We consider a well-known partial order of Prikry for producing a collapsing function of minimal degree. Assuming $MA + \neq CH$, every new real constructs the collapsing map.

We consider a well-known partial order of Prikry for producing a collapsing function of minimal degree. Assuming MA + ≠ CH, every new real constructs the collapsing map.

Uncountable superperfect forcing is tree forcing on regular uncountable cardinals κ with κ<κ=κ, using trees in which the heights of nodes that split along any branch in the tree form a club set, and such that any node in the tree with more than one immediate extension has measure-one-many extensions, where the measure is relative to some κ-complete, nonprincipal normal filter F. This forcing adds a generic of minimal degree if and only if F is κ-saturated.