The problem of aggregating a set of ordinal rankings of n alternatives has given rise to a number of consensus models. Among the most common of these models are those due to Borda and Kendall, which amount to using average ranks, and the /1 and /2 distance models. A common criticism of these approaches is their use of ordinal rank position numbers directly as the values of being ranked at those...

We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. The generalized Turing machine is able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the ...

We apply set-theoretical ideas to an iteration problem of dynamical systems. Among other results, we prove that these iterations never stabilise later than the first uncountable ordinal; for every countable ordinal we give examples in Baire space and in Cantor space of an iteration that stabilises exactly at that ordinal; we give an example of an iteration with recursive data which stabilises e...

With the enormous scale of massive open online courses (MOOCs), peer grading is vital for addressing the assessment challenge for open-ended assignments or exams while at the same time providing students with an effective learning experience through involvement in the grading process. Most existing MOOC platforms use simple schemes for aggregating peer grades, e.g., taking the median or mean. T...

An ordinal calculator has been developed as an aid for understanding the countable ordinal hierarchy and as a research tool that may eventually help to expand it. A GPL licensed version is available in C++. It is an interactive command line calculator and can be used as a library. It includes notations for the ordinals uniquely expressible in Cantor normal form, the Veblen hierarchies and a for...

We present a reduction framework from ordinal regression to binary classification based on extended examples. The framework consists of three steps: extracting extended examples from the original examples, learning a binary classifier on the extended examples with any binary classification algorithm, and constructing a ranking rule from the binary classifier. A weighted 0/1 loss of the binary c...

We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic well-founded partial order is bounded by ω; 2) The ordinal heights of automatic well-founded re...