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$\Pi_2$ large cardinal notions; generalizations of weakly hyper-Woodin and hyper-Woodin cardinals

I'll answer the title question, which I will interpret as being about direct implications, that is, whether all $$\theta$$ cardinals are $$\sigma$$ cardinals. There are some general rules for which large cardinal properties imply each other depending on their Lévy complexity:

• If $$\theta$$ is weaker than $$\sigma$$, then, of course, $$\theta$$ does not imply $$\sigma$$.
• If $$\theta$$ is stronger than $$\sigma$$ and the complexity of $$\theta$$ is less than or incomparable to that of $$\sigma$$, then $$\theta$$ generally does not imply $$\sigma$$. Exception: inaccessible cardinals are worldly even though inaccesibility is $$\Pi_1$$-definable and wordliness is only $$\Delta_2$$-definable (if wordly cardinals were $$\Pi_1$$-definable, worldliness would be downward absolute to inner models, and it is not); I think there are other exceptions involving virtual large cardinals but I don't know them that well.
• If $$\theta$$ is stronger than $$\sigma$$ and the complexity of $$\theta$$ is greater than or equal to that of $$\sigma$$ and at least $$\Sigma_2$$ or $$\Pi_2$$, then $$\theta$$ generally implies $$\sigma$$. Exception: Enhanced supercompact cardinals are not generally $$C^{(2)}$$-superstrong even though both enhanced supercompactness and $$C^{(2)}$$-superstrongness both have complexity $$\Sigma_3$$.
• If $$\theta$$ is stronger than $$\sigma$$ and both have complexity $$\Delta_2$$ or $$\Pi_1$$, $$\theta$$ may or may not imply $$\sigma$$.

For $$\Delta_2$$- or $$\Pi_1$$-definable large cardinal properties, a finer hierarchy is useful. We can call it the extended local Lévy hierarchy and it is based on alternation of quantifiers over the elements of $$V_{\kappa + n}$$ (subsets of $$V_{\kappa + n - 1}$$), where quantifiers over lower ranks are treated as bounded quantifiers. Large cardinal properties generally don't imply other properties that are higher in this hierarchy.

Here's a list. Most of these properties are described in Cantor's Attic.

I'll start with $$\Delta_2$$-definable properties. I'm not sure which of them are $$\Pi_1$$-definable but I note where I'm sure that they are. Additionally I note their complexity in the extended local Lévy hierarchy.

• The weakest large cardinal property is worldly [$$\Delta^1_1 (V_\kappa)$$].
• An inaccessible cardinal [$$\Pi_1$$/$$\Pi^1_1 (V_\kappa)$$] is worldly.
• An $$\alpha$$-inaccessible cardinal [$$\Pi_1$$ at least for finite $$\alpha$$/$$\Pi^1_1 (V_\kappa)$$] More generally, an inaccessible cardinal of non-trivial Carmody degree [$$\Pi^1_1 (V_\kappa)$$] is inaccessible of every lesser degree.
• A (boldface) $$\Sigma_2$$-Mahlo cardinal (equivalently $$\Delta_2$$-Mahlo) [$$\Pi^1_1 (V_\kappa)$$] is inaccessible of every Carmody degree.
• A (boldface) $$\Pi_2$$-Mahlo (equivalently $$\Sigma_3$$-Mahlo) [$$\Pi^1_1 (V_\kappa)$$] is $$\Sigma_2$$-Mahlo. A $$\Pi_{n+1}$$-Mahlo cardinal (equivalently $$\Sigma_{n+2}$$-Mahlo) [$$\Pi^1_1 (V_\kappa)$$] is $$\Pi_n$$-Mahlo. A lightface $$\Pi_{n+2}$$-Mahlo cardinal [$$\Pi^1_1 (V_\kappa)$$] is lightface $$\Pi_{n+1}$$-Mahlo and boldface $$\Pi_n$$-Mahlo but not generally boldface $$\Pi_{n+1}$$-Mahlo. A boldface $$\Pi_n$$-Mahlo is lightface $$\Pi_n$$-Mahlo.
• A cardinal that is $$\Pi_n$$-Mahlo for every finite $$n$$ is said to be $$\Pi_\omega$$-Mahlo [$$\Pi^1_1 (V_\kappa)$$].
• A Mahlo cardinal [$$\Pi_1$$/$$\Pi^1_1 (V_\kappa)$$] is $$\Pi_\omega$$-Mahlo.
