Selection and Uniformization Problems in the Monadic Theory of Ordinals: A Survey

Abstract

A formula ψ(Y) is a selector for a formula ϕ(Y) in a structure $\mathcal{M}$ if there exists a unique Y that satisfies ψ in $\mathcal{M}$ and this Y also satisfies ϕ. A formula ψ(X,Y) uniformizes a formula ϕ(X,Y) in a structure $\mathcal{M}$ if for every X there exists a unique Y such that ψ(X,Y) holds in $\mathcal{M}$ and for this Y, ϕ(X,Y) also holds in $\mathcal{M}$. In this paper we survey some fundamental algorithmic questions and recent results regarding selection and uniformization, when the formulas ψ and ϕ are formulas of the monadic logic of order and the structure $\mathcal{M}=(\alpha,<)$ is an ordinal α equipped with its natural order. A natural generalization of the Church problem to ordinals is obtained when some additional requirements are imposed on the uniformizing formula ψ(X,Y). We present what is known regarding this generalization of Church’s problem.