In Kanamori's The Higher Infinite a diagram is included towards the end of the book which illustrates the large cardinal hierarchy by listing many large cardinal axioms and drawing their direction implications/relative consistency implications.

I am wondering if there is anywhere available a similar diagram, or maybe even preferably a list, dealing with the interpretability/consistency hierarchy (here I am assuming them both to be equivalent, which I believe just limits it to theories that are reflexive and conservative over $\Pi^0_1$ sentences, correct?).

Of course, such a list could easily be arbitrarily long ($T < T+Con(T)< etc.$) and it could also be arbitrarily short ($PRA < PA < Z_2 < ZFC$). I guess what I am looking for is something that does list many of the useful or well studied fragments of $PA$, fragments of $Z_2$, and fragments of $ZFC$ without falling victim to arbitrary sequences of direct assumptions of consistency. Things like Presburger arithmetic, $Q$, $I\Delta_0 + EXP$, the big five of reverse mathematics, $KP$ set theory, are what I am looking for.

My main motivation for asking this question is that I very often have seen answers to questions here or elsewhere that involved to me less familiar fragments of $PA$ or $Z_2$ and not known where to look them up or where their place is in the, I guess, larger reverse mathematics program (Simpson's Subsystems of Second Order Arithmetic was often but not always helpful, and there, as far as I'm aware, is not such list or diagram of the systems discussed). I have a vague memory that there existed a list that was along these lines in a paper on ordinal analysis; I thought that it was written by Rathjen, however I reviewed many of his papers looking for it and was unable to find what I thought I recalled.

A list, a diagram, or even an article or book which contains the information I am looking for would be a satisfactory answer.

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    $\begingroup$ A table of the sort you are looking for can be found on pages 39-40 of Harvey Friedman's paper "Interpretations according to Tarski,: available at the link cpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/… $\endgroup$
    – Ali Enayat
    Sep 7, 2018 at 8:22
  • $\begingroup$ @AliEnayat that is exactly the paper I thought I remembered seeing, thank you. $\endgroup$
    – Not_Here
    Sep 7, 2018 at 22:11


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