Boundedness Theorems for Flowers and Sharps

Abstract. We show that the Sigma11 - and Sigma12 -boundedness theorems extend to the category of continuous dilators. We then apply these results to conclude the corresponding theorems for the category of sharps of real numbers, thus establishing another connection between Proof Theory and Set Theory, and extending work of Girard-Normann and Kechris-Woodin

Boundedness theorems for dilators and ptykes

The main theorem of this paper is: If ƒ is a partial function from ℵ_1 to ℵ_1 which is ∑^1_1-bounded, then there is a weakly finite primitive recursive dilator D such that for all infinite α ϵ dom(ƒ), ƒ(α) ⩽ D(α). The proof involves only elementary combinatorial constructions of trees. A generalization to ptykes is also given