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There are two natural orders on large cardinal axioms.

(a) Consistency strength order

$\sigma \leq_C \theta \Longleftrightarrow ZFC\vdash Con(ZFC+\theta)\longrightarrow Con(ZFC+\sigma)$

(b) Implication strength order

$\sigma \leq_I \theta \Longleftrightarrow ZFC\vdash \theta\longrightarrow \sigma$

It is well-known that these are not same and many large cardinals (e.g. Woodin cardinals) have different positions in large cardinal tree when we endow it with different orderings.

Question 1. Are there any other important orderings on the tree of large cardinal axioms?

Question 2. What is the shape of large cardinal tree in implication strength order? Is there any diagram somewhere in the texts which summarizes the results on direct implication of large cardinal axioms?

I hope somebody give me an explicit diagram which shows the differences between large cardinal tree in implication and consistency strength orders or at least based on the answers one be able to design a diagram for large cardinal tree in implication order and post it as an answer which summarizes the answers of the others.

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  • $\begingroup$ Could you clarify what you are saying about Woodin cardinals? I suspect you refer to the fact that Woodin cardinals need not be measurable (for example.) But the axiom "there is a measurable cardinal" is both consistent relative to, and implied by, the axiom "there is a Woodin cardinal," so this situation does not seem to be an instance of the orderings (a) and (b) of large cardinal axioms being different. $\endgroup$ Feb 20, 2014 at 1:27
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    $\begingroup$ Why do you think it is a tree? Of course, if it is linear, as some set theorists assert, then it will be a tree, but this linearity view seems to be connected with a certain more stringent vision of what counts as a large cardinal axiom. With a more relaxed view, we seem very likely to have a partial order that is not a tree. $\endgroup$ Feb 20, 2014 at 1:29
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    $\begingroup$ But a tree is a special kind of partial order, one for which the predecessors of every node are linear; so once a node splits, the branches never grow together again. But do you think the large cardinal order is like that? $\endgroup$ Feb 20, 2014 at 2:08
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    $\begingroup$ "..set theorists usually use it in the literature" Can you give an example? $\endgroup$
    – bof
    Sep 2 at 23:01
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    $\begingroup$ My recent paper arxiv.org/abs/2208.07445 is an extended discussion of the issue whether the hierarchy of large cardinal consistency strength is linear or indeed well-ordered. It is not actually linear, nor well-ordered, but rather densely ordered with many antichains, although some people argue that it is linear for so-called "natural" theories. I aimed to provide counterexamples. $\endgroup$ Sep 3 at 6:18

3 Answers 3

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As an answer to the question (1), one can produce a series of natural orderings on large cardinal axioms using the operator $Con(ZFC+\dots)$.

$\sigma \leq_0 \theta \Longleftrightarrow ZFC\vdash \theta \longrightarrow \sigma$

$\sigma \leq_1 \theta \Longleftrightarrow ZFC\vdash Con(ZFC+\theta) \longrightarrow Con(ZFC+\sigma)$

$\sigma \leq_2 \theta \Longleftrightarrow ZFC\vdash Con(ZFC+Con(ZFC+\theta)) \longrightarrow Con(ZFC+Con(ZFC+\sigma))$

$\cdots$

We have $\leq_0 \subseteq \leq_1 \subseteq \leq_2\subseteq \cdots$ and one can think about an animated diagram which shows the change of positions of each large cardinal axiom in the tree when its ordering varies over $\langle \leq_i:i\in \omega\rangle$.

An important property of these orderings on large cardinal axioms is that by the definitions we have $\forall \sigma,\theta \in \text{Large Cardinal Axioms}~~\forall i,j\in \omega~~((i\leq j~~\wedge~~\sigma=_i \theta )\Longrightarrow \sigma =_j \theta)$ and so maybe there is a (possibly ordinal valued) step $k$ such that all main large cardinal axioms are equivalent in the sense of $\leq_k$ ordering. Note that the same phenomenon happens between $WI$ (existence of a weakly inaccessible cardinal) and $SI$ (existence of a strongly inaccessible cardinal) because we have $WI<_0 SI$ but $\forall i\geq 1~~WI=_i SI$.