• An $$\alpha$$-Mahlo cardinal [$$\Pi_1$$ for finite $$\alpha$$/$$\Pi^1_1 (V_\kappa)$$] is $$\beta$$-Mahlo for every $$\beta \lt \alpha$$, where 0-Mahlo is the same as Mahlo. A cardinal $$\kappa$$ that is $$\alpha$$-Mahlo for every $$\alpha \lt \kappa^+$$ is said to be greatly Mahlo [$$\Pi^1_1 (V_\kappa)$$] (if I understand correctly).
• A weakly compact cardinal [$$\Pi^1_2 (V_\kappa)$$] is greatly Mahlo.
• A $$\Pi^n_m$$-indescribable cardinal [$$\Pi^n_{m+1} (V_\kappa)$$?] is $$\Pi^n_k$$-indescribable for all $$k \lt m$$ and thus $$\Pi^i_k$$-indescribable for every $$i \lt n$$ and every $$k \lt \omega$$, for $$\Pi^{i+1}_0$$-indescribable is the same as $$\Pi^i_k$$-indescribable for every $$k \lt \omega$$. Weakly compact is equivalent to $$\Pi^1_1$$-indescribable. A cardinal that is $$\Pi^n_m$$-indescribable for every $$n, m \lt \omega$$ is said to be totally indescribable [$$\Pi^\omega_2 (V_\kappa)$$]
• A $$\gamma$$-strongly unfoldable cardinal $$\kappa$$ [$$\Pi^{\gamma+1}_1 (V_\kappa)$$?] is $$\eta$$-strongly unfoldable for every $$\eta$$ such that $$\kappa \le \eta \lt \gamma$$. For finite $$n$$, $$n$$-strongly unfoldable is equivalent to $$\Pi^{n+1}_1$$-indescribable (this follows from Hauser's characterization of indescribable cardinals).
• A subtle cardinal [$$\Pi_1$$/$$\Pi^1_1 (V_\kappa)$$] is greatly Mahlo but not generally weakly compact.
• An almost ineffable cardinal [$$\Pi^1_2 (V_\kappa)$$] is subtle and weakly compact but not generally $$\Pi^1_2$$-indescribable.
• An ineffable cardinal [$$\Pi^1_3 (V_\kappa)$$] is almost ineffable and $$\Pi^1_2$$-indescribable but not generally $$\Pi^1_3$$-indescribable.
• An $$n$$-subtle cardinal [$$\Pi_1$$/$$\Pi^1_1 (V_\kappa)$$] is $$m$$-subtle for all $$m \lt n$$, where subtle is the same as 1-subtle or 2-subtle, depending on your convention, but not generally weakly compact.
• An $$n$$-almost ineffable cardinal [$$\Pi^1_2 (V_\kappa)$$] is $$n$$-subtle and $$m$$-almost ineffable for all $$m \lt n$$, but not generally $$\Pi^1_2$$-indescribable.
• An $$n$$-ineffable cardinal [$$\Pi^1_3 (V_\kappa)$$] is $$n$$-almost ineffable and $$m$$-ineffable for all $$m \lt n$$, but not generally $$\Pi^1_3$$-indescribable.
• A cardinal that is $$n$$-subtle for all $$n \lt \omega$$ may be called totally subtle [$$\Pi^1_1 (V_\kappa)$$]. A totally subtle cardinal is not generally weakly compact.
• A cardinal that is $$n$$-almost ineffable for all $$n \lt \omega$$ may be called totally almost ineffable [$$\Pi^1_2 (V_\kappa)$$]. A totally almost ineffable cardinal is not generally $$\Pi^1_2$$-indescribable.
• A cardinal that is $$n$$-ineffable for all $$n \lt \omega$$ is called totally ineffable [$$\Pi^1_3 (V_\kappa)$$]. A totally ineffable cardinal is not generally $$\Pi^1_3$$-indescribable.
• A completely ineffable cardinal [$$\Delta^2_1 (V_\kappa)$$ if I remember correctly] is totally ineffable and $$\Pi^2_0$$-indescribable but not generally $$\Pi^2_1$$-indescribable.
• A weakly Ramsey cardinal [$$\Pi^1_2 (V_\kappa)$$] is totally almost ineffable but not generally $$\Pi^1_2$$-indescribable.
• An $$\alpha$$-iterable cardinal [$$\Pi^1_2 (V_\kappa)$$] is $$\beta$$-iterable for every $$\beta \lt \alpha$$, where 1-iterable is the same as weakly Ramsey, but not generally $$\Pi^1_2$$-indescribable. The iterable hierarchy ends at $$\omega_1$$-iterable.
• An $$\alpha$$-club Erdős cardinal (for infinite $$\alpha$$) [$$\Pi^1_1 (V_\kappa)$$] is totally subtle but not generally weakly compact. An $$\alpha+1$$-iterable cardinal (for countable $$\alpha$$) is $$\alpha$$-club Erdős (lemma 4.5 of Gitman and Schindler).
