_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\lambda, C^{(n)}\cap\alpha+1,\{\gamma\})_{\gamma\le\zeta}$$, where $$\gamma_\alpha$$ is a limit point of $$C^{(n)}$$ with uncountable cofinality, $$\lambda\in C^{(n)}$$, and there is no $$\kappa’\le\alpha$$ that is $$C^{(m)}-n-$$fold $$\lambda-$$extendible. This is $$\Pi_{m+1}-$$definable. Then, if $$\alpha$$ is the critical point of $$e_{0,1}: \mathfrak M_{\zeta_0}\rightarrow\mathfrak M_{\zeta_1}$$. Then each $$e_{i,i+1}(\kappa’)\in C^{(m)}$$, and so $$\kappa’$$ is $$C^{(m)}-n-$$fold $$\lambda-$$extendible.◼

Theorem: The following are conservative:

$$(i)$$ For every $$\Gamma$$, $$VP_\omega(\Gamma)$$.

$$(ii)$$ For every $$m$$, there is an $$\omega-C^{(m)}-$$fold extendible cardinal; alternatively, for every $$m$$, there is a $$C^{(m)}-I3$$ cardinal.

$$(iii)$$ For every $$A$$, there is $$\omega-$$fold extendible for $$A$$ cardinal.

$$Proof.$$ $$(ii)$$ extends $$(i)$$. We show with $$(ii)$$ as the weaker characterization. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be some $$\Sigma_n$$ definable class, with parameter $$p$$. Let $$V_{\kappa+\eta}\ni p$$ be $$\Sigma_n$$ elementary in $$V$$, $$\zeta_0=\kappa+\eta$$, and let $$\zeta_i$$ be a sequence witnessing $$\omega-$$fold $$\eta-$$extendibility. $$\alpha_i\lt\kappa_{i+1}$$ and $$\alpha\gt\kappa_i$$. As $$\kappa_i\in C^{(n)}$$, then $$V_{\zeta_i}\prec V_{\zeta_j}$$. Then $$e_{i,j}$$ restricts to an embedding $$k_{i,j}: \mathfrak M_\alpha^{V_{\zeta_i}}\rightarrow \mathfrak M_{j(\alpha)}^{V_{\zeta_j}}$$. But $$\mathfrak M_\alpha^{V_{\zeta_i}}=\mathfrak M_\alpha$$.

$$(iii)$$ extends $$(i)$$. Let $$A=\{\mathfrak M_\alpha|\alpha\in Ord\}$$, and $$F(\beta)=\text{sup}\{\alpha\lt\beta|rank(\mathfrak M_\alpha)\lt\beta\}$$. Then we have non-trivial elementary embeddings from some $$(V_{\zeta_i},\in,\mathfrak M_\alpha)_{\alpha\lt F(\zeta_i)}$$ into $$(V_{\zeta_j},\in,\mathfrak M_\alpha)_{\alpha\lt F(\zeta_j)}$$.

$$(i)$$ extends to $$(ii)$$. Finally, it remains to verify that every model $$M_0$$ of $$(i)$$ is a model of $$(ii)$$ and $$(iii)$$. We show with $$(ii)$$ as the stronger characterization. First, for $$(ii)$$, let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$C^{(m)}-\omega-$$fold $$\eta-$$extendible, and $$\kappa$$ otherwise. Let $$C=\{\lambda\in C^{(m)}|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$\mathfrak N_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha)$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$. We show that if $$\kappa$$ is the critical point of $$e_{0,1}: \mathfrak N_{\zeta_0}\rightarrow \mathfrak N_{\zeta_1}$$, then $$\kappa$$ is $$C^{(m)}-\omega-$$fold extendible. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Since $$\kappa_i\lt\gamma_{\zeta_i}$$ and $$\gamma_{\zeta_i}\in C$$, $$g(\kappa_i)\lt\gamma_{\zeta_i}$$. It follows that $$e_{i,j}\restriction V_{g(\kappa_i)}\rightarrow V_{e_{i,j}(g(\kappa_i))}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt e_{i,j}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But then, these properties show that $$\kappa$$ is $$C^{(m)}-\omega-$$fold $$\eta-$$extendible.

