Sorteeritavus

Sorteeritavus, known in English as sortability, is a concept primarily found in logic and computability theory. It refers to the property of certain logical theories or formal systems to be decidable in a specific way. Specifically, a theory is sortable if there exists an algorithm that can decide, for any given formula, whether that formula is provable in the theory. This decidability implies that there is a mechanical procedure to determine the truth or falsehood of statements within the framework of that theory.

The concept is closely related to the idea of recursive enumerability. A set is recursively enumerable if there's an algorithm that lists all its members. A theory is sortable if the set of its provable sentences is recursively enumerable. This means we can write a program that, given enough time, will eventually output any valid theorem of the theory. However, sortability doesn't guarantee that the algorithm will always halt for sentences that are *not* provable.

Many fundamental logical theories, such as first-order Peano arithmetic, are known not to be sortable. This non-sortability is a significant result, demonstrating inherent limitations in what can be mechanically proven or decided within these systems. The exploration of sortability helps us understand the boundaries of formal reasoning and computation.