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When studying Computational Logic last year, we had to prove a theorem called "Compactness Theorem" ("Teorema de Compacidad in Spanish) which states that a set of formulae $\Gamma$ is satisfiable if and only if every finite subset of $\Gamma$ is satisfiable

I am now going over Rudin's Principles of Mathematical Analysis to "master the basics" and came upon a theorem that states that for a collection of compact sets $\lbrace K_\alpha \rbrace$ if the intersection of every finite subcollection is nonempty, then the intersection of the whole collection is nonempty.

These two theorems seem to state extremely similar things, so I was wondering how they're connected

Arturo Magidin
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Francisco José Letterio
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1 Answers1

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The "compactness" theorem in mathematical logic alludes to the "compactness" of the topological space $S(B(\text{Sent}))$ generated by the boolean algebra $B(\text{Sent})$ of the set of sentences $\text{Sent}$. More specifically:

  1. The Boolean Algebra $B(\text{Sent})$ is the set of logical equivalence classes of $\text{Sent}$, with the relation $\leq$ defined by $$\varphi \leq \psi \iff \models \varphi \rightarrow \psi$$.

  2. We define $$S(B(\text{Sent}))=\{U \subset B(\text{Sent}) \colon \text{ $U$ is an ultrafilter with respect to $\leq$} \},$$ with the topology generated by the sets $$[\varphi]=\{U \in S(B(\text{Sent})) \colon \varphi \in U \}.$$

NOTE: Slight abuse of notation to account for when I write "$\varphi$" since what I really mean is the logic equivalence class since ultrafilters here are sets of logic equivalence classes.

An equivalent statement of the "compactness theorem" in logic is the statement "$S(B(\text{Sent}))$ is compact".

There is an equivalent definition of compactness--that is of similar flavor to the theorem you mention in baby Rudin--stating that a topology $K$ is compact if a collection $\{C_\alpha\}_{\alpha \in J}$ of closed sets such that $C_{\alpha_1} \cap \cdots \cap C_{\alpha_n}$, for any $\alpha_1, \ldots, \alpha_n \in J$ is nonempty implies $\bigcap_{\alpha \in J} C_\alpha$ is nonempty.

This definition of compactness is useful to see that $S(B(\text{Sent}))$ is compact. Since what this says is for any theory $\Gamma$, we have $$\forall \Sigma \subset_{\text{fin}} \Gamma \left( \bigcap_{\varphi \in \Sigma} [\varphi] \neq \emptyset \right) \implies \bigcap_{\varphi \in \Gamma} [\varphi] \neq \emptyset,$$ and it's pretty straightforward (once you know the necessary definitions) to see that $\Gamma$ is satisfiable if and only if $\bigcap_{\varphi \in \Gamma} [\varphi] \neq \emptyset$ (i.e., there exists an ultrafilter $U$ that contains all of $\Gamma$, up to logical equivalence)

This is more or less the idea, but I know that there's a lot of terminology to unpack so I will edit the answer later if anything I said needs clarity.

Dark Logician
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