Also one can ask some natural questions about the interactions of these orderings with each other as follows.

Definition. Let $\sigma,\theta$ be two large cardinal axioms. Define the convergence number of $\sigma, \theta$ ($\alpha_{\sigma, \theta}$) to be $min\{\gamma\in Ord~|~\sigma =_\gamma \theta\}$ if $\{\gamma\in Ord~|~\sigma =_\gamma \theta\}\neq \emptyset$ and $\infty$ if$\{\gamma\in Ord~|~\sigma =_\gamma \theta\}=\emptyset$.

Example. $\alpha_{WI,SI}=1$

Question 1. Is $\alpha_{\sigma,\theta}<\infty$ for each two large cardinal axioms $\sigma, \theta$?

Question 2. Is there an ordinal $\alpha_0$ such that for each two large cardinal axiom $\sigma,\theta$ we have $\alpha_{\sigma,\theta}\leq \alpha_0$?

Question 3. What is the convergence number of "existence of a strongly compact cardinal" and "existence of a superstrong cardinal"?

Question 4: Consider $\mathbb{L}$ be the set of large cardinal axioms. Is there any ordinal $\alpha$ such that the $\langle \mathbb{L}, \leq_{\alpha}\rangle$ be a linear order?

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    $\begingroup$ How do you iterate your Con() operator past $\omega_1^{\mathrm{CK}}$? My understanding is that there's no clearly-defined notion past about that point; mathoverflow.net/questions/153272/… isn't quite the same, but the iterated-consistency operator there should be much stronger than even the one here... $\endgroup$ Feb 20, 2014 at 2:05
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    $\begingroup$ If $\sigma$ says "there is a strongly inaccessible cardinal" and "$\theta$ says "there is a Mahlo cardinal", then from $\mathsf{ZFC} + \theta$ we can prove $\sigma + \text{Con}(\sigma) + \text{Con}(\sigma + \text{Con}(\sigma)) + \text{Con}(\ldots) + \cdots$, iterated up to any recursive ordinal (and as Steven points out it is not clear what it would mean to continue farther) so the convergence you are talking about does not seem to occur... $\endgroup$ Feb 20, 2014 at 2:53
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    $\begingroup$ Moreover, as far as I know the same thing happens with any pair of large cardinal axioms $\sigma$ and $\theta$ such that $\mathsf{ZFC} + \theta$ proves $\sigma + \text{Con}(\sigma)$. (That is, if such statements have different consistency strength, then the difference in their consistency strength is very large compared to the differences in consistency strength we get by tacking on "Con" statements.) $\endgroup$ Feb 20, 2014 at 2:56
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    $\begingroup$ If $1 \le i < \omega$ then the ordering $\le_i$ on all the large cardinal axioms that I can think of ("natural" large cardinal axioms, not metamathematical statements like consistency statements) seems to be the same as the ordering $\le_1$. Do you have any counterexample to this? $\endgroup$ Feb 20, 2014 at 4:20
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    $\begingroup$ Regarding the possibility of differences between the orderings $\le_i$ for $i \ge 1$: Do you know the proof that $\mathsf{ZFC} + {}$"there is a strongly inaccessible cardinal" implies $\text{Con}(\mathsf{ZFC} + \text{Con}(\mathsf{ZFC} + \cdots)\cdots)$? Similar things happen higher up in the large cardinal hierarchy, and to me this makes the idea that the orderings are different seem unlikely. In particular, it is not hard to show that "there is a supercompact cardinal" is stronger than any iterated consistency statement obtained from "there is a Woodin cardinal". $\endgroup$ Feb 20, 2014 at 21:58
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There is also:

c) interpretability strength order:

$\sigma \leq_F \theta \Longleftrightarrow \exists f\ \forall \psi\ ZFC\vdash Con(ZFC+\theta+f(\psi))\longrightarrow Con(ZFC+\sigma+\psi)$

for some suitable interpretation $f$. This article on Independence and Large Cardinals gives more exposition.

