Frege's construction was flawed. Russell discovered that Basic Law V is inconsistent (this is Russell's paradox ). Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead . They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop so much of mathematics, such as an " axiom of reducibility ". Even Russell said that this axiom did not really belong to logic.