$$(i)$$ extends to $$(iii)$$. Let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$\omega-$$fold $$\eta-$$extendible for $$A$$, and $$\kappa$$ otherwise. Let $$C=\{\lambda\in C^{(m)}|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$\mathfrak N_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha, A\cap V_{\gamma_\alpha})$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$. We show that if $$\kappa$$ is the critical point of $$e_{0,1}: \mathfrak N_{\zeta_0}\rightarrow \mathfrak N_{\zeta_1}$$, then $$\kappa$$ is $$\omega-$$fold extendible for $$A$$. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Since $$\kappa_i\lt\gamma_{\zeta_i}$$ and $$\gamma_{\zeta_i}\in C$$, $$g(\kappa_i)\lt\gamma_{\zeta_i}$$. It follows that $$e_{i,j}\restriction V_{g(\kappa_i)}\rightarrow V_{e_{i,j}(g(\kappa_i))}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt e_{i,j}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But then, these properties show that $$\kappa$$ is $$\omega-$$fold $$\eta-$$extendible for $$A$$.◼

Theorem: The following are equivalent:

$$(i)$$ $$\kappa$$ is $$\omega-$$fold Vopěnka.

$$(ii)$$ For every $$A\subseteq V_\kappa$$, there is some $$\kappa’\lt\kappa$$ $$\omega-$$fold $$\lt\kappa-$$extendible for $$A$$.

$$(iii)$$ For every $$A\subseteq V_\kappa$$, there is some $$\kappa’\lt\kappa$$ $$\omega-$$fold $$\lt\kappa-$$extendible for $$A$$.

$$Proof.$$ $$(i)\rightarrow (ii)$$. Let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$\omega-$$fold $$\eta-$$extendible for $$A$$, and $$\kappa$$ otherwise. Let $$C=\{\lambda|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$\mathfrak N_\alpha=(V_{\gamma_\alpha},\in,\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha, A\cap V_{\gamma_\alpha})$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$. We show that if $$\kappa$$ is the critical point of $$e_{0,1}: \mathfrak N_{\zeta_0}\rightarrow \mathfrak N_{\zeta_1}$$, then $$\kappa$$ is $$n-$$fold extendible for $$A$$. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Since $$\kappa_i\lt\gamma_{\zeta_i}$$ and $$\gamma_{\zeta_i}\in C$$, $$g(\kappa_i)\lt\gamma_{\zeta_i}$$. It follows that $$e_{i,j}\restriction V_{g(\kappa_i)}\rightarrow V_{e_{i,j}(g(\kappa_i))}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt e_{i,j}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But then, these properties show that $$\kappa$$ is $$\omega-$$fold $$\eta-$$extendible for $$A$$.

$$(ii)\rightarrow (iii)$$. Let $$\lambda\gt\kappa’$$. Then, let $$e_{i,k}=j^k\restriction V_{j^i(\kappa’)}$$. By closure for $$M$$, we can easily get that $$M\vDash(j(\kappa’)\text{ is }n-\text{fold }\lambda-\text{supercompact for }A)$$, and the rest follows by elementarity.

$$(iii)\rightarrow (i)$$. Let $$A=\{\mathfrak M_\alpha|\alpha\in Ord\}$$, and $$F(\beta)=\text{sup}\{\alpha\lt\beta|rank(\mathfrak M_\alpha)\lt\beta\}$$. Then we have non-trivial elementary embeddings from some $$(V_{\zeta_i},\in,\mathfrak M_\alpha)_{\alpha\lt F(\zeta_i)}$$ into $$(V_{\zeta_j},\in,\mathfrak M_\alpha)_{\alpha\lt F(\zeta_j)}$$.◼

