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The lattice of all sets of natural numbers
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A Panorama of Logic

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.

Joel David Hamkins's avatar
Joel David Hamkins
Jan 24, 2024
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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!

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