• An almost Ramsey cardinal [$$\Pi^1_1 (V_\kappa)$$] is a beth fixed point but not generally worldly.
• A cardinal $$\kappa$$ that is $$\alpha$$-club Erdős for all $$\alpha \lt \kappa$$ [$$\Pi^1_1 (V_\kappa)$$] is almost Ramsey but not generally weakly compact.
• A pre-Ramsey cardinal [$$\Pi^1_1 (V_\kappa)$$?] is $$\alpha$$-club Erdős for all $$\alpha \lt \kappa$$ (at least I think so) but not generally weakly compact.
• A Ramsey cardinal [$$\Pi^1_2 (V_\kappa)$$] is pre-Ramsey and $$\omega_1$$-iterable but not generally $$\Pi^1_2$$-indescribable.
• An ineffably Ramsey cardinal [$$\Pi^1_3 (V_\kappa)$$] is Ramsey and totally ineffable but not generally $$\Pi^1_3$$-indescribable.
• A $$\Pi_n$$-Ramsey cardinal [$$\Pi^1_{n+2} (V_\kappa)$$] is $$\Pi_m$$-Ramsey for $$m \lt n$$ and $$\Pi^1_{n+1}$$-indescribable but not generally $$\Pi^1_{n+2}$$-indescribable. $$\Pi_0$$-Ramsey is the same as Ramsey and $$\Pi_1$$-Ramsey is the same as ineffably Ramsey.
• A $$\Pi_\alpha$$-Ramsey cardinal [$$\Delta^2_1 (V_\kappa)$$?] is $$\Pi_\beta$$-Ramsey for $$\beta \lt \alpha$$ but probaly not generally $$\Pi^2_1$$-indescribable. If $$\kappa$$ is $$\Pi_\alpha$$-Ramsey for all $$\alpha \lt {(2^{\kappa})}^+$$, it is said to be completely Ramsey [$$\Pi^2_1 (V_\kappa)$$?].
• An almost fully Ramsey cardinal [$$\Pi^2_1 (V_\kappa)$$] is completely Ramsey but probaly not generally $$\Pi^2_1$$-indescribable.
• A strongly Ramsey cardinal [$$\Pi^1_2 (V_\kappa)$$] is Ramsey but not generally $$\Pi^1_2$$-indescribable.
• A super Ramsey cardinal [$$\Delta^2_1 (V_\kappa)$$] is strongly Ramsey and $$\Pi_\omega$$-Ramsey but not generally $$\Pi^2_1$$-indescribable and probably not generally $$\Pi_{\omega+1}$$-Ramsey
• A fully Ramsey cardinal [$$\Pi^2_1 (V_\kappa)$$] but probaly not generally $$\Pi^2_1$$-indescribable.
• A locally measurable cardinal [$$\Pi^1_2 (V_\kappa)$$] is strongly Ramsey but not generally $$\Pi^1_2$$-indescribable.
• A measurable cardinal [$$\Sigma^2_1 (V_\kappa)$$] is locally measurable, fully Ramsey and $$\Pi^2_1$$-indescribable (=+1-strongly unfoldable) but not generally $$\Pi^2_2$$-indescribable.
• A $$+\gamma$$-strong cardinal [$$\Sigma^{\gamma+1}_1 (V_\kappa)$$?] is $$+\eta$$-strong for every $$\eta \lt \gamma$$ and $$+\gamma$$-strongly unfoldable but not generally $$+\gamma+1$$-strongly unfoldable (or $$\Pi^{\gamma+1}_2$$-indescribable for finite $$\gamma$$). Measurable is equivalent to +1-strong.
• A Woodin cardinal [$$\Pi^1_1 (V_\kappa)$$] is pre-Ramsey (any cardinal in which the pre-Ramsey cardinals are stationary is pre-Ramsey) but not generally weakly compact.
• A weakly hyper-Woodin cardinal [$$\Sigma^2_1 (V_\kappa)$$] is Woodin and measurable but not generally $$\Pi^2_2$$-indescribable.
• A hyper-Woodin cardinal [$$\Sigma^2_1 (V_\kappa)$$] is weakly hyper-Woodin but not generally $$\Pi^2_2$$-indescribable.
• A subcompact cardinal [$$\Sigma^2_1 (V_\kappa)$$] is Woodin, locally measurable and fully Ramsey but not generally $$\Pi^2_1$$-indescribable.
