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Rank-into-rank axioms were initially thought to be close to inconsistency, but over time evidence accumulated that in a sense they are just one point in a natural hierarchy of symmetry principles. Ku …

Typically, large cardinals with stronger consistency strength are larger, but there is an important general exception: If a large cardinal axiom is $Σ_{k+1}$, then typically the least example is large …

For every c.e theory $T$ extending KP (Kripke-Platek) with a model $M$ of height $α<ω_1$, the intersection of all such $M$ is a subset of $L_{α^{+,\mathrm{CK}}}$. This holds since the existence of su …

BTEE is conservative over the stationary reflection principle (SRP), i.e. ZFC + (schema) {there is $n$-subtle cardinal}$_{n∈\mathbb{N}}$. Using $n$-ineffable in the schema is equivalent. Note that we …

In one sense, closure under sharps is itself a standard point in the hierarchy of consistency strengths. Just like the exact consistency strength of "ZFC + measurable" is "ZFC + measurable", so is the …

The consistency strength is above that of $n$-iterable cardinals for finite $n$. Thus, despite the seeming weakness of the statement, the $ω$-Erdős upper bound given by Reinhardt is fairly close to t …

Overall, the large cardinal axiom hierarchy is very similar between ZC (ZFC minus replacement; we are including regularity) and ZFC. A large cardinal axiom (unprovable in ZFC) satisfied by $κ$ typica …

No, it is not; existence of a proper class of extendible cardinals is not provable in ZFC from any consistent $Σ^V_5$ statement. Assume a proper class of extendibles and let $φ$ be a true $Σ_5$ state …

Here is another even simpler refutation. Recall that $δ$ is Berkeley iff for every predicate $A$ and $α≥δ$, there is a nontrivial elementary self-embeding of $(V_α,∈,A)$ with critical point $<δ$. (T …

The Löwenheim number (compared to LS, this uses sentences rather than theories) for the second order logic $L^2$ is the least $κ$ such that $V_κ$ satisfies all true $Σ_2$ sentences. This $κ$ has cofi …

The first chromatic cardinal is the first Mahlo cardinal. (Per the connection with the reflection in the question, I assume that in the definition, $α$ and the first argument of $c_i$ need not be ina …

The examples in this thread are interesting as curiosities, but while one never knows where fundamental explorations will lead, we should be skeptical of their physical applicability. In physics, the …

Large cardinals offer a detailed coherent picture — with a single principle, that of symmetry, reaching even (essentially) the strongest large cardinals. They continually offer new results — without …