Operations on the set of large cardinal axioms

Question

Here's a question from a non-set-theorist, but a sometime-user of large cardinals.

The name Cantor's attic is pretty evocative for the collection of large cardinal axioms: looking through the pages there really feels like I'm in an old, neglected attic, full of boxes which were labeled with names at various times and then stored haphazardly here and there. In order to understand the relationships between different large cardinal axioms, it seems there's no alternative to opening each and every box and comparing their contents yourself. (This sense of disorder is ironic, given that one of the chief features of the attic is that it is well-ordered by consistency strength!)

But there are patterns to be found.

• For instance, from a large cardinal axiom $\phi$ we can generate more large cardinal axioms by asking for a proper class of $\phi$-cardinals in $V_\kappa$, or by asking for the class of $\phi$-cardinals to be stationary in $V_\kappa$. Call the latter operation Mahlofication.

• There seem to also be "operations" of superfication, Woodinification, and Shelahfication, but I don't know much about them. One complication is that it seems one often needs to modify a large cardinal by introducing various parameters into its definition in order to apply some of these operations.

Question 1: What does it mean to Mahlofy, superfy, Woodenify, or Shelahfy a large cardinal notion? Is it correct to say that these operations all ask for cardinal $\kappa$ such that $V_\kappa$ has "a lot" of the original type of large cardinals? I think these operations generally increase consistency strength -- is it by "a lot" or "a little"? And are these operations "idempotent", or can they be "iterated"?

Question 2: Are there other important "operations" on large cardinal notions?

Qustion 3: Is the list of large cardinal notions at Cantor's attic, say, generated by a substantially smaller list under some collection of "operations"?

Question 4: What about relations between "operations" -- for instance, are there distinct large cardinal notions which become identified under Mahlofication or something? If you restrict attention to large cardinal notions fixed by Mahlofication or something, does the attic look a bit neater?

Obviously these questions are all broad and ill-defined and possibly nonsensical. But if there's anything here which is not nonsense and can be addressed in a precise or at least meaningful manner, I'd be grateful to hear about it!

Answer 1

I know several ways to modify large cardinal notions:

If a family of large cardinal notions is defined with an ordinal parameter (call it $\gamma$-$\phi$), one can define the property of being $\gamma$-$\phi$ for all $\gamma$:

• If $\kappa$ is $\gamma$-shrewd for all $\gamma$, it is said to be shrewd.
• If $\kappa$ is $\gamma$-strongly unfoldable for all $\gamma$, it is said to be strongly unfoldable (equivalent to being shrewd).
• If $\kappa$ is $\gamma$-strong for all $\gamma$, it is said to be strong.
• If $\kappa$ is $\lambda$-supercompact for all $\lambda$, it is said to be supercompact.
• If $\kappa$ is $\gamma$-extendible for all $\gamma$, it is said to be extendible.

If the definition of a large cardinal notion $\phi(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a large cardinal notion asserting that there are unboundedly many such $\theta$, that is, for every ordinal $\gamma$ there is such a $\theta \gt \gamma$ (is that what you mean by superfication?):

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be 0-extendible or otherwordly; if additionally $\kappa$ is inaccessible, it is said to be 0-pseudo-uplifting; and if additionally $\kappa$ and $\theta$ are both inaccessible, it is said to be 0-uplifting. If there are unboundely many such $\theta$, $\kappa$ is said to be totally otherwordly, pseudo-uplifting or uplifting, respectively.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \cup \{A\} \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. If for every $\gamma$ and every $A \subseteq V_\kappa$ there exists such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be superstrongly unfoldable.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. If for every $\gamma$ there is a high jump embedding $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be super-high-jump. One can similarly define super-high-jump for strongness cardinals but this property is equivalent to being globally superstrong just like high jump for strongness is equivalent to superstrong.
• The definitions of superstrong, almost huge and huge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and certain other properties. If for every $\gamma$ there is such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be globally superstrong, super-almost-huge, or superhuge, respectively. One can define a similar strengthening of the definition of $\gamma$-extendible cardinals (there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$); this strengthening doesn't appear to have a name but one could call such cardinals globally $\gamma$-extendible.
• An elementary embedding $j: V_\lambda \to V_\lambda$ is called a rank into rank, $I_3$ or $E_0$ embedding. Such an elementary embedding extends to a $\Sigma_0$-elementary embedding $\hat{j}: V_{\lambda+1} \to V_{\lambda+1}$; if $\hat{j}$ is $\Sigma_{2n}$-elementary (equivalently $\Sigma_{2n-1}$-elementary) it is called an $E_n$ embedding (for $n \lt \omega$). An elementary embedding $j: V \to M$ such that $j^n(\kappa) \subset M$ for all $n \lt \omega$ is called an $I_2$ embedding (any $I_2$ embedding restricts to an $E_1$ embedding and conversely any $E_1$ embedding extends to an $I_2$ embedding) and the critical point of an $I_2$ embedding is sometimes said to be $\omega$-fold superstrong. A further strengthening of $E_n$, involving a fully elementary embedding $j: V_{\lambda+1} \to V_{\lambda+1}$, is called $I_1$ or $E_\omega$. If for every $\gamma$ there is an $E_n$ embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be a P-$E_n$ cardinal; P-$E_0$ cardinals are also called $\omega$-fold extendible. If for every $\gamma$ there is an $I_2$ ($\omega$-fold superstrong) embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be $\omega$-fold strong (which is of course equivalent to P-$E_1$). Similiarly we can define P-$E_\omega$ cardinals.

