= Solution
\b[No such set of propositions exists.] Suppose that $S$ had precisely the total injections as its models, fix $x_0\in X$, and adjoin
$$
\neg p_{x_0,y}\qquad(y\in Y)
$$
to $S$. Every finite subset of the enlarged theory has a model. Indeed, it excludes only finitely many possible values $F\subset Y$ for $x_0$. Starting with any injection $h:X\to Y$, either $h(x_0)\notin F$, or one may choose $x_1\ne x_0$ with $h(x_1)\notin F$ and swap the values of $x_0$ and $x_1$.
The <propositional compactness theorem> would therefore give a model of the whole enlarged theory. Its relation $f_v$ is a model of $S$ but has no value at $x_0$, contradicting the assumed description of the models of $S$. This is the <compactness obstruction to expressing totality over an infinite codomain>.
Back to article page
