[C1] Weak systems of Gandy, Jensen and Devlin in Set Theory: Centre de Recerca Matemàtica, Barcelona 2003-4 edited by Joan Bagaria and Stevo Todorčević, Trends in Mathematics, Birkhäuser Verlag, Basel, 2006, 149-224. .ps .dvi

(contains a variety of constructions proving the independence of various natural statements in various weak systems of set theory and shedding light on flaws in Devlin's treatise Constructibility. The first third of the paper develops versions of the model building techniques of my paper Slim Models, applicable to the systems considered in the paper. The heavily syntactic middle third of the paper examines the effect of adding the axiom ``the class of finite subsets of any set is a set" to the various systems proposed by Devlin and by Gandy; and closes with the suggestion that the addition, instead, of the slightly weaker form ``for each positive integer k the class of subsets of size k of any given set is a set", mightly be precisely the elusive optimal strengthing of Devlin's system BS for the purposes to hand. The final third returns to model-building mode to answer other questions about these systems.)

The flaws in Devlin's book were noticed, but not corrected, soon after its publication in 1984. The assorted insights, which I accumulated over the next nineteen years, turned during my Barcelona sabbatical of 2003/4 into an apparatus criticus for curing the problem.

The ideas underlying the next two papers emerged slowly over fifteen years, and the paper [B4], which was written in 2008, draws heavily on them. In 2009 mature drafts of the papers [C2 and 3] were circulated and placed on my web-site, and submitted for publication; but in October 2010, when I tested them in a course to Cambridge graduate mathematicians given in the Michaelmas Term, Nathan Bowler, who attended the course, suggested important improvements, and accordingly the first of the two papers will now appear under joint authorship.

[C2] (with Nathan Bowler) Rudimentary recursion, gentle functions and provident sets Notre Dame Journal of Formal Logic, special issue on set theory and higher order logic, 56 (2015) 3-60

(We introduce the collections of rudimentarily recursive and gentle functions, and study those sets, which we call provident, which are transitive and closed under all rudimentarily recursive functions, allowing parameters from within the set in question. We identify a single rudimentary recursion, with parameter, to instances of which all others reduce; we obtain various characterizations of provident sets, showing in particular that the segment J_ν of the Jensen hierarchy is provident if and only if ων is an indecomposable ordinal; and we find strong uniform bounds on the rate of growth of rudimentarily recursive functions.)

[C3] Provident sets and rudimentary set forcing Fundamenta Mathematicæ 230 (2015) 99-148.

(Using the theory of rudimentary recursion and provident sets developed in the previous paper, we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen's J_ω. Much of the development is rudimentary or at worst given by rudimentary recursions with parameter the notion of forcing under consideration. Our development eschews the power set axiom. We show that the forcing relation for restricted wffs is propagated through our hierarchies by a rudimentary function, and we show that the construction of names for the values of rudimentary and rudimentarily recursive functions is similarly propagated. Our main result is that a set-generic extension of a provident set is provident.)