The relativistic dynamics of a thin spherical shell of dust collapsing under its own weight is developed on the hypothesis that the shell rebounds elastically after imploding to a point. The resulting description involves certain causal anomalies. The question of whether these anomalies can lead to actual logical contradictions is investigated. Possible similarities between the results obtained for the special case considered here and the characteristics of general asymmetric collapse are discussed. Reasons are given for believing (i) that singularities do not arise in the course of an asymmetric collapse except in special cases; (ii) that collapse of a mass M does not proceed irreversibly to zero volume, but is an oscillatory phenomenon in which the object never becomes smaller than a critical linear size of order GMc2; (iii) that when the object is near this critical size, a region develops in its interior where past and future cannot be globally distinguished; and (iv) that since the law of baryon conservation cannot be globally valid in such a pathological region, largescale annihilation of baryons is possible.