Representation of Lindenbaum-Tarski algebras as a field of sets

Question

I don't know if this changes anything, but for simplicity let's say we are only talking about propositional calculus.

Here are some facts that I've gathered when there are finite variables and I'm looking for their equivalent ones with infinite variables. Assume there are only $n$ variables, then:

1. The Lindenbaum-Tarski algebra $L$ is isomorphic to $2^{2^n}$
2. The atoms are of the form $(\lnot) p_1 \wedge (\lnot) p_2 \wedge \dots \wedge(\lnot) p_n$, where $(\lnot$) means that there may or may not be a $\lnot$. I think of this as only one row being true in a truth table.
3. For any formula $\phi \in L$, if the length of any chain from $\phi$ to $0$ is $k$, then it's the join of $k$ atoms. I think of this as only $k$ rows being true in a truth table.
4. The only ultrafilters $U$ are the principal ones. Hence the realizations $L/U \to \bf 2$ must take exactly one of the atoms to $1$. I interpret this as picking one row in a truth table.

I have tried to find some equivalences in the infinite case, but have not been able to do so. Assume there are infinite variables, then:

1. The Lindenbaum-Tarski algebra $L$ is isomorphic to $?$ By Stone's representation theorem there should be some subset of $f(\alpha)$, where $\alpha$ is the set of variables and $f$ is some function.
2. It is atomless and coatomless.
3. $-$? How do I think of the truth tables?
4. The only ultrafilters $U$ are the non-principal ones. How do these look? How do their realizations look as truth tables?

I am sorry these are many questions but I thought they were too related to make a question for each.

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If you know some more basic facts like these, I'd appreciate them for building my intuition.

Answer 1

Let $L(S)$ denote the Lindenbaum-Tarski algebra of propositional calculus with a set $S$ of variables. Note that $L(S)$ is just the free Boolean algebra on the set $S$ (since the axioms of Boolean algebra are equivalent to imposing all propositional tautologies; this is a consequence of Stone's representation theorem). Its Stone space is the Cantor space $\{0,1\}^S$: an ultrafilter on $L(S)$ is just an assignment of a truth value to each variable, i.e. a function $S\to\{0,1\}$. So, elements of $L(S)$ can be thought of as clopen subsets of $\{0,1\}^S$. This is just like the finite case--an element of $\{0,1\}^S$ is a row of an infinite "truth table" where you assign a truth value to each variable and use that to determine the truth value of any proposition. The difference is that when $S$ is finite, $\{0,1\}^S$ is just a finite discrete space, so every set of rows in the truth table can be realized by some proposition, whereas if $S$ is infinite, then $\{0,1\}^S$ is not discrete, and only certain sets of rows are realized by propositions (those which are clopen subsets).

To get a more concrete understanding of what elements of $L(S)$ look like, it is helpful to observe that $L(S)$ is just the direct limit of $L(F)$ where $F$ ranges over all finite subsets of $S$ (and the inclusions $L(F)\to L(S)$ are all injective). So given an element $x\in L(S)$, it comes from $L(F)$ for some finite subset of $S$, and so you can understand it in terms of the atoms of $L(F)$. If you add a new element $s\in S$ to $F$, this just splits each of the atoms in two pieces, one obtained by taking a meet with $s$ and one obtained by taking a meet with $\neg s$. To compute a join or meet of two elements of $L(S)$, you just find an $L(F)$ that contains both of them and compute the join or meet in $L(F)$.

If $S=\{s_1,s_2,\dots\}$ is countable, you can visualize this in terms of an infinite descending binary tree. At the top of the tree is $1$, which then splits into the two atoms $s_1$ and $\neg s_1$ of $L(\{s_1\})$, which then split into the four atoms $s_1\wedge s_2,s_1\wedge\neg s_2,\neg s_1\wedge s_2,\neg s_1\wedge\neg s_2$ of $L(\{s_1,s_2\})$, and so on. Each level of the tree consists of the atoms of $L(\{s_1,\dots,s_n\})$ for some $n$, and each element of $L(S)$ can be written as a join of atoms at some level of the tree (and thus also at each lower level, by splitting the atoms down into the lower level atoms). To compute a join or meet of two elements of $L(S)$, you write both of them in terms of the atoms at some level of the tree and take the union or intersection of those sets of atoms.