• A cardinal $$\kappa$$ that is $$2^\kappa$$-supercompact is subcompact, hyper-Woodin and +2-strong. More generally, a cardinal $$\kappa$$ that is $$\beth_{\kappa+\gamma}$$-supercompact is $$\lambda$$-supercompact for every $$\lambda$$ such that $$\kappa \le \eta \lt \beth_{\kappa+\gamma}$$ and $$+\gamma+1$$-strong but not generally $$+\gamma+2$$-strongly unfoldable (or $$\Pi^{\gamma+2}_2$$-indescribable for finite $$\gamma$$).
• A Vopěnka scheme cardinal [$$\Delta^1_1 (V_\kappa)$$] is worldly but not generally inaccesible.
• A Vopěnka cardinal (equivalently a 2-fold Woodin cardinal, equivalently a Woodin for supercompactness cardinal) [$$\Pi^1_1 (V_\kappa)$$] is a Vopěnka scheme cardinal and a Woodin cardinal (any cardinal in which Woodin cardinals are stationary is Woodin) but not generally weakly compact. More generally, an n-fold Vopěnka cardinal (equivalently an n+1-fold Woodin cardinal, equivalently an n-fold Woodin for supercompactness cardinal) [$$\Pi^1_1 (V_\kappa)$$] is n-fold Woodin but not generally weakly compact.
• We may define that $$\kappa$$ is an $$n$$-fold weakly hyper-Woodin cardinal if, for every function $$f:\kappa \to \kappa$$, there is a measure $$U_f$$ on $$\kappa$$ that concentrates on the set of ordinals $$\alpha \lt \kappa$$ such that $$f"\alpha \subseteq \alpha$$ and there is (an extender for) an elementary embedding $$j: V \to M$$ such that $$V_{j^{n-1}(j(f)(\alpha))} \subset M$$} (I believe we can require without loss of equivalence that $$j^{n-1}(f\upharpoonleft \alpha)=f\upharpoonleft j^{n-1}(\alpha)$$). We may define that $$\kappa$$ is an $$n$$-fold weakly hyper-Woodin for supercompactness cardinal if, for every function $$f:\kappa \to \kappa$$, there is a measure $$U_f$$ on $$\kappa$$ that concentrates on ordinals $$\alpha \lt \kappa$$ such that $$f"\alpha \subseteq \alpha$$ and there is (a fine measure for) an elementary embedding $$j: V \to M$$ such that $$M^{j^{n-1}(j(f)(\alpha))} \subset M$$ (I believe we can require without loss of equivalence that $$j^{n-1}(f\upharpoonleft \alpha)=f\upharpoonleft j^{n-1}(\alpha)$$). We may define that $$\kappa$$ is an $$n$$-fold weakly hyper-Woodin for extendibility cardinal if, for every function $$f:\kappa \to \kappa$$, there is a measure $$U_f$$ on $$\kappa$$ that concentrates on the ordinals $$\alpha \lt \kappa$$ such that $$f"\alpha \subseteq \alpha$$ and there is an elementary embedding $$j: V_{j^{n-1}(j(f)(\alpha))} \to V_{j^n(j(f)(\alpha))}$$ (I believe we can require without loss of equivalence that $$j^{n-1}(f\upharpoonleft \alpha)=f\upharpoonleft j^{n-1}(\alpha)$$). A cardinal is $$n+1$$-fold weakly hyper-Woodin iff it is $$n$$-fold weakly hyper-Woodin for supercompactness iff it is $$n$$-fold weakly hyper-Woodin for extendibility (If $$\alpha$$ is $$n+1$$-fold $$j(f)(\alpha)$$-strong, it is $$n$$-fold $$j(f)(\alpha)$$-extendible. If $$\alpha$$ is $$n$$-fold $$j(f)(\alpha)+1$$-extendible, it is $$n+1$$-fold $$j(f)(\alpha)$$-strong and $$n$$-fold $$\beth{j(f)(\alpha)+1}$$-supercompact. If there is an $$n$$-fold $$f+1$$-weakly hyper-Woodin for supercompactness measure $$U_{f+1}$$ on $$\kappa$$, $$N \vDash \text{"\kappa is n-fold j(f)(\kappa)-supercompact"}$$, where $$N$$ is the ultrapower by $$U_{f+1}$$, and the measure generated by the $$n$$-fold $$j(f)(\kappa)$$-supercompact embedding $$i:N \to M$$ is an $$n$$-fold $$f$$-weakly hyper-Woodin for extendibility measure below $$U_{f+1}$$ in the Mitchell order by proposition 8.5 of Sato 2007) An $$n+1$$-fold weakly hyper-Woodin cardinal is $$n+1$$-fold Woodin and $$n$$-fold hyper-Woodin (an $$n+1$$-fold weakly hyper-Woodin cardinal has a measure concentrating on $$n+1$$-fold supercompact cardinals, so there is a $$n$$-fold hyper-Woodin below that in the Mitchell order) but not generally $$\Pi^2_2$$-indescribable.
