User blog:Alemagno12/Huge Ordinal Analysis

I am making a new blog post series in which I will analyze progressively stronger theories and try to find lower bounds for their strength. The goal of this series is to try to unify the higher portion of the two branches of computable googology, so we can know if, say, BMS is stronger than Friedman's functions derived from finite promise games.

Here are the theories that I am planning on analyzing: After that, I will analyze the previous systems with projective determinancy added to them; and then, I am not sure in what order I will analyze the next systems, but I am planning on analyzing ZF, systems of the form ZF + there exists a (some cardinal), and systems of the form Παω-CA.
 * Z2
 * Z3 (I need help with the definition of this one)
 * Higher Order Arithmetic (I need help with the definition of this one)
 * ZFC
 * ZFC + there exists an inaccessible cardinal
 * ZFC + some extensions to inaccessible cardinals
 * ZFC + there exists a Mahlo cardinal
 * ZFC + there exists a rank-into-rank cardinal
 * ZFC + there exists a rank-into-rank cardinal

(btw, I already started analyzing Z2: will make a blog post with the analysis once I've progressed far enough)