# 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!

## Keep reading with a 7-day free trial

Subscribe to Infinitely More to keep reading this post and get 7 days of free access to the full post archives.