In Brainiac: Science Abuse, there is a wonderful category called How Hard is Your Thing? (see [1]). Thaila Zucchi makes seemingly hard things undergo some of the ultimate stress tests: Can they stand the heat of thermite? The abrasion of an angle grinder? The impact of a ton of bricks?

There were several occasions in the history of mathematics when mathematicians had to answer to the question: how hard is your maths?

« Oh, it looks pretty hard but just how hard is it? »

Here are three examples of stress tests that have led to foundational crises:

Pouring the thermite of Russell’s paradox on it: the set of all sets which are not members of themselves, does it contain itself?

Grinding it with Gödel’s first incompleteness theorem: a logical theory sufficiently expressive to describe the arithmetic of the natural numbers is either inconsistent or incomplete.

Throwing the continuum on it: the question whether there is a set whose cardinality is strictly between that of the natural numbers and that of the real numbers is independent of Zermelo–Fraenkel set theory.

Intrigued? Want some more? Go get a quick overview at Wikipedia’s page on the foundations of mathematics.