Definition: $$\kappa$$ is $$\omega-$$fold Woodin if and only if for every $$A\subseteq V_\kappa$$, there is some $$\alpha\lt\kappa$$ such that $$\omega-$$fold $$\lt\kappa-$$strong. $$\kappa$$ is $$\omega-$$fold Woodin for supercompactness if and only if for every $$A\subseteq V_\kappa$$, there is some $$\alpha\lt\kappa$$ such that $$\omega-$$fold $$\lt\kappa-$$supercompact.

Theorem: $$\kappa$$ is $$\omega-$$fold Woodin for sueprcompactness if and only if $$\kappa$$ is $$\omega-$$fold Vopěnka. Similarly, for every class $$A$$, there is some $$\kappa$$ $$\omega-$$fold strong for $$A$$ is conservative over for every $$n\lt\omega$$ there is some $$\kappa$$ $$C^{(n)}-\omega-$$fold strong.

$$Proof.$$ The first part is trivial in the context of the previous theorem. For the second part:

$$(ii)$$ extends $$(i)$$. We can assume every class is definable. Let $$A$$ be some $$\Sigma_n$$ definable class, with parameter $$p$$. Let $$\lambda\gt rank(p)$$ be such that $$\lambda\in C^{(n)}$$ (As there is a cofinal in $$\lambda$$ sequence of such cardinals, we can get such a $$\lambda$$). Then $$A\cap V_\lambda$$ is definable in $$V_{\lambda+1}$$ and so $$j^+(A\cap V_\lambda)=V_\lambda$$.

$$(i)$$ extends to $$(ii)$$. Let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$I1(\kappa,\gamma)_n$$ for $$\gamma\gt\eta$$, and $$\kappa$$ otherwise. Let $$C=\{\lambda\in C^{(m)}|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$A=(\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha)$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$.. We show that if $$\kappa$$ is the critical point of $$j: V\rightarrow M$$ for arbitrarily large $$\alpha$$ (Which it is easy to see exists), then $$\kappa$$ is $$I2(\kappa,\gamma)_n$$ for arbitarily large $$\gamma$$. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Let $$g(\kappa)=\kappa+\eta$$, and $$\alpha\gt g(\kappa)$$. Since $$\kappa_i\lt\gamma_\alpha$$ and $$\gamma_\alpha\in C$$, $$g(\kappa_i)\lt\gamma_\alpha$$. It follows that $$j^{n+}(j): V_{\gamma_\alpha}\rightarrow V_{\gamma_\alpha}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt j^{(k)}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But these properties show $$\kappa$$ is $$I1(\kappa,\gamma)_n$$ for $$\gamma\gt\eta$$.◼

Definition: The $$I1-$$tower principle holds if and only if for each $$A\subseteq V_\kappa$$, the exists some $$\kappa$$ such that for each $$\alpha$$ there is a $$\lambda\gt\alpha$$ such that $$I1(\kappa,\lambda)$$ (As witnessed by $$j$$) and $$j(A\cap V_\lambda)=A\cap V_\lambda$$, and $$\alpha\lt\kappa$$.

We say the $$I1-$$tower principle holds if and only if $$Ord$$ is an $$I1-$$tower, with $$A$$ being arbitrary classes.

