Although there is an abundance of known properties independent of the usual axioms of set theory ZFC, only relatively few of them concern the objects taught in calculus courses. In this talk I like to discuss two of such examples. The first concerns multivariable calculus and is related to Fubini theorem: When the existence of iterated integrals of a function f(x,y) implies that the resulting value is independent of the order of integration? The second concerns continuously differentiable functions from the interval I=[0,1] to itself: Is it possible that a set of such functions that cover the unit square has fewer elements than the set of real numbers? The talk will be fairly elementary.