State the completeness theorem for propositional logic. Explain briefly how the proof of this theorem changes from the usual proof in the case when the set of primitive propositions may be uncountable.

State the compactness theorem and the decidability theorem, and deduce them from the completeness theorem.

A poset $(X,<)$ is called two-dimensional if there exist total orders $<_{1}$ and $<_{2}$ on $X$ such that $x<y$ if and only if $x<_{1} y$ and $x<_{2} y$. By applying the compactness theorem for propositional logic, show that if every finite subset of a poset is two-dimensional then so is the poset itself.

[Hint: Take primitive propositions $p_{x, y}$ and $q_{x, y}$, for each distinct $x, y \in X$, with the intended interpretation that $p_{x, y}$ is true if and only if $x<_{1} y$ and $q_{x, y}$ is true if and only if $x<2 y .]$