Quite a number of other paradoxes arose in the early years following Burali-Forti's result. Some of these modern paradoxes, falling more or less within the context of set theory, can be seen to be related to several ancient paradoxes of logic. This statement, if true, makes the speaker a liar for telling the truth. The quoted statement can neither be true nor false without entailing a contradiction. A simplification of Richard's paradox is due to G. G. Berry in 1906.
Various modifications exist.
K. Grelling and L. Nelson called attention in 1908 to a paradox which they regarded as only a variant of Russell's paradox. The Grelling paradox can be stated quite simply. One of the more interesting paradoxes appeared in 1924 and was due to two distinguished Polish mathematicians, S. Banach and A. Tarski. Banach's and Tarski's work extended the implications of a paradoxical theorem due to the German mathematician Felix Hausforff. Denote the large sphere by S and the small sphere by P .