A linear order is also likely here.

If we found $\theta$ and $\phi$ to be incompatible, we would probably say "$\theta$ is a large cardinal axiom, about the size of the set-theoretic universe, and $\phi$ is more about the structure of the universe, so it's not really a large cardinal axiom."

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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 $\theta$ 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 inaccessibility 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.

  • An inaccessible cardinal [$\Pi_1$/$\Pi^1_1 (V_\kappa)$] is worldly [$\Delta^1_1 (V_\kappa)$].
  • 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 inaccessible.
  • 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.
  • 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 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, equivalently 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.
  • 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 $n$-fold hyper-Woodin and $m$-fold Magidor for $m \lt n$.

The following large cardinal properties have complexity $\Pi_3$:

  • A totally otherwordly cardinal is otherworldly and $\Sigma_3$-correct.
  • A cardinal that is inaccessible and $\Sigma_3$-correct is said to be $\Sigma_3$-reflecting. A $\Sigma_3$-reflecting cardinal is $\Pi_2$-Mahlo but not generally totally otherwordly.
  • A cardinal that is inaccessible and totally otherwordly is said to be pseudo-uplifting. A pseudo-uplifting cardinal is $\Pi_\omega$-Mahlo.
  • An uplifting cardinal is pseudo-uplifting.
  • A superstrongly unfoldable, equivalently strongly uplifting, cardinal is weakly superstrong, strongly unfoldable, and uplifting.
  • A globally superstrong cardinal is superstrong, strong and superstrongly unfoldable.
  • A cardinal is extendible, that is $+\gamma$-extendible for every ordinal $\gamma$, if an only if it is 2-fold strong, that is 2-fold $+\gamma$-strong for every ordinal $\gamma$. An extendible cardinal is hypercompact (Let $\lambda_0$ be any wordly cardinal greater than $\kappa$ and define $\lambda_{n+1}$ as the least cardinal such that there exists an extendible embedding $j_n: V{\lambda_n} \to V_{\lambda_{n+1}}$ with critical point $\kappa$ and $\lambda_\omega$ as the supremum of the $\lambda_n$. By a factor embedding argument combining lemma 3.5 of Sato 2007 and this Mathoverflow answer by Gabe Goldberg, the existence of $j_n: V_{\lambda_n} \to V_{\lambda_{n+1}}$ implies, for every $\gamma \lt \lambda_n$, that $\kappa$ is the critical point of a $\beth_{\kappa+\gamma}$-supercompact embedding $i_n: V \to M$ such that $j_n = k_n \circ i_n \upharpoonleft V_{\lambda_n}$ for some elementary embedding $k_n$ with critical point greater than $\beth_{\kappa+\gamma}$, and if $H_{j_n(\beth_{\kappa+\gamma})} \vDash \text{"$\kappa$ is $j_n(\alpha)$-hypercompact"}$, we have $H_{\beth_{\kappa+\gamma}} \vDash \text{"$\kappa$ is $\alpha$-hypercompact"}$ by elementarity of $k_n$ and because hypercompactness is $\Pi_2$-definable and thus downward absolute between hereditary cardinality ranks. Thus, if $j_n$ and $\lambda_{n+1}$ exist, we have $V_{\lambda_n} \vDash \text{"$\kappa$ is supercompact"}$ and if additionally $V_{\lambda_{n+1}} \vDash \text{"$\kappa$ is $j_n(\alpha)$-hypercompact"}$, then $V_{\lambda_n} \vDash \text{"$\kappa$ is $\alpha$-hypercompact"}$. If $\lambda_\omega$ exists, then by transfinite induction, $V_{\lambda_n} \vDash \text{"$\kappa$ is hypercompact"}$ for any $n$. If $\kappa$ is extendible, there are unboundedly many ranks in which it is hypercompact, and since hypercompactness is $\Pi_2$-definable, that means that $\kappa$ really is hypercompact.) and globally superstrong.

I can't finish this list because I have hit the length limit.

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