Theorem: The $$I1-$$tower principle is conservative over the assertion that for each $$n$$, there is some $$\kappa$$ such that for each $$\alpha$$, for some $$\gamma\gt\alpha$$ $$I1(\kappa,\gamma)_n$$.

$$Proof.$$ $$(ii)$$ extends $$(i)$$. We can assume every class is definable. Let $$A$$ be some $$\Sigma_n$$ definable class, with parameter $$p$$. Let $$\lambda\gt rank(p)$$ be such that $$\lambda\in C^{(n)}$$ (As there is a cofinal in $$\lambda$$ sequence of such cardinals, we can get such a $$\lambda$$). Then $$A\cap V_\lambda$$ is definable in $$V_{\lambda+1}$$ and so $$j^+(A\cap V_\lambda)=V_\lambda$$.

$$(i)$$ extends to $$(ii)$$. Let $$g(\kappa)$$ be the least $$\kappa+\eta$$ such that $$\kappa$$ is not $$I1(\kappa,\gamma)_n$$ for $$\gamma\gt\eta$$, and $$\kappa$$ otherwise. Let $$C=\{\lambda\in C^{(m)}|g\restriction\lambda: \lambda\rightarrow\lambda\}$$. Let $$\{\mathfrak M_\alpha|\alpha\in Ord\}$$ be a natural sequence such that for any $$j: \mathfrak M_\alpha\rightarrow \mathfrak M_\beta$$ with critical point $$\lambda$$, $$\lambda\in C$$.

Let $$\mathfrak N_\alpha=(V_{\gamma_\alpha+1},\in,\{\alpha\},\mathfrak M_\alpha, C\cap\gamma_\alpha)$$, where $$\gamma_\alpha$$ is the least limit point of $$C$$ above every ordinal in the domain of $$\mathfrak M_\alpha$$. We show that if $$\kappa$$ is the critical point of $$j: \mathfrak N_\alpha\rightarrow \mathfrak N_\alpha$$ for arbitrarily large $$\alpha$$ (Which it is easy to see exists), then $$\kappa$$ is $$I1(\kappa,\gamma)_n$$ for arbitarily large $$\gamma$$. Now assume to the contrary $$g(\kappa)\gt\kappa$$.

Let $$g(\kappa)=\kappa+\eta$$, and $$\alpha\gt g(\kappa)$$. Since $$\kappa_i\lt\gamma_\alpha$$ and $$\gamma_\alpha\in C$$, $$g(\kappa_i)\lt\gamma_\alpha$$. It follows that $$j^{n+}(j): V_{\gamma_\alpha}\rightarrow V_{\gamma_\alpha}$$ is an elementary embedding with critical point $$\kappa_i$$. Then $$\kappa_i\in C$$ as $$\mathfrak M_\alpha$$ is encoded in $$\mathfrak N_\alpha$$. Therefore $$g(\kappa_i)\lt j^{(k)}(\kappa_i)$$, where $$g(\kappa_0)=\kappa+\eta$$. But these properties show $$\kappa$$ is $$I1(\kappa,\gamma)_n$$ for $$\gamma\gt\eta$$.◼

Theorem (Gabe Goldberg$$^8$$): If there exists a non-trivial elementary embedding $$j: L_1(V_{\lambda+1})\rightarrow L_1(V_{\lambda+1})$$ with critical point $$\kappa$$, then $$\kappa$$ is an $$I1-$$tower.

8. The Reinhardt hierarchy.

Definition: $$\kappa$$ is weakly Reinhardt if and only if there exists a non-trivial elementary embedding $$j: V_{\lambda+1}\rightarrow V_{\lambda+1}$$ with critical point $$\kappa$$, and $$V_\kappa\prec V$$.

Theorem: The existence of a weakly Reinhardt cardinal is conservative over the assertion that for every definable club $$C$$, there is some $$\kappa\in C$$ such that $$\kappa$$ is $$I1$$. In particular, $$I0$$ implies the consistency of a weakly Reinhardt cardinal.

