Dear Harvey,

You are right, to get strength from reflection one not only needs higher-order logic (lengthenings) but also class (2nd order) parameters. As I said in what I wrote, I chose to ignore parameters to simplify the discussion. In fact allowing more than 2nd-order parameters will lead to inconsistency unless one treats them carefully using embeddings, what I call “Magidor reflection”. But obviously I wanted to avoid this subtle discussion of parameters to bring out the main point: Lenthenings are required to derive an inaccessible from reflection.

Best,

Sy