In calculus, one learns several subtle "pathologies" involving the real number line, such as convergent but not absolutely convergent infinite series, differentiable functions with discontinuous derivative, and infinitely-differentiable functions whose Taylor series has zero radius of convergence. A little thought reveals that some of these concepts seem to be much more complicated than others. For example, while in calculus one learns to approximately "check" if a function is continuously differentiable (just draw a graph and zoom in!), it seems absolutely hopeless to check plain differentiability in any systematic way.

This talk will give a gentle introduction to Descriptive Set Theory, a subfield of mathematical logic providing the tools to make such statements precise, which can be variously thought of as "infinitary propositional logic", "low-level set theory", "real analysis without approximations", or "computability modulo countable uncomputability".