$$Proof.$$ Let $$M$$ be some model of the existence of a weakly Reinhardt cardinal $$\kappa$$. Then, if $$n$$ is a natural, $$\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(\gamma\text{ is }I1\land\Sigma_n-\text{reflecting}))$$, and so $$\forall\alpha\lt\kappa(V_\kappa\vDash\exists\gamma\gt\alpha(\gamma\text{ is }I1\land\Sigma_n-\text{reflecting}))$$, and so $$V_\kappa\vDash\forall\alpha(\exists\gamma\gt\alpha(\gamma\text{ is }I1\land\Sigma_n-\text{reflecting}))$$, and so $$\forall\alpha(\exists\gamma\gt\alpha(\gamma\text{ is }I1\land\Sigma_n-\text{reflecting}))$$. Then, if $$C=\{\alpha|\phi(\alpha,p)\}$$, let $$\gamma\gt rank(p)$$ and $$\gamma$$ is $$\Sigma_n-$$reflecting be $$I1$$, where $$\phi(x,y)$$ is $$\Sigma_n$$. Then $$C$$ is club below $$\gamma$$, and so $$\gamma\in C$$. Therefore $$M$$ is a model of the assertion that for every definable club $$C$$, there is some $$\kappa\in C$$ such that $$\kappa$$ is $$I1$$.

For the other direction, let $$M$$ be some model such that in $$M$$ for every definable club $$C$$, there is some $$\kappa\in C$$ such that $$\kappa$$ is $$I1$$, and add to the elementary diagram of $$M$$ a constant symbol $$\kappa$$, together with the axioms that $$\kappa$$ is $$I1$$ and $$\Sigma_n-$$reflecting for every $$n$$. Then every finite fragment of that theory is consistent, so by Compactness it has some model $$M’$$. It is easy to generate an elementary embedding $$j: M\rightarrow M’$$.

To get the last result, simply use the fact that if $$j: L(V_{\lambda+1})\rightarrow L(V_{\lambda+1})$$ is elementary, then so is $$j\restriction L_1(V_{\lambda+1}): L_1(V_{\lambda+1})\rightarrow L_1(V_{\lambda+1})$$, so that in particular every $$I0$$ cardinal is an $$I1$$ tower.◼

The following cardinals are defined in the $$ZF$$ context.

Definition: $$\kappa$$ is strongly Reinhardt if and only if there there exists a non-trivial elementary embedding $$j: V_{\lambda+2}\rightarrow V_{\lambda+2}$$ with critical point $$\kappa$$, and $$\lambda=\lim_{n\rightarrow\omega}j^n(\kappa)$$ and $$V_\lambda\prec V_\gamma$$ for some $$\gamma\gt\lambda$$.

Theorem: If $$\kappa$$ is strongly Reinhardt, then there is a normal measure $$D$$ on $$\kappa$$, such that $$\{\beta\lt\kappa|\forall n\lt\omega,\alpha\lt\kappa(\exists\gamma’\lt\kappa(\gamma’\gt\alpha\land I1(\beta,\gamma’)\land V_\gamma\prec V_\kappa))\}\in D$$.

$$Proof.$$ Let $$j: V_{\lambda+2}\rightarrow V_{\lambda+2}$$, and $$V_\lambda\prec V_\gamma$$. Then each $$V_{j^n(\kappa)}\prec V_\lambda$$, and so $$V_\gamma\vDash\forall n\lt\omega,\alpha\lt\lambda(\exists\gamma'(\gamma’\gt\alpha\land I1(\kappa,\gamma’)_n))$$, and the same holds in $$V_\lambda$$ and so in $$V_{j(\kappa)}$$. Then let $$D$$ be the measure generated by $$j$$.◼

Definition: $$\kappa$$ is superstrongly Reinhardt(=Reinhardt) if and only if there is a non-trivial elementary embedding $$j: V\rightarrow V$$ with critical point $$\kappa$$.

Definition: $$\kappa$$ is Reinhardt $$\alpha-$$many times if and only if there is some increasing sequence $$\{\lambda_\beta|\beta\le\alpha\}$$ such that for each $$\beta\le\alpha$$, $$\kappa$$ is Reinhardt with target $$\lambda_\beta$$.

