How do ordinals measure the strength and computational power of formal theories and what information can be gleaned from this correlation? This will be the guiding question of this talk. The connection between ordinal representation systems and theories is established in ordinal analysis, a central area of proof theory. Ordinal-theoretic proof theory came into existence in 1936. Gentzen showed ...

We present an overview of the meaningful aggregation functions mapping ordinal scales into an ordinal scale. Three main classes are discussed, namely order invariant functions, comparison meaningful functions on a single ordinal scale, and comparison meaningful functions on independent ordinal scales. It appears that the most prominent meaningful aggregation functions are lattice polynomial fun...

In this paper, we present an ordinal feature based method for face recognition. Ordinal features are used to represent faces. Hamming distance of many local sub-windows is computed to evaluate differences of two ordinal faces. AdaBoost learning is finally applied to select most effective hamming distance based weak classifiers and build a powerful classifier. Experiments demonstrate good result...

It is true in the Solovay model that every countable ordinal-definable set of sets of reals contains only ordinal-definable elements.

The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the secondorder theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordin...

An α-automaton (for α some ordinal) is an automaton similar to a Muller automaton that processes words of length α. A structure is called α-automatic if it can be presented by α-automata (completely analogous to the notion of automatic structures which can be presented by the well-known finite automata). We call a structure ordinal-automatic if it is α-automatic for some ordinal α. We continue ...

In this paper we address the problem of aggregating outranking statements from multiple preference criteria of ordinal significance. The concept of ordinal concordance of a global outranking situation is defined and an operational test for its presence is developed. Finally, we propose a new kind of robustness analysis for global outranking statements integrating classical dominance, ordinal an...

We study the problem of ordinal embedding: given a set of ordinal constraints of the form distance(i, j) < distance(k, l) for some quadruples (i, j, k, l) of indices, the goal is to construct a point configuration x̂1, ..., x̂n in R that preserves these constraints as well as possible. Our first contribution is to suggest a simple new algorithm for this problem, Soft Ordinal Embedding. The key fe...