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Concept

Equivalence relation

A relation that is reflexive, symmetric and transitive, and therefore cuts a set into classes with every element in exactly one. It is the standard way of declaring two things the same for a stated purpose without them being identical.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The classes, a selection from them, and the translates that cannot have a length. Points of several classes of the unit interval under translation by rationals, drawn one class per row, above rows showing rational translates of a selection that never overlap.

A set that has no size at all

Slide the unit interval along itself by every rational and the points fall into classes. Choose one point from each and the resulting set has no length — not zero, not positive, none: countably many disjoint copies of it would have total length nought or infinity, and the union needs something in between.

analysis · Measure Two models the modal language cannot separate, and two it can. Four Kripke models in two pairs: the upper pair joined by a bisimulation and agreeing on every formula, the lower pair separated by a formula found by search.

Two diagrams the language cannot tell apart

A modal formula sees a diagram of worlds and arrows through a very narrow window. Exactly how narrow is settled by a game: where one player can answer every move, no formula whatever separates the two starting worlds, however different the diagrams look.

logic · Modal logic

Named alongside it

The objects these essays reach for when they reach for this one.

AccessibilityAxiom of choiceBisimulationContradictionCountabilityDecision procedureDense setExhaustive searchExpressive powerKripke modelMeasureMeasure zero

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