Theorem: If $$\kappa$$ is Reinhardt $$\alpha+\gamma-$$many times with target $$\lambda_\alpha$$, then $$\lambda_\alpha$$ is Reinhardt $$\gamma-$$many times.

$$Proof.$$ Let $$j’$$ witness $$\kappa$$ is Reinhardt $$\alpha+\gamma-$$many times with target $$\lambda_\alpha$$, and $$j$$ witness $$\kappa$$ is Reinhardt $$\alpha+\gamma-$$many times with target $$\lambda_{\alpha+\gamma}$$. Then let $$k=j^+(j’)$$.◼

Definition: $$\kappa$$ is super Reinhardt if and only if $$\kappa$$ is Reinhardt $$\alpha-$$many times for every $$\alpha$$.

A simple corollary to the previous theorem is that, if $$\kappa$$ is super Reinhardt, then there is a proper class of super Reinhardt cardinals. In particular, if $$\lambda$$ is a target for $$\kappa$$, $$\lambda$$ is super Reinhardt.

Theorem: If $$\kappa$$ is super Reinhardt, then $$V_\kappa\prec V$$.

$$Proof.$$ Let $$V_\kappa$$ reflect $$\phi(x,x_0…x_n)$$. Assume $$\exists x(\phi(x,x_0…x_n))$$, and let $$z$$ be a witness to that. Then for $$x_0…x_n\in V_\kappa$$ and $$j(\kappa)\gt rank(z)$$, $$V_\kappa\vDash\exists x(\phi(x,x_0…x_n))$$ if and only if $$V_{j(\kappa)}\vDash\exists x(\phi(x,x_0…x_n))$$. But $$z\in V_{j(\kappa)}$$.◼

Theorem: If $$\kappa$$ is super Reinhardt, then there is a normal measure $$D$$ on $$\kappa$$, such that $$\{\beta\lt\kappa|\beta\text{ is strongly Reinhardt}\}$$.

$$Proof.$$ Let $$j: V\rightarrow V$$ be a non-trivial elementary embedding with critical point $$\kappa$$. Then for every $$m$$, $$\kappa\in C^{(n)}$$ if and only if $$j^m(\kappa)\in C^{(n)}$$ if and only if $$\lambda\in C^{(n)}$$, where $$\lambda$$ is the least fixed point above $$\kappa$$ (I.e. The supremum of the critical sequence). Therefore $$V_\lambda\prec V$$, and $$j\restriction V_{\lambda+2}: V_{\lambda+2}\rightarrow V_{\lambda+2}$$, and if $$k: V\rightarrow V$$ is a non-trivial elementary embedding with critical point $$\kappa$$, with $$k(\kappa)\gt\lambda$$, the same logic establishes $$V_\lambda\prec V_\gamma$$. furthermore, the property of being strongly Reinhardt is first-order expressible.◼

Theorem: If $$\kappa$$ is super Reinhardt, then there is an unbounded class of inaccessibles $$\gamma\lt\kappa$$ such that $$(V_\gamma,V_{\gamma+1})\vDash\exists\kappa'(\kappa’\text{ is Reinhardt}\land\exists D(D\text{ is a normal measure}\land\{\lambda’\lt\kappa’|\lambda\text{ is Reinhardt}\}\in D))$$.

