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Martin Sleziak
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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.