A critique of one of Galileo's mental experiments and an explanation on why bodies fall with equal acceleration through the vacuum

In the famous thought experiment studied in this article, Galileo attempted to refute the Aristotelian hypothesis that heavier bodies should fall more quickly than lighter ones. After pointing out some inconsistencies in Galileo’s approach, we show, through the design of two alternative but equivalent experiments, that from his imaginary experiment, it is not possible to reach the conclusion that all bodies fall simultaneously. We show why, to explain the result of this type of experience, it is necessary to establish the equivalence between inertial and gravitational masses derived exclusively from experience.