Lesson 6

After the failure of Cantor's set theory, the race was on to put mathematics on solid footing. Some mathematicians, philosophers, and logicians wanted to axiomatize all of mathematics; they wanted to find a set of axioms from which all mathematical truths could be derived, and they wanted these axioms to be consistent.

Bertrand Russell was arguably the leader of this movement. He published his first book on the subject, The Principles of Mathematics, in 1903, in which he detailed his paradox. In 1910, he and Alfred North Whitehead published the first volume of their attempt to axiomatize all of mathematics: Principia Mathematica.

In Principia Mathematica, Russell and Whitehead defined a system of types in which one object could not contain another of the same type, thus avoiding Russell's paradox. (Under these axioms, there was no question as to whether the set \(S\) contains itself because nothing can contain itself.) After defining this new kind of set, Principia Mathematica undertook the task of proving what they could about specific sets under these axioms–e.g. the natural numbers, the integers, the real numbers, etc.

It would be an extreme understatement to call this a slow and tedious process. On page 379, Principia Mathematica finally manages to prove that \(1+1=2\). (The authors famously commented “The above proposition is occasionally useful.”) Two more volumes were published in 1912 and 1913 respectively. Whitehead and Russell planned to continue beyond the third, but the tedium eventually tired them out.

Many mathematicians believed that Russell and Whitehead had accomplished their task of finding axioms that could in principle be used to develop all of mathematics, though many were not happy with the specific axiom scheme that they chose (nor with their notation).