• We may define that $$\kappa$$ is an $$n$$-fold hyper-Woodin cardinal if there is a measure $$U$$ on $$\kappa$$ that, for every function $$f:\kappa \to \kappa$$, concentrates on the set of ordinals $$\alpha \lt \kappa$$ such that $$f"\alpha \subseteq \alpha$$ and there is (an extender for) an elementary embedding $$j: V \to M$$ such that $$V_{j^{n-1}(j(f)(\alpha))} \subset M$$} (I believe we can require without loss of equivalence that $$j^{n-1}(f\upharpoonleft \alpha)=f\upharpoonleft j^{n-1}(\alpha)$$). We may define that $$\kappa$$ is an $$n$$-fold hyper-Woodin for supercompactness cardinal if there is a measure $$U$$ on $$\kappa$$ that, for every function $$f:\kappa \to \kappa$$, concentrates on ordinals $$\alpha \lt \kappa$$ such that $$f"\alpha \subseteq \alpha$$ and there is (a fine measure for) an elementary embedding $$j: V \to M$$ such that $$M^{j^{n-1}(j(f)(\alpha))} \subset M$$ (I believe we can require without loss of equivalence that $$j^{n-1}(f\upharpoonleft \alpha)=f\upharpoonleft j^{n-1}(\alpha)$$). We may define that $$\kappa$$ is an $$n$$-fold hyper-Woodin for extendibility cardinal if there is a measure $$U_f$$ on $$\kappa$$ that, for every function $$f:\kappa \to \kappa$$, concentrates on the ordinals $$\alpha \lt \kappa$$ such that $$f"\alpha \subseteq \alpha$$ and there is an elementary embedding $$j: V_{j^{n-1}(j(f)(\alpha))} \to V_{j^n(j(f)(\alpha))}$$ (I believe we can require without loss of equivalence that $$j^{n-1}(f\upharpoonleft \alpha)=f\upharpoonleft j^{n-1}(\alpha)$$). A cardinal is $$n+1$$-fold hyper-Woodin iff it is $$n$$-fold hyper-Woodin for supercompactness iff it is $$n$$-fold hyper-Woodin for extendibility (If $$\alpha$$ is $$n+1$$-fold $$j(f)(\alpha)$$-strong, it is $$n$$-fold $$j(f)(\alpha)$$-extendible. If $$\alpha$$ is $$n$$-fold $$j(f)(\alpha)+1$$-extendible, it is $$n+1$$-fold $$j(f)(\alpha)$$-strong and $$n$$-fold $$\beth{j(f)(\alpha)+1}$$-supercompact. Now suppose that $$\kappa$$ is $$n$$-fold hyper-Woodin for supercompactness. For any $$f:\kappa \to \kappa$$, define $$g:\kappa \to \kappa$$ by as the function enumerating the ordinals $$\beta$$ such that $$\langle V_\beta, \in, f\upharpoonleft\beta \rangle \prec \langle V_\kappa, \in, f \rangle$$. Let $$U$$ denote an $$n$$-fold hyper-Woodin for supercompactness measure and $$j: V \to M$$ denote the ultrapower embedding by $$U$$. Then $$i: M \to N$$ witnesses that $$\kappa$$ is $$j(g)(\kappa)$$-supercompact in $$M$$, $$i(g)(\kappa)=j(g)(\kappa)$$, and $$i(f)\upharpoonleft j(g)(\kappa)= j(f)\upharpoonleft j(g)(\kappa)$$. By the proof of proposition 8.5 of Sato 2007, $$\kappa$$ is $$i(g)(\kappa)$$-extendible, and thus $$i(f)(\kappa)$$-extendible, in $$N$$. Then $$\langle V_{i(g)(\kappa)}, \in, i(f)\upharpoonleft j(g)(\kappa) \rangle \prec \langle V_{j(\kappa)^N, \in, i(f) \rangle$$ by definition of $$g$$, so $$V_{i(g)(\kappa)} \vDash \text{"\kappa is i(f)(\kappa)-extendible"}$$ and since $$V_{i(g)(\kappa)^M=V_{i(g)(\kappa)^N, \kappa is i(f)(\kappa)-extendible in M. Thus U concentrates on n-fold f-extendible cardinals. Since this works for every f:\kappa \to \kappa, U is an n-fold hyper-Woodin for extendibility measure.) An n-fold hyper-Woodin cardinal is n-fold weakly hyper-Woodin. - If a cardinal is n-fold Woodin for every n \lt \omega, equivalently n-fold Vopěnka for every n, equivalently n-fold Woodin for every n we may call it \lt \omega-fold Woodin or \lt \omega-fold Vopěnka [\Pi^1_1 (V_\kappa)]. A \lt \omega-fold Woodin cardinal is not generally weakly compact. - An \omega-fold Vopěnka cardinal [\Pi^1_1 (V_\kappa)] is \lt \omega-fold Vopěnka but not generally weakly compact. - An \omega-fold Woodin (W-E_1) cardinal [\Pi^1_1 (V_\kappa)] is \omega-fold Vopěnka (any cardinal in which \omega-fold Vopěnka cardinals are stationary is \omega-fold Vopěnka) but not generally weakly compact. A W-E_{n+1} cardinal [\Pi^1_1 (V_\kappa)] is W-E_n$$ but not generally weakly compact.

