Philosophy of Mathematics Seminar

Joel David Hamkins, Notre Dame

Natural instances of illfoundedness and nonlinearity in the hierarchy of consistency strength

There is an unexplained logical mystery in the foundations of mathematics, namely, our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Why should it be? The phenomenon is thought to carry significance for foundations, perhaps pointing us, some have argued, toward the ultimately correct mathematical theories, the “one road upward.” And yet, we know as a purely formal matter that the hierarchy of consistency strength is not well-ordered. It is ill-founded, densely ordered, and nonlinear. The statements commonly used to illustrate these features, however, are often dismissed as unnatural or as Gödelian trickery. In this talk, however, I aim to rebut that criticism by presenting a variety of natural hypotheses that reveal ill-foundedness in consistency strength, density in the hierarchy of consistency strength, and incomparability in consistency strength. This will lead to discussion of the role and meaning of “natural” in the foundations of mathematics.