Ordinal Conditional Functions (OCFs) are one of the predominant frameworks to define belief change operators. In his original paper Spohn defines OCFs as functions from the set of worlds to the set of ordinals. But in subsequent paper by Spohn and others, OCFs are just used as functions from the set of worlds to natural numbers (plus eventually +∞). The use of transfinite ordinals in this frame...

In this paper a variable-free parametric representation of manifolds is discussed, using transfinite interpolation or approximation, i.e. function blending in some functional space. This is a powerful approach to generation of curves, surfaces and solids (and even higher dimensional manifolds) by blending lower dimensional vector-valued functions. Transfinite blending, e.g. used in Gordon-Coons...

Making an extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorph...

Transfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define interpolants of values given on a smooth planar contour. When the data is two-dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some ...

The purpose of this paper is to show how the concept of transfinite computations relative to certain functionals of type 3 can be used to construct topologies and transfinite proof systems adding extra structure to some sets transfinitly definable over the continuum. Our aim is to initiate a fine-structure analysis of sets of the form Lκ(HC). We will motivate this below. Before doing so, let us...

This article provides an ordinal analysis of Σ1 transfinite dependent choice.