If the definition of a large cardinal notion $\phi(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a $C^{(n)}$ variant, additionally asserting that $\theta$ is $\Sigma_n$-correct.

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be otherwordly. One can define $C^{(n)}$-otherwordly cardinals by additionally requiring that $\theta$ is $\Sigma_n$-correct.
• The definitions of superstrong, $2$-fold $\gamma$-strong, almost huge, huge and $\lambda$-hyperhuge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and the definition of $\gamma$-extendible cardinals says that there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$. If additionally $\theta$ is $\Sigma_n$-correct, $\kappa$ is said to be $C^{(n)}$-superstrong, $C^{(n)}$-almost huge, $C^{(n)}$-huge, of $C^{(n)}$-$\gamma$-extendible, respectively, and one can similarly define $C^{(n)}$-$2$-fold $\gamma$-strong and $C^{(n)}$-$\lambda$-hyperhuge cardinals.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $C^{(n)}$-high jump for strongness and $C^{(n)}$-high jump cardinals by additionally requiring that $\theta$ is $\Sigma_n$-correct. Just like high jump for strongness is equivalent to superstrong, $C^{(n)}$-high jump for strongness is equivalent to $C^{(n)}$-superstrong.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \cup \{A\} \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. One can define $C^{(n)}$-weakly superstrong cardinals by requiring that for every $A$ there is such a $j$ with $j(\kappa)$ $\Sigma_n$-correct.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If $M^{j(f)(\kappa)} \subset M$, $\kappa$ is said to be Shelah for supercompactness. One can define $C^{(n)}$-Shelah cardinals as follows: for every function $f: \kappa \to \kappa$ such that all ordinals in the range of $f$ are $\Sigma_n$-correct in $V_\kappa$, there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$ and $j(f)(\kappa)$ is $\Sigma_n$-correct. One can similarly define $C^{(n)}$-Shelah for supercompactness cardinals.
• If there is an $E_i$ embedding $j: V_\lambda \to V_\lambda$ with critical point $\kappa$ such that $j^m(\kappa)$ is $\Sigma_n$-correct, then $\kappa$ is said to be $m$-$C^{(n)}$-$E_i$; if $\lambda$ is $\Sigma_n$-correct, $\kappa$ is said to be $\omega$-$C^{(n)}$-$E_i$. 1-$C^{(n)}$-$E_i$ cardinals are also called $C^{(n)}$-$E_i$, 1-$C^{(n)}$-$E_0$ cardinals are also called $C^{(n)}$-$I_3$, and $\omega$-$C^{(n)}$-$E_i$ cardinals are also called $C^{(n)+}$-$I_3$. $m$-$C^{(n)}$-$I_1$ cardinals (which could also be called $m$-$C^{(n)}$-$E_\omega$) are defined similarly and 1-$C^{(n)}$-$I_1$ and $\omega$-$C^{(n)}$-$I_1$ are also called $C^{(n)}$-$I_1$ and $C^{(n)+}$-$I_1$, respectively.