The following large cardinal properties have complexity $$\Sigma_2$$:

• An otherworldly cardinal (also known as 0-extendible) is worldly.
• A cardinal is said to be 0-pseudo-uplifting if it is inaccessible and otherworldly. A 0-pseudo-uplifting cardinal is $$\Pi_\omega$$-Mahlo in addition to otherworldly.
• A 0-uplifting cardinal (that is, inaccessible and otherworldly to an inaccessible target) is, of course, 0-pseudo-uplifting.
• A weakly superstrong cardinal $$\kappa$$ is $$\gamma$$-strongly unfoldable for any $$\gamma$$ less than its weakly superstrong witnessing ordinal (the least $$\lambda$$ such that $$V_\lambda \vDash \text{"\kappa is weakly superstrong"}$$) and 0-uplifting.
• A Shelah cardinal $$\kappa$$ is weakly hyper-Woodin, $$\gamma$$-strong for every $$\gamma$$ less than its Shelah witnessing ordinal, and weakly superstrong (Define $$f:\kappa \to \kappa$$ by $$f(\alpha)=$$ the weakly superstrong witnessing ordinal of $$\alpha$$ if $$\alpha$$ is weakly superstrong in $$V_\kappa$$ and the least $$\beta \gt \alpha$$ that is $$\Sigma_2$$-correct in $$V_\kappa$$ otherwise. Since $$\kappa$$ is Shelah, there is an elementary embedding $$j: V \to M$$ such that $$V_{j(f)(\kappa)} \subset M$$. If we can prove that $$\kappa$$ is weakly superstrong in $$M$$, $$j(f)(\kappa)$$ is its witnessing ordinal in $$M$$ by definition of $$f$$ and thus in $$V$$ since $$V_{j(f)(\kappa)} \subset M$$. To prove that $$\kappa$$ is weakly superstrong in $$M$$, factor $$j$$ as $$k \circ j_U$$, where $$U$$ is the measure defined by $$j$$ and $$j_U: V \to N$$ is the ultrapower embedding by $$U$$. Since $$M^\kappa \subset M$$, we have $$j_U(P) \in N$$ for every $$\kappa$$-model $$P$$, $$\kappa$$ is weakly superstrong in $$N$$, witnessed by the embeddings $$J \upharpoonleft P: P \to j(P)$$. Since $$k: N \to M$$ is an elementary embedding whose critical point is greater than $$\kappa$$, $$\kappa$$ is weakly superstrong in $$M$$.).
• A superstrong cardinal is hyper-Woodin and Shelah.
• A +1-extendible cardinal is superstrong (this is a special case of a result listed below as superstrong is the same as 2-fold +0-strong).
• A quasicompact cardinal is subcompact and +1-extendible.
• A 2-fold +1-strong cardinal $$\kappa$$ is $$2^\kappa$$-supercompact and quasicompact. More generally, a 2-fold $$+\gamma$$-strong cardinal $$\kappa$$ is $$\beth_{\kappa+\gamma}$$-supercompact, unless $$\gamma$$ is a limit ordinal of cofinality less than $$\kappa$$ (that follows from the proof in this Mathoverflow answer by Gabe Goldberg; note that we only need the elementary embedding $$j : V_{\kappa+\eta+1} \to V_{j(\kappa)+j(\eta)+1}$$ to be $$\Delta_0$$-elementary, which a restriction of a 2-fold $$+\gamma$$-strong embedding is), and $$+\gamma$$-extendible. Even more generally, an $$n+1$$-fold $$+\gamma$$-strong cardinal $$\kappa$$ is $$n$$-fold $$\beth_{\kappa+\gamma}$$-supercompact, unless $$\gamma$$ is a limit ordinal of cofinality less than $$\kappa$$, and $$n$$-fold $$+\gamma$$-extendible.