$$Proof.$$ Let $$j: V\rightarrow V$$ be a non-trivial elementary embedding with critical point $$\kappa$$, and $$\lambda$$ the supremum of the critical sequence, so that $$j(\lambda)=\lambda$$. Then let $$\gamma_0$$ be the least inaccessible above $$\lambda$$, so that $$\gamma_0$$ is definable from $$\lambda$$ and so $$j(\gamma_0)=\gamma$$. Then $$k=j\restriction V_{\gamma_0+1}$$ is a non-trivial elementary embedding for $$V_{\gamma_0+1}$$ into itself with critical point $$\kappa$$, so that $$(V_{\gamma_0},V_{\gamma_0+1})\vDash\kappa\text{ is Reinhardt}$$, and as $$k$$ preserves second order properties, if $$D$$ is the measure generated by $$j$$, $$\{\lambda’\lt\kappa|(V_{\gamma_0},V_{\gamma_0+1})\vDash\kappa\text{ is Reinhardt}\}\in D$$. Now, let $$\phi(\gamma)$$ be the statement that $$(V_\gamma,V_{\gamma+1})\vDash\exists\kappa'(\kappa’\text{ is Reinhardt}\land\exists D(D\text{ is a normal measure}\land\{\lambda’\lt\kappa’|\lambda\text{ is Reinhardt}\}\in D))$$. Then $$\forall\alpha\lt\kappa(\exists\gamma\gt\alpha(\phi(\gamma))$$ if and only if $$\forall\alpha\lt\kappa(\exists\gamma\gt\alpha,\lt\kappa(\phi(\gamma))$$.◼

9. Many open problems.

Question: If $$\kappa$$ is hyper $$n-$$huge, then is there a normal measure on $$\kappa$$ concentrating on ultra $$n-$$huge cardinals? What is the consistency strength of hyper $$n-$$huge cardinals relative to ultra $$n-$$huge cardinals?

Question: Where do the variants of stationarily superhugeness* sit? Asserting the set of $$\alpha$$ is stationary, the set of $$j^i(\kappa)$$ is stationary, the set of $$j^{n+1}(\kappa)$$ is stationary, or $$j^{n+1}(\kappa)\gt\alpha$$.

Question: If $$\kappa$$ is ultra $$n-$$huge, then is there a normal measure on $$\kappa$$ concentrating on almost ultra $$n-$$huge cardinals? What about superhuge cardinals?

Question: If $$\kappa$$ is almost ultra $$n-$$huge, then is there a normal measure on $$\kappa$$ concentrating on super almost $$n-$$huge cardinals? Is $$\kappa$$ necessarily super $$n-$$huge? What is the consistency strength of almost ultra $$n-$$huge cardinals?

Question: If $$\kappa$$ is $$C^{(m+1)}-$$super $$n-$$huge, then is there a normal measure on $$\kappa$$ concentrating on $$C^{(m)}-$$super $$n-$$huge cardinals? What about ultrahuge, hyperhyge, superhuge* cardinals? What about super almost huge cardinals or almost ultrahuge cardinals?

Question: Can the least $$IE$$ cardinal be $$I2$$? Is the consistency strength of $$I2$$ above $$IE$$?

Question: What is the relation between $$\omega-$$fold supercompact and $$\omega-$$fold extendible cardinals?

Question: Where does $$I0$$ sit in the Reinhardt hierarchy?

Question: Where does a Reinhardt cardinal sit next to a strongly Reinhardt in terms of consistency strength? What about size?

Question: If $$\kappa$$ is Reinhardt, is $$\kappa$$ necessarily $$C^{(n)}-$$Reinhardt for some $$n\gt 1$$?

Question: If there is a non-trivial elementary embedding $$j: V\rightarrow V$$, does it follow $$0^‡$$ exists.

10. References.

$$^1$$ Elementary Chains and $$C^{(n)}-$$cardinals, Konstantinos Tsaprounis

$$^2$$ Ultrahuge cardinals, Konstantinos Tsaprounis

$$^3$$ Virtual large cardinals, Victoria Gitman and Ralf Schindler

$$^4$$ Double helix in large large cardinals and iteration of elementary embeddings, Sato Kentaro

$$^5$$ Many times huge and superhuge cardinals, Julius B. Barbanel, Carlos A. Diprisco, and It Beng Tan

$$^6$$ The Higher Infinite, Akihiro Kanamori

$$^7$$ The iterability hierarchy above I3, Alessandro Andretta and Vincenzo Dimonte

$$^8$$ Where does this strengthening of I1 stand? MathOverflow (Answer), Gabe Goldberg