• One can similarly define $m$-$C^{(n)}$ variants of $k$-fold variants (see below): $\kappa$ is $m$-$C^{(n)}$-$k$-superstrong ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\gamma$-extendible ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\gamma$-strong ($1 \le m \le k-1$), $m$-$C^{(n)}$-$k$-fold high jump for extendibility ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold high jump for strongness ($1 \le m \le k-1$), $m$-$C^{(n)}$-almost-$k$-huge ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold 0-extendible ($0 \le m \le k-1$; see below for $n=k$), $m$-$C^{(n)}$-$k$-huge ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\gamma$-ultrahuge ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\lambda$-hyperhuge ($1 \le m \le k$) or $m$-$C^{(n)}$-$k$-fold high jump ($1 \le m \le k-1$) if there is an elementary embedding $j$ witnessing that $\kappa$ is $k$-superstrong, $k$-fold $\gamma$-extendible, $k$-fold $\gamma$-strong, $k$-fold high jump for extendibility, $k$-fold high jump for strongness, almost-$k$-huge, $k$-fold 0-extendible, $k$-huge, $k$-fold $\gamma$-ultrahuge, $k$-fold $\lambda$-hyperhuge, or $k$-fold high jump, respectively, and additionally $j^m(\kappa)$ is $\Sigma_n$-correct.
• For $k$-fold high jump one can additionally define the following $m^+$-$C^{(n)}$ variants: $\kappa$ is $m^+$-$C^{(n)}$-$k$-fold high jump ($0 \le m \le k-1$), $m^+$-$C^{(n)}$-$k$-fold high jump for strongness ($0 \le m \le k-1$), or $m^+$-$C^{(n)}$-$k$-fold high jump for extendibility ($0 \le m \le k-1$; see below for $n=k$) if there is an elementary embedding $j$ witnessing that $\kappa$ is $k$-fold high jump, $k$-fold high jump for strongness, or $k$-fold high jump for extendibility, respectively, such that $j^m(\theta)$ is $\Sigma_n$-correct, where $\theta$ is the clearance of $j$.
• One could similarly define $m$-$C^{(n)}$-$k$-fold Shelah ($1 \le m \le k-1$) and $m^+$-$C^{(n)}$-$k$-fold Shelah ($0 \le m \le k-1$) cardinals, but for a given $k$ they all turn out to be equivalent, so such cardinals should simply be called $C^{(n)}$-$k$-fold Shelah. Similarly, there are many possible definitions of $C^{(n)}$-$k$-fold Shelah for supercompactness and $C^{(n)}$-$k$-fold Shelah for extendibility cardinals, all of which are equivalent to being $C^{(n)}$-$k$+1-fold Shelah.
• One can define that $\kappa$ is $k$-fold high jump for extendibility if it is the critical point of and elementary embedding $j: V_{j^k(\theta)} \to V_\eta$, where $\theta$ is the clearance of $j$. One can define $k^+$-$C^{(n)}$-$k$-fold high jump for extendibility cardinals by additionally requiring that $\eta$ (and thus the supremum of the ordinals in the range of $j$, if that isn't $\eta$) is $\Sigma_n$-correct. Similarly, one can define that $\kappa$ is $k^+$-$C^{(n)}$-$k$-fold 0-extendible if it is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$ such that $\eta$ (and thus the supremum of the ordinals in the range of $j$, if that isn't $\eta$) is $\Sigma_n$-correct.
• This method can be combined with those described above to define $C^{(n)}$-totally otherwordly, $C^{(n)}$-pseudo-uplifting, $C^{(n)}$-uplifting, $C^{(n)}$-superstrongly unfoldable, $C^{(n)}$-globally superstrong, $C^{(n)}$-extendible, $C^{(n)}$-super-high-jump, $C^{(n)}$-super-almost-huge, $C^{(n)}$-superhuge, $C^{(n)}$-ultrahuge and $C^{(n)}$-hyperhuge cardinals.

For most large cardinal notions stronger than measurable and weaker than wholeness axioms, one can define $n$-fold variants:

• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma} \subset M$, $\kappa$ is said to be $\gamma$-strong. If $V_{j^{n-1}(\kappa+\gamma)} \subset M$, $\kappa$ is said to be $n$-fold $\gamma$-strong.
• If $\kappa$ is $\gamma$-strong for all $\gamma$, it is said to be strong. If $\kappa$ is $n$-fold $\gamma$-strong for all $\gamma$, it is said to be $n$-fold strong.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ whose critical point, which we will call $\mu$, is less than $\kappa$ and such that $f"\mu \subset \mu$, $V_{j(f)(\mu)} \subset M$, and $j(f)(\mu)=f(\mu)$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be a Woodin cardinal. If for every such $f$ there are such $j: V \to M$ and $\mu$ such that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent), then $\kappa$ is said to be $n$-fold Woodin.
• If for every function $f: \kappa \to \kappa$ there is an measure $U_f \subset \mathcal{P}(\kappa)$ such that for $U_f$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be weakly hyper-Woodin. One can define $n$-fold weakly hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If for every such $f$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(j^{n-1}(\kappa))} \subset M$, then $\kappa$ is said to be a $n$-fold Shelah cardinal.
• If there is an measure $U \subset \mathcal{P}(\kappa)$ such that for every function $f: \kappa \to \kappa$, $U$ satisfies that for $U$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{f(\mu)} \subset M$ and $j(f)(\mu)=f(\mu)$, then $\kappa$ is said to be hyper-Woodin (contrary to what I previously said, we can't drop the requirement that $j(f)(\mu)=f(\mu)$, even if we replace $V_{f(\mu)} \subset M$ with $V_{j(f)(\mu)} \subset M$). One can define $n$-fold hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{f(j^{n-1}(\mu))} \subset M$ and for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If there are unboundedly many cardinals that are clearances of high jump for strongness embeddings with critical point $\kappa$, $\kappa$ can be said to be super high jump for strongness One can define $n$-fold high jump for strongness cardinals by requiring $V_{j^{n-1}(\theta)} \subset M$, and one can similarly define $n$-fold super high jump for strongness cardinals. For $n \ge 2$, unlike for $n=1$, $n$-fold high jump for strongness and $n$-fold super high jump for strongness are not equivalent to $n$-fold superstrong and $n$-fold globally superstrong but weaker than $n-1$-fold almost huge and stronger than $n$-fold super-high-jump.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$, then $\kappa$ is said to be superstrong. If $V_{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-superstrong.
• One can define $n$-fold globally superstrong as follows: for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$.
• If there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $\gamma$-extendible. If there is an elementary embedding $j: V_{j(\kappa+\gamma)} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $n$-fold $\gamma$-extendible.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^\lambda \subset M$, $\kappa$ is said to be $\lambda$-supercompact. If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^{n-1}(\lambda)} \subset M$, $\kappa$ is said to be $n$-fold $\lambda$-supercompact.
• If $\kappa$ is $\lambda$-supercompact for all $\gamma$, it is said to be supercompact. If $\kappa$ is $n$-fold $\lambda$-supercompact for all $\gamma$, it is said to be $n$-fold supercompact.
• If $\kappa$ is $\gamma$-extendible for all $\gamma$, it is said to be extendible. If $\kappa$ is $n$-fold $\gamma$-extendible for all $\gamma$, it is said to be $n$-fold extendible. Being $n$-fold extendible is equivalent to being $n+1$-fold strong.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $n$-fold high jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$, and clearance $\theta$ such that $M^\theta \subset M$ and $\theta \gt \gamma$, then $\kappa$ is said to be super-high-jump. One can define $n$-fold super-high-jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$ (and still $\theta \gt \gamma$).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$, then $\kappa$ is said to be almost huge. If $M^{\lt j^n(\kappa)} \subset M$, $\kappa$ is said to be almost $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-huge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-$n$-huge
• An elementary embedding $j: V_{j(\kappa)} \to V_\eta$ with critical point $\kappa$ is called an $A_2$ embedding. The critical point can also be called a 2-fold 0-extendible cardinal. More generally, if $\kappa$ is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$, one can call it an $n$-fold 0-extendible cardinal.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$, then $\kappa$ is said to be huge. If $M^{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be superhuge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-$n$-huge
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $V_{j({\kappa+\gamma})} \subset M$, $\kappa$ is said to be $\gamma$-ultrahuge. One can define $n$-fold $\gamma$-ultrahuge by $M^{j^n(\kappa)} \subset M$ and $V_{\kappa+\gamma} \subset M$.
• If $\kappa$ is $\gamma$-ultrahuge for all $\gamma$, it is said to be ultrahuge. Thus one can define $n$-fold $\gamma$-ultrahuge to mean $n$-fold ultrahuge for all $\gamma$.
• 2-fold $\lambda$-supercompact cardinals are also called $\lambda$-hyperhuge and 2-fold supercompact cardinals are also called hyperhuge. Thus one can alternatively refer to $n+1$-fold $\lambda$-supercompact as $n$-fold $\lambda$-hyperhuge and $n+1$-fold supercompact cardinals as $n$-fold hyperhuge. For $n \ge 1$, a cardinal is $n$-fold hyperhuge iff it is $n+1$-fold extendible (thus iff it is $n+2$-fold strong).

