Ordinalisia

Ordinalisia is a theoretical construct in order theory and logic that generalizes the concept of ordinal numbers to multi-dimensional ordered systems. It provides a formal framework for comparing elements that carry several independent order criteria, enabling coherent ranking beyond a single linear order.

An ordinalisia consists of a set O together with a family of total orders {≤_i} indexed by a set I, and a rule to derive a global order ≤_G. Common aggregation rules include lexicographic extension, product orders, and prioritized or layered orders. Variants may emphasize different aggregation schemes or coherence conditions among the component orders.

In simple cases, an ordinalisia with two criteria on pairs (α,β) of ordinals can use lexicographic comparison: (α1,β1) ≤_G (α2,β2) if α1 < α2 or (α1 = α2 and β1 ≤ β2). The properties of the global order depend on the chosen aggregation, with well-foundedness and transfinite induction preserved under specific, well-behaved constructions.

Applications appear in proof theory, model theory, and multi-criteria decision analysis, where hierarchical or multi-dimensional ranking is needed. Limitations include potential loss of information when reducing multiple orders to a single one, and increased complexity in defining and working with the global order.

See also: ordinal, well-order, lexicographic order, multi-criteria decision analysis, order theory.