  The analogue of Hilbert's tenth problem over $\mathbb{Q}$ asks for an
algorithm to decide the existence of rational points in algebraic varieties
over this field. This remains as one of the main open problems in the area of
undecidability in number theory. Besides the existence of rational points,
there is also considerable interest in the problem of effectivity: one asks
whether the sought rational points satisfy determined height bounds, often
expressed in terms of the height of the coefficients of the equations defining
the algebraic varieties under consideration. We show that, in fact, Hilbert's
tenth problem over $\mathbb{Q}$ with (finitely many) height comparison
conditions is undecidable.
