For if we start from the primary aggregate of "all conceivable things," its infinity can certainly not be transcended by any other infinity. But according to the above theorem the "aggregate of all partial aggregates" would have a greater potentiality, although it itself cannot extend further than to the conception of the maximum of all conceivable things. We thus arrive at an insoluble paradox, a typical example of how, in the system of conceptions involved, something is insufficient or not in conformity with logical thought. And this sceptical view receives support from various remarks of Descartes, Locke, Leibniz, and particularly Gauss, who, long before the advent of the Theory of Aggregates, raised a protest against inexact definitions of infinity. In another case, however, the same theory seems to arise by perfectly logical processes, although it again leads to a statement that does not seem correct to "common sense." For it shows by a very subtle and ingenious method that all the surface-points of a surface infinitely extended in all directions may be brought to correspond in a reversible single manner to the linear points of a line, however small; so that to every point of the unlimited plane there corresponds a definite point of the line, and vice versa. The same theorem may be extended to three-dimensional space, with the result that we have to reconcile ourselves with the incredible fact that, expressed in popular language, a straight line of however small length exhibits the same potentiality with regard to the number of its points, as all the points in the universe. For my own part, I must confess that no means suggests itself to me to make this paradox intelligible. But the _sacrificium intellectus_ comes within dangerous proximity. Einstein, who values and marvels at the theory of aggregates as a science, or perhaps more as a work of art built up from the materials of science, gives whole-hearted support to the proof. He refuses to accept the notion of a paradox--that is, he recognizes a contradiction not in our process of reasoning, but only in a habit of thought that is open to correction. I should give much to discover the means of correction! * * * * * * * * A third example arises out of the special theory of relativity. It has a mysterious paradoxical character that vanishes when a clear view of the relationships involved has been obtained.