TITLE: Proving Tychonoff's theorem with the Compactness theorem of logic
QUESTION [6 upvotes]: It seems to be known that Tychonoff's Theorem for Hausdorff spaces and the Compactness theorem of first order logic are both equivalent over ZF to the ultrafilter lemma. Does anyone know a slick proof for the implication "Compactness Theorem $\rightarrow$ Tychonoff for Hausdorff spaces" (without using the ultrafilter lemma as an intermediate step)?

REPLY [6 votes]: Suppose that we have a family of compact Hausdorff space $(X_i, \tau_i)$ where $\tau_i$ is the topology and $i$ ranges over an index set $I$. Assume that the product $X = \prod_i X_i$ is not compact, so there is an open cover $(W_\alpha: \alpha < \kappa)$ of $X$ which does not have a finite subcover. We define a language and a consistent theory and then apply the compactness theorem to get a point in $X - \bigcup_\alpha W_\alpha$.
W.l.o.g, assume that each $W_\alpha = \prod_i W_{\alpha,i}$ for $W_{\alpha,i} \in \tau_i$ and $spt(W_\alpha) = \{i: W_{\alpha,i} \neq X_i\}$ is finite. 
The language consists of

constants $c_i$ for $i \in I$;
for each $i$ and each $U \in \tau_i$, a unary predicate $P_{i,U}$.

The intended meaning of $(c_i: i \in I)$ is a (non-standard) point in $X - \bigcup_\alpha W_\alpha$; and $P_{i,U}(x)$ means $x \in U$.
The theory $T$ consists of

$P_{i,X_i}(c_i)$;
for each $\alpha$, $\bigvee_{i \in spt(W_\alpha)} \neg P_{i,W_{\alpha,i}}(c_i)$ (i.e., $(c_i: i \in I) \not\in W_\alpha$);
for each finite $F \subset \tau_i$, if $\bigcup_{U \in F} U = X_i$ then the sentence $(\forall x) (P_{i,X_i}(x) \to \bigvee_{U \in F} P_{i,U}(x))$ is in $T$;
if $C = X_i - U$ is closed and $U_0, U_1 \in \tau_i$ are s.t. $U_i \cap C \neq \emptyset$ and $U_0 \cap U_1 \cap C = \emptyset$, then the sentence $\neg P_{i,U_0}(c_i) \vee \neg P_{i,U_1}(c_i)$ is in $T$.

By the assumption that $W_\alpha$'s do not have a finite subcover, $T$ is consistent. So take a model $M$ of $T$, let $\mathcal{C}_i$ be the set of $X_i - U$ s.t. $U \in \tau_i$ and $M \models \neg P_{i,U}(c_i)$. Then $\mathcal{C}_i$ has finite intersection property and by compactness $\bigcap \mathcal{C}_i$ is non-empty. By the last set of sentences in $T$, $\bigcap \mathcal{C}_i$ is a singleton $\{p_i\}$ (here we need Hausdorff-ness). The point $(p_i: i \in I)$ is not covered by any $W_\alpha$.