Boundedness theorems for dilators and ptykes

Abstract

The main theorem of this paper is: If ƒ is a partial function from ℵ₁ to ℵ₁ which is ∑₁¹-bounded, then there is a weakly finite primitive recursive dilatorD such that for all infinite $\text{αϵdom(ƒ)}$, $\text{ƒ(α) ⩽ D(α)}$. The proof involves only elementary combinatorial constructions of trees. A generalization to ptykes is also given.