Inexact limits. On the heap-paradox, Darwinism and recursive grammar. The heap-paradox (sorites) can be reinforced by a combination with the basic idea of Achilles (the paradox of his running-match with the tortoise). The logical pattern of the reinforced heap-paradox will be analysed in a new manner by distinguishing between limited and unlimited transitivity. This analysis makes explicit the constructive (and by no means descriptive) character of the paradox. Finally it is shown that the (invalid) logical pattern of the heap-paradox is applied in popular presentations of Darwinism (Russell, Wuketits), in the debate about abortion (Singer) and in the foundation of recursive theories of grammar (Chomsky, Davidson, Grice, Searle).