Mathematicians interested in understanding the directions of current research in set theory will not want to overlook this book, which contains the proceedings of the AMS Summer Research Conference on Axiomatic Set Theory, held in Boulder, Colorado, June 19–25, 1983. This was the first large meeting devoted exclusively to set theory since the legendary 1967 UCLA meeting, and a large majority of the most active research mathematicians in the field participated. All areas of set theory, including constructibility, forcing, combinatorics and descriptive set theory, were represented; many of the papers in the proceedings explore connections between areas. Readers should have a background of graduate-level set theory.

There is a paper by S.~Shelah applying proper forcing to obtain consistency results on combinatorial cardinal “invariants” below the continuum, and papers by R.~David and S.~Freidman on properties of $0^\#$. Papers by A.~Blass, H.-D.~Donder, T.~Jech and W.~Mitchell involve inner models with measurable cardinals and various combinatorial properties. T.~Carlson largely solves the pin-up problem, and D.~Velleman presents a novel construction of a Souslin tree from a morass. S.~Todorcevic obtains the strong failure of the \qedprinciple from the Proper Forcing Axiom and A.~Miller discusses properties of a new species of perfect-set forcing. H.~Becker and A.~Kechris attack the third Victoria Delfino problem while W.~Zwicker looks at combinatorics on $P_\kappa(\lambda)$ and J.~Henle studies infinite-exponent partition relations. A.~Blass shows that if every vector space has a basis then $AC$ holds. I.~Anellis treats the history of set theory, and W.~Fleissner presents set-theoretical axioms of use in general topology.