*Question:* What is the consistency strength of existence of a $(κ^{++})^L$-Mahlo cardinal $κ$?

I am particularly interested in how the strength compares to weakly compact cardinals (and other levels of indescribability), and I am looking for equiconsistencies between an $α$-Mahlo $κ$ (for some sense of $α$-Mahlo) and existence of weakly compact $κ$.

The motivation for the question is that in set theory, strong levels of one axiom schema naturally morph into weak versions of the next schema. For example:

* The replacement schema corresponds to Ord being inaccessible, and a strong limit $κ$ is inaccessible iff for every $U⊂V_κ$, $V_κ$ satisfies replacement for formulas in $(V_κ,∈,U)$.

* A schema $∃(\text{regular }λ) \, V_λ ≺_{Σ_n} V$ corresponds to Ord being Mahlo, and a strong limit $κ$ is Mahlo iff $∀U⊂V_κ \, ∃(\text{regular }λ<κ) \, (V_λ,∈,U∩V_λ) ≺ (V_κ,∈,U)$.

* After iterations of Mahloness, the next step is weak compactness, prompting the question: What is the correspondence between strong levels of Mahloness and weak compactness?

The definition of $α$-Mahlo $κ$ for $α≤κ^+$ is standard (or see below). Beyond $(κ^+)^L$, we have different choices:

a. require $α≤κ^+$, so $(κ^+)^L < κ^+$

b. use the definition for $α≤κ^+$ beyond $κ^+$ (allowing GCH to fail at $κ$)

c. use ordinal representation systems for diagonalization beyond $κ^+$ (not sure how far this works)

d. $κ$ being $α$-Mahlo in a sufficiently large generic collapse $\mathrm{Coll}(κ,λ)$

(and for more choices, the above also makes sense for weakly Mahlo cardinals).

My guess is that these possibilities have the same strength, but in any case, any of these will be fine for an answer.

*Definition of $α$-Mahlo cardinals*

If the reader is unfamiliar with greatly Mahlo cardinals, here is a definition and some properties.

A strong limit cardinal $κ$ is 0-Mahlo iff it is regular uncountable.

$κ$ is $α+1$-Mahlo iff the set of $α$-Mahlo cardinals is stationary below $κ$.

For limit $α≤κ$, $κ$ is $α$-Mahlo iff it is $α'$ Mahlo for every $α'<α$.

Beyond $κ$-Mahlo, we rely on diagonalization, and it turns out that the iteration is well-defined up to $κ^+$.

Define $M_α$ modulo the nonstationary ideal on $κ$:

$M_0$ = $κ$

$M_{α+1}$ = {$λ<κ$: $\mathrm{cf}(λ)>ω$ and $M_α$ is stationary below $λ$}

$M_α$ for limit $α<κ^+$ is greatest lower bound of $(M_{α'}:α'<α)$ (so $M_α \setminus M_{α'}$ is nonstationary).

A strong limit $κ$ is $α$-Mahlo ($α≤κ^+$) iff for every $α'<α$, $M_{κ+α}$ is stationary below $κ$.

Also, the greatest lower bound above is defined modulo the nonstationary ideal (which suffices). If desired, given a choice of fundamental sequences, we can pick a canonical representative of $M_α$ as follows:

$\mathrm{cf}(α)<κ$ -- $M_α$ is the intersection of $M_{α'}$ for $α'$ on the fundamental sequence of $α$.

$\mathrm{cf}(α)=κ$ -- $M_α$ is the diagonal intersection of $M_{α'}$ for $α'$ on the fundamental sequence of $α$. For example, $M_κ$ consists of (modulo a nonstationary set) regular cardinals below $κ$.

Weakly compact cardinals are greatly Mahlo (i.e. $κ^+$-Mahlo) and more. For example, the property that every stationary subset of $κ$ reflects (i.e. is stationary below some $λ<κ$) is consistency-wise (and in terms of the least such cardinal in L) strictly between greatly Mahlo and weakly compact.