An Application of Logic to Combinatorial Geometry: How Many Tetrahedra are Equidecomposable with a Cube?

It is known (see Rapp 9]) that elementary geometry with the additional quantifier “there exist uncountably many” is decidable. We show that this decidability helps in solving the following problem from combinatorial geometry: does there exist an uncountable family of pairwise non-congruent tetrahedra that are n-equidecomposable with a cube? Mathematics Subject Classification: 03B25, 03C80, 51M04, 52B05, 52B10.