Question

David Lewis was one of the great philosophers of our time. He was a genuine philosopher, his focus was on theoretical metaphysics. And he had something to say about mathematics. His last book - he only wrote four - was Parts of Classes (1991). It had a strong set theoretical - and as you may guess mereological - impact.

I wonder if there is a thorough mathematical analysis of - some of - Lewis' concepts and arguments, subsequently or unknowingly (see Michael's answer below).

I am especially interested in his concept of natural properties which he introduced in New Work for a Theory of Universals (1983), and whether and how this concept might be applicable to mathematics.

Very brief summary (just a teaser): To each set corresponds a property. Such properties are abundant. To the definable sets correspond a restricted but still very large family of properties. Natural properties in turn are very sparse, play a prominent role (at least in Lewis' metaphysics), have to be grasped somehow intuitively, and it's not clear, how they and their corresponding sets can be characterized.

PS: I found an announcement of a talk on Natural Properties in Mathematics. Does anyone have a transcript of this talk or something like that?

PS 2: Here is an older question of mine asking for natural properties in arithmetics.

Answer 1

Lewis introduced the concept of common knowledge (I know that you know that I know that you know that...) in his book Convention. The concept has been formalized in a partitional model of knowledge in the simple and elegant paper Agreeing to Disagree by Robert Aumann. Aumann wasn't aware of the prior work of Lewis. The concept of common knowledge became one of the building blocks of modern non-cooperative game theory and has been extensively studied and generalized. A survey can be found here.