User blog comment:Meowzz/Question: 0=1 Cardinal/@comment-26428969-20180422163440/@comment-78.192.45.127-20180423193306

The forcing-related meaning includes all "local" LCAs

It includes λ-supercompactness for any particular λ, but not full supercompactness. It includes λ-strongness for any particular λ, but not full strongness. And so on.

In general that definition covers everything that can be locally verified in some V-rank. Measurability of κ can be verified in V_κ+2 (as the ultrafilter witnessing measurability is an element of V_κ+2). Mahloness and weak compactness can be verified in V_κ+1 (for Mahlos: every club C⊆V_κ must contain a λ such that V_κ satisfies "λ is inaccessible", for weak compactness: every tree T composed of elements of V_κ and of length κ must have a cofinal branch, which is itself going to be an element of V_κ+1).

A very related result: for all sentences φ, "V_α satisfies φ" is a Σ2 sentence, for any ordinal α. This is partly why the "local" LCAs are precisely the Σ2 sentences (satisfying some additional properties (forcing related properties, it turns out))