The lattice of sets of natural numbers is rich | Joel David Hamkins

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We find in this lattice every possible set of natural numbers $A\of\N$, and we consider them with respect to the subset relation $A\of B$. This is a lattice order because the union of two sets $A\union B$ is their least upper bound with respect to inclusion and the intersection $A\intersect B$ is their greatest lower bound. Indeed, it is a distributive lattice in light of $A\intersect(B\union C)=(A\intersect B)\union(A\intersect C)$, but furthermore, as a power set algebra it is a Boolean algebra and every Boolean algebra is a distributive lattice.

The empty set $\emptyset$ is the global least element at the bottom of the lattice, of course, and the whole set $\N$ is the greatest element, at the top. The singletons $\set{n}$ are atoms, one step up from $\emptyset$, while their complements $\N\setminus\set{n}$ are coatoms, one step down from $\N$. One generally conceives of the finite sets as clustering near the Earthly bottom of the lattice, finitely close to $\emptyset$, whereas the cofinite sets soar above, finitely close to the heavenly top. In the vast central regions between, in contrast, we find the infinite-coinfinite sets, including many deeply interesting sets such as the prime numbers, the squares, the square-free numbers, the composites, and uncountably many more.

Question. Which familiar orders can we find as suborders in the power set lattice $P(\N)$ of all sets of natural numbers?

We can easily find a copy of the natural-number order $\$ in the power set lattice $P(\N)$, simply by considering the chain of initial segments:

$$\emptyset\of\set{0}\of\set{0,1}\of\set{0,1,2}\of\cdots$$

This sequence climbs like a ladder up the left side of the main figure. Curious readers will notice that although every one of these sets is finite and hence conceived as clinging to Earth, as we had said, nevertheless there is no upper bound for this sequence in the lattice other than the global maximum $\N$—any particular natural number is eventually subsumed. Thus one climbs to the heavens on this ladder, which stretches to the very top of the lattice, while remaining close to Earth all the while on every rung.

The complements of those sets similarly form a infinite descending sequence:

$$\cdots\ \of\ \N\setminus\set{0,1,2}\ \of\ \N\setminus\set{0,1}\ \of\ \N\setminus\set{0}\ \of\ \N$$

This chain of cofinite sets dangles like a knotted rope from the heavens down the right side of the main figure. Every knot on this rope is a cofinite set, which we conceived as very near the top, and yet the rope dangles temptingly down to Earth, for there is no lower bound except $\emptyset$.

Can we find a copy of the discrete integer order $\$ in the power set lattice?

Yes, we can. An example is already visible in the red chain at the right side of main lattice diagram, a sky crane hanging in space. Standing at the set of odd numbers, we can ascend in discrete steps by adding additional even numbers one at a time; or we can descend similarly by removing successive odd numbers. The result is a discrete chain isomorphic to the integer order $\Z$, and it is unbounded except by the extreme points $\emptyset$ and $\N$.

Can we find a dense order, say, a copy of the rational line $\$?

What? You want to find a dense order in the lattice of sets of natural numbers? That might seem impossible, if one thinks of the lattice as having a fundamentally discrete nature. The nearest neighbors of any set, after all, are obtained by adding or removing exactly one element—doing so makes a discrete step up or down with no strictly intermediate set. Does the essentially discrete nature of the lattice suggest that we cannot embed the dense order $\$?

No! The surprising fact is that indeed we can embed the rational line into the power set lattice $P(\N)$. Indeed, much more is true—it turns out that we can embed a copy of any countable order whatsoever into the lattice $P(\N)$. Every countable order that occurs anywhere at all occurs in the lattice of sets of natural numbers.

Theorem. The power set lattice on the natural numbers $P(\N)$ is universal for all countable orders. That is, every order relation on a countable domain is isomorphic to a suborder of $\$.

Proof. Let us prove first that every order relation $\$ is isomorphic to the subset relation $\of$ on a collection of sets. Namely, for each $q\in L$ let $q{\downarrow}$ be the down set of $q$, the set of predecessors of $q$:

$$q{\downarrow}=\set{p\in L\mid p\preccurlyeq q}\of L$$

This is a subset of $L$, and what I claim is that

$$p\preccurlyeq q\quad\text{ if and only if }\quad p{\downarrow}\of...