• A ($$n$$-fold) $$+\gamma+1$$-extendible cardinal is 2-fold ($$n+1$$-fold) $$+\gamma$$-strong (proved in this Mathoverflow answer by Joel David Hamkins).
• If $$\gamma$$ is a limit ordinal of cofinality less than $$\kappa$$, then $$\kappa$$ is 2-fold ($$n+1$$-fold) $$+\gamma$$-strong if and only if it is ($$n$$-fold) $$+\gamma$$-extendible (proved by Joel David Hamkins in the same post linked above).
• A 2-fold ($$n+1$$-fold) Shelah cardinal, eqivalently a ($$n$$-fold) Shelah for supercompactness cardinal (the equivalence was proved by me in this Mathoverflow post) is 2-fold ($$n+1$$-fold) weakly hyper-Woodin and $$+\gamma+1$$-extendible, 2-fold ($$n+1$$-fold) $$+\gamma$$-strong and ($$n$$-fold) $$\beth_{\kappa+\gamma}$$-supercompact for all $$\gamma$$ less than its 2-fold ($$n+1$$-fold) Shelah/($$n$$-fold) Shelah for supercompactness witnessing ordinal.
• A ($$n$$-fold) high jump cardinal (by $$n$$-fold high jump cardinal, I mean the critical point of an elementary embedding $$j: V \to M$$ such that $$M^{j^{n-1}(\lambda)} \subset M$$, where $$\lambda$$ is the clearance of $$j$$) is 2-fold ($$n+1$$-fold) weakly hyper-Woodin and ($$n$$-fold) Shelah for supercompactness.
• A 2-fold ($$n+1$$-fold) high jump for strongness cardinal is ($$n$$-fold) high jump (this is a special case of a result listed elsewhere).
• A ($$n$$-fold) almost huge cardinal is 2-fold ($$n+1$$-fold) high jump for strongness (Suppose $$j: V \to M$$ is an almost huge embedding with critical point $$\kappa$$. Denote the clearance of the embedding by $$\lambda$$. Since $$2^\lambda \lt j(\kappa)$$, we have $$M^{2^\lambda} \subset M$$, so $$j \upharpoonleft V_{\lambda+1} \in M$$. Thus $$M \vDash \text{"\kappa is 2-fold (n+1-fold) high jump for strongness"}$$ for the same reason as a $$n$$-fold $$+\gamma+1$$-extendible cardinal is $$n+1$$-fold $$+\gamma$$-strong. Since $$V_\lambda \prec V_{j(\kappa)}$$, $$\kappa$$ is 2-fold ($$n+1$$-fold) high jump for strongness in $$V_\lambda$$ and thus in $$V$$.)
• A 2-fold ($$n+1$$-fold) 0-extendible cardinal (by $$n$$-fold 0-extendible I mean the critical point $$\kappa$$ of an elementary embedding $$j: V_{j^{n-1}(\kappa)} \to \lambda$$; Solovay, Reinhardt and Kanamori 1978 calls 2-fold 0-extendible cardinals $$A_2$$) is ($$n$$-fold) almost huge (theorem 8.3 of that paper).
• A ($$n$$-fold) huge cardinal $$\kappa$$ is 2-fold ($$n+1$$-fold) 0-extendible (as ($$n$$-fold) huge is the same as ($$n$$-fold) $$\kappa$$-hyperhuge, this is a special case of the fact that ($$n$$-fold) $$\beth_{\kappa+\gamma}$$-hyperhuge cardinals are $$n+1$$-fold $$+\gamma$$-extendible).
• A 2-fold ($$n+1$$-fold) superstrong cardinal is ($$n$$-fold) huge.
• A ($$n$$-fold) $$\beth_{\kappa+\gamma}$$-hyperhuge cardinal $$\kappa$$, also called 2-fold ($$n+1$$-fold) $$\beth_{\kappa+\gamma}$$-supercompact, is 2-fold ($$n+1$$-fold) $$+\gamma$$-extendible (proved by me in this Mathoverflow post). In particular, a cardinal $$\kappa$$ that is ($$n$$-fold) $$2^\kappa$$-hyperhuge is 2-fold ($$n+1$$-fold) +1-extendible and thus 2-fold ($$n+1$$-fold) superstrong.
• A cardinal that is $$n$$-superstrong for every $$n \lt \omega$$ can be called $$\lt \omega$$-fold superstrong. Similarly we can define $$\lt \omega$$-fold huge, $$\lt \omega$$-fold almost huge, $$\lt \omega$$-fold Shelah, $$\lt \omega$$-fold Shelah for supercompactness and $$\lt \omega$$-fold high jump cardinals. These are all equivalent.
