ABSTRACT
Returning once again to measure theory, I unearth the root of the failed attempts at solutions. The way that informative measures are developed argues for a distinction between state spaces that have a natural measure of their own and state spaces that don’t. Bertrand’s original four paradoxes fall into two groups on either side of the naturalness distinction, plenitude paradoxes and paucity paradoxes, which distinction allows me to explain why the chord paradox is rightly renownedly known as Bertrand’s paradox, even if not so known for this reason. Plenitude paradoxes have a clear strategy for their solution: to argue that there is a unique, or most, natural measure. The chord paradox, by contrast, has no natural measure, no measure of its own, and this is why no one has solved it. Lacking a measure of its own, although there are various measures related to their properties that have some naturalness that can be used in the crucial theorem to produce candidate normative measures, none is the natural measure to use.
