During the second half of the twentieth century, many mathematicians lost interest in the foundations of mathematics. One of the reasons for this decline was an increasingly popular view that general set theory and Gödel-style incompleteness and independence results do not have much effect on mathematics as it is actually practiced. That is, as long as mathematicians study relatively concrete mathematical objects, they can avoid all foundational issues by appealing to a vague hybrid of philosophical positions including Platonism, formalism, and sometimes even social constructivism. Harvey Friedman (born 1948) has continually fought this trend, and in 1984 he received the National Science Foundation's Alan T. Waterman Award for his work on revitalizing the foundations of mathematics.

One of Friedman's methods of illustrating the importance of foundational issues is to isolate pieces of mathematics that either display the incompleteness phenomenon or require substantial set...