Concept

Impossibility

A theorem stating that no object of a described kind exists, as against a report that none has been found. What makes one is an argument covering every candidate at once, usually a quantity a candidate would have to change and provably cannot.

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

3 consistent judges, and a majority that is not. A table of judges against three questions, every judge's row internally consistent, with the majority answer to each question underneath forming a combination no judge holds.

The court that contradicts itself

Three judges each answer three questions, and each answers them consistently. Take the majority on each question separately and the answers no longer hang together — the body as a whole asserts a combination no member of it holds, and no rearrangement of the procedure removes the problem.

applied · judgement aggregation

A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

Equal area is enough, and equal volume is not

Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.

computation · scissors congruence

Named alongside it

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

Aggregation Area Axiom Congruence Consistency Dihedral angle Dissection Exhaustive search Invariant Logical connective Majority Operation set

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