The objections which have been raised against the existence of   the totality of all real numbers and infinite sets generally lose   all their justification once one has adopted the view stated above   [the axiomatic method]. By the set of the real numbers we do not   have to imagine something like the totality of all possible laws   governing the development of a fundamental series, but rather, as   has been set out, a system of things whose mutual relations are   given by the finite and closed systems of axioms I–IV [for   complete ordered fields] given above, and about which statements   only have validity in the case where one can derive them via a   finite number of inferences from those axioms. (Hilbert 1900b:   184)[10]