• An $$I_3$$, also called $$E_0$$, critical point, equivalently an $$\omega$$-fold $$\gamma$$-extendible cardinal for some $$\gamma$$, equivalently $$\omega$$-fold $$\gamma$$-extendible for every $$\gamma$$ less than the critical supremum of the $$E_0$$ embedding, is $$\lt \omega$$-fold huge.
• An $$IE_\omega$$ embedding is an $$E_0$$ embedding and its critical point is $$\omega$$-fold Vopěnka. An $$IE_\alpha$$ embedding for limit $$\alpha \le \omega_1$$ is $$IE_\beta$$ for $$\beta \lt \alpha$$. $$IE$$ is equivalent to $$IE_{\omega_1}$$.
• An $$E_1$$ embedding, equivalently the restriction of an $$I_2$$ embedding to $$V_{\lambda+1}$$, where $$\lambda$$ is the supremum of the critical sequence (the critical point of an $$I_2$$ embedding is also called an $$\omega$$-fold superstrong cardinal) is an $$IE$$ embedding.
• An $$E_{n+1}$$ embedding is an $$E_n$$ embedding and its critical point is $$W-E_n$$ so in particular, the critical point of an $$E_2$$ embedding is $$\omega$$-fold Woodin.
• An $$E_\omega$$ embedding, that is one that is $$E_n$$ for every $$n \lt \omega$$, is also called $$I_1$$
• The restriction of an $$I_0$$ embedding to $$V_{\lambda+1}$$, where $$\lambda$$ is the supremum of the critical sequence, is an $$I_1$$ embedding.

The following large cardinal properties have complexity $$\Pi_2$$:

• A cardinal that is inaccessible and $$\Sigma_2$$-correct is said to be $$\Sigma_2$$-reflecting. A $$\Sigma_2$$-reflecting cardinal $$\kappa$$ is $$\Sigma_2$$-Mahlo (for any $$\Sigma_2$$-definable club $$C$$, $$\text{"There is an inaccessible cardinal in C"}$$ is true in $$V$$, as it is witnessed by $$\kappa$$, and it is $$\Sigma_2$$, so by $$\Sigma_2$$-correctness, it is true in $$V_\kappa$$).
• A strongly unfoldable cardinal (that is, one that is $$\gamma$$-strongly unfoldable for every $$\gamma$$) is $$\Sigma_2$$-reflecting in addition to totally indescribable.
• A strong cardinal (that is, one that is $$\gamma$$-strong for every $$\gamma$$) is strongly unfoldable in addition to measurable.
• A supercompact cardinal (that is, a cardinal $$\kappa$$ that is $$\kappa+\gamma$$-supercompact for every $$\gamma$$) is strong in addition to subcompact and hyper-Woodin.
• An $$\alpha$$-hypercompact cardinal is $$\beta$$-hypercompact for every $$\beta \lt \alpha$$. 1-hypercompact is equivalent to supercompact and a cardinal that is $$\alpha$$-hypercompact for every $$\alpha$$ is said to be hypercompact.
• Define an $$n$$-fold Magidor cardinal as a cardinal $$\kappa$$ such that, for every $$\gamma \gt \kappa$$, there is an elementary embedding $$j: V_\zeta \to V_\gamma$$ such that $$\zeta \lt \kappa$$ and $$\kappa= j^n(crit(j))$$. A cardinal is 1-fold Magidor iff it is supercompact. A 2-fold Magidor cardinal is hypercompact (Fix a $$\Sigma_2$$-correct $$\gamma \gt \kappa$$. There is an elementary embedding $$j$$ as in the definition. Restrictions of $$j$$ witness that $$V_\kappa \vDash \text{"crit(j) is \beta-extendible"}$$ for every $$\beta \lt j(crit(j))$$ and by elementarity of $$j(j\upharpoonleft V_{crit(j) +1} : V_{crit(j)+1} \to V_{\kappa+1}$$ (whose critical point is $$j(crit(j))$$), the same holds in $$V_{j(crit(j))}$$, so $$V_{j(crit(j))} \vDash \text{"crit(j) is extendible"}$$, and by elementarity of $$j$$, $$V_\kappa \vDash \text{"j(crit(j)) is extendible"}$$. Since extendible cardinals are hypercompact (which I will prove below) and hypercompactness is $$\Pi_2$$-definable, $$V_\zeta \vDash \text{"crit(j) is hypercompact"}$$, so by elementarity, $$V_\gamma \vDash \text{"\kappa is hypercompact"}$$ and by $$\Sigma_2$$-correctness, that is true in $$V$$.) An $$n$$-fold Magidor cardinal is $$m$$-fold Magidor for $$m \lt n$$.

I hope I'll finish this list later.