This book is intended as a sequel or partial update of Subsystems of Second Order Arithmetic (SOSOA). It consists of 24 original research papers exhibiting some of the exciting new developments in reverse mathematics and subsystems of second order arithmetic, subsequent to 1998.

1. Possible ra-diagrams of models of arithmetic 2. Weak theories of nonstandard arithmetic and analysis 3. Notions of compactness in weak subsystems of second order arithmetic 4. Proof-theoretic strength of the stable marriage theorem and other problems 5. Free sets and reverse mathematics 6. Interpreting arithmetic in the r.e. degrees under Z4-induction 7. Reverse mathematics, Archimedean classes, and Hahn’s Theorem 8. The Baire category theorem over a feasible base theory 9. Basic applications of weak Konig’s lemma in feasible analysis 10. Maximal nonfinitely generated subalgebras 11. Metamathematics of comparability 12. A note on compactness of countable sets 13. A survey of the reverse mathematics of ordinal arithmetic 14. Reverse mathematics and ordinal suprem 15. Did Cantor need set theory? 16. Models of arithmetic: Quantifiers and complexity 17. Higher order reverse mathematics 18. Arithmetic saturation 19. WQO and BQO theory in subsystems of second order arithmetic 20. Reverse mathematics and graph coloring: Eliminating diagonalization 21. Undecidable theories and reverse mathematics 22. II1 sets and models of WKLo 23. Manipulating the reals in RCA0 24. Reverse mathematics and weak systems of 0-1 strings for feasible analysis