The lattice of sets of natural numbers is rich
This article dives deep into the intricate world of the power set lattice of natural numbers, revealing its surprisingly rich and complex structure. It demonstrates how this seemingly discrete mathematical object can embed not only countable orders like the integers but also dense orders like the rationals and even uncountable chains isomorphic to the real numbers. Hacker News readers will appreciate this rigorous exploration of abstract mathematical properties, challenging intuitions about infinite sets and order theory.
The Lowdown
The article explores the fascinating properties of the lattice of all sets of natural numbers, P(ω), ordered by the subset relation. It establishes P(ω) as a Boolean algebra, with the empty set and the set of all natural numbers as its extreme elements, and categorizes sets into finite, cofinite, and infinite-coinfinite regions. The core of the discussion revolves around the types of order relations that can be found within this vast structure.
- Embedding Familiar Orders: The author first shows how basic orders like the natural numbers and integers can be easily embedded into P(ω) through specific chains of sets.
- Universality for Countable Orders: A pivotal theorem is introduced, proving that P(ω) is "universal for all countable orders." This means any countable order relation can be found as a suborder within P(ω), a surprising result given the lattice's discrete nature.
- Dense Orders: Counter-intuitively, the article demonstrates that P(ω) can embed dense orders, such as the rational line (Q), despite its discrete 'steps' between elements.
- Uncountable Chains: Even more astonishingly, P(ω) is shown to contain an uncountable chain isomorphic to the real continuum (R). This is achieved by completing the embedding of the rational line.
- Uncountable Antichains: The article further proves the existence of an uncountable antichain within P(ω)—a collection of continuum many pairwise incomparable sets, constructed using paths through a binary tree.
- Homogeneity Properties: P(ω) exhibits remarkable homogeneity. The lattice structure 'above' any cofinite set and 'below' any infinite set is isomorphic to the entire P(ω). Furthermore, any two infinite coinfinite sets are structurally indistinguishable, being automorphic images of one another.
The author concludes that the power set lattice on the natural numbers is a structure of profound complexity and homogeneity. Its true nature is best understood through its overarching structural properties, such as its universality for orders and its various forms of homogeneity, rather than focusing on the characteristics of individual sets within it. This discussion is presented as a selection from the author's upcoming book, "Topics in Logic."