Rising Entropy

What is classification?

Classification is one of the most basic human activities. We wake up to a world of vibrant experience and immediately begin structuring it, organizing it into objects and actions, people and animals, edible and non-edible, friend and foe, and so on.

What is definability?

A major theme of descriptive set theory is the following identification, which I’d like to try to motivate: DEFINABLE = BOREL A definition is a syntactic thing. It’s a sentence with a free variable, like “x is the 15th digit in the decimal expansion of π” or “f is the identity function on ℝ”. To precisely state what definability is, we must specify a formal language to work in.

Part 2: How big is 𝒫(ω)?

Now the pieces are all in place to start applying forcing to prove some big results. Everything that follows assumes the existence of a countable transitive model M of ZFC. First, a few notes on terminology. The language of ZFC is very minimalistic. All it has on top of the first-order-logic connectives is a single binary relation symbol ∈.

What is Countable Saturation?

I’ve spent a lot of time in the last posts gushing about how cool and powerful Łoś’s theorem is. But Łoś’s theorem is only the second most powerful property of ultraproducts that I know. The most powerful is what’s known as countable saturation. Countable saturation is a property of structures, and it’s all about definability and parameters.