The Lattice of Sets of Natural Numbers Is Rich

Explore the power set lattice of natural numbers, a Boolean algebra where every possible set of natural numbers exists. This structure is universal for all countable orders, meaning it contains a copy of every countable order, including the rationals. Surprisingly, it also embeds the real continuum and contains an uncountable antichain. Moreover, the lattice exhibits homogeneity: above any cofinite set or below any infinite set, it looks the same, and there are automorphisms mapping any infinite coinfinite set to any other.