The lattice of all sets of natural numbers

Let us explore the vast and densely populated expanses of the lattice of all sets of natural numbers—the power set lattice ⟨P(ℕ),⊆⟩—seeking monsters and wild suborders inhabiting this land.

Let us explore the vast and densely populated expanses of the lattice of all sets of natural numbers—the power set lattice ⟨P(ℕ), ⊆ ⟩, depicted below. We shall find there every possible set of natural numbers A ⊆ ℕ, ordered amongst each other by the subset relation A ⊆ B. We shall seek out monsters and wild animals inhabiting this land, crazy uncountable suborders of the power-set lattice, including copies of the real order ℝ stretched up through the lattice.

Let us explore!