Thomas Kuhn  has presented a view of the history of science as a succession of “paradigms” which are not completely comparable with each other. Karl Popper  has attacked this view as being relativistic and denying that there is objective scientific truth. Popper seems to be saying that in order for science to be objectively true, every two scientific theories must be completely comparable. In taking this position, Popper is making a claim for a kind of completeness that is ruled out for theories strong enough to be “interesting” by Gödel’s Incompleteness Theorem. In this paper, Gödel’s Incompleteness Theorem and its proof will be used to draw conclusions about this conflict between Kuhn and Popper. In particular, it will be argued that if Popper had taken Gödel’s Incompleteness Theorem into account, he would have wound up with a position consistent with that of Kuhn. Gödel’s Incompleteness Theorem will also be used to argue that Feyerabend  does not really have an argument that there is no objective scientific method. The paper will close with some remarks on what might really count as objective scientific truth. ∗This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.