One can also define new large cardinal notions by replacing strongness embeddings ($j: V \to M$ with $V_\zeta \subset M$) by supercompactness embeddings ($j: V \to M$ with $M^\lambda \subset M$):

• ($n$-fold) Woodin for supercompactness cardinals are defined in analogy with ($n$-fold) Woodin cardinals; $n$-fold Woodin for supercompactness is equivalent to $n+1$-fold Woodin.
• ($n$-fold) Shelah for supercompactness cardinals are defined in analogy with ($n$-fold) Shelah cardinals; $n$-fold Shelah for supercompactness is equivalent to $n+1$-fold Shelah.
• One can also define ($n$-fold) weakly hyper-Woodin for supercompactness cardinals in analogy with ($n$-fold) weakly hyper-Woodin cardinals; $n$-fold weakly hyper-Woodin for supercompactness is equivalent to $n+1$-fold weakly hyper-Woodin.
• One can define ($n$-fold) hyper-Woodin for supercompactness cardinals in analogy with ($n$-fold) hyper-Woodin cardinals; $n$-fold hyper-Woodin for supercompactness is equivalent to $n+1$-fold hyper-Woodin.

Similarly, one can define new large cardinal notions by replacing supercompactness embeddings by strongness embeddings:

• High jump for strongness cardinals are defined in analogy with high jump cardinals.
• One can in the same way define ($n$-fold) super-high-jump for strongness in analogy with ($n$-fold) super-high-jump cardinals.

Additionally, one can define new large cardinal notions by replacing strongness or supercompactness embeddings by extendibility embeddings ($j: V_\zeta \to V_\eta$):

• If one thus defines ($n$-fold) Woodin for extendibility cardinals in analogy with ($n$-fold) Woodin cardinals, one gets a simplified definition of ($n$-fold) Vopenka cardinals; $n$-fold Woodin for extendibility/$n$-fold Vopenka is equivalent to $n+1$-fold Woodin and to $n$-fold Woodin for supercompactness.
• In the same way, one can define ($n$-fold) Shelah for extendibility cardinals in analogy with ($n$-fold) Shelah cardinals; $n$-fold Shelah for extendibility is equivalent to $n+1$-fold Shelah and to $n$-fold Shelah for supercompactness.
• One can define ($n$-fold) high jump for extendibility cardinals in analogy with high jump cardinals; $n$-fold high jump for extendibility is weaker than $n$-fold high jump but stronger than $n+1$-fold hyper-Woodin.
• One can define ($n$-fold) hyper-Woodin for extendibility or ($n$-fold) hyper-Vopenka cardinals in analogy with ($n$-fold) hyper-Woodin cardinals; $n$-fold hyper-Vopenka is equivalent to $n+1$-fold hyper-Woodin and to $n+1$-fold hyper-Woodin for supercompactness.
• One can define ($n$-fold) weakly hyper-Woodin for extendibility or ($n$-fold) weakly hyper-Vopenka cardinals in analogy with ($n$-fold) weakly hyper-Woodin cardinals; $n$-fold weakly hyper-Vopenka is equivalent to $n+1$-fold weakly hyper-Woodin and to $n$-fold weakly hyper-Woodin for supercompactness.

Answer 2

At least for Mahlo-ness, things are pretty simple to describe:

Given a property $P(\kappa)$ of cardinals implying strong inaccessibility, let $P^{\mathrm{Mahlo}}(\kappa)$ be the property "$\kappa$ has property $P$ and the set of cardinals $<\kappa$ with property $P$ is stationary in $\kappa$."

I don't think this, or any other, definition is standard in the literature, but the basic idea is fairly implicit.

Working above strong inaccessibility prevents certain combinatorial subtleties from cropping up which I don't think are germane to the question. There is also a weakening of the above gotten by dropping the requirement that $\kappa$ itself have property $P$, but at a glance it seems less natural (although perhaps more important for fine-grained analyses of consistency strength). It's immediate from the definition that $P^{\mathrm{Mahlo}}$ is always strictly stronger than $P$ itself since the smallest $P$-having cardinal can't have $P^{\mathrm{Mahlo}}$, so this answers your idempotency question for Mahlo-ification. As long as $P$ is reasonable (e.g. whether $P(\kappa)$ holds is computed correctly in $V_{\kappa+n}$ for $n$ finite) it also addresses the relative consistency strength issue.