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Source. Ramsey (1930), Part II, printed p. 276 (PDF, physical p. 13).

Use the forms and system $P$ from the preceding reduction. Let $N$ be an integer with $1\leq N\leq n$.

The universal sentence has a model on exactly $N$ elements if and only if $P$ contains an $N$-form $A_N$ together with every smaller form involved in $A_N$.

Necessity

Suppose a model has universe $U=\{u_1,\ldots,u_N\}$. Evaluate every relation atom on this ordered list. The resulting complete canonical alternative lies in one $N$-form $A_N$. Since the universal sentence holds, the $N$-row of $P$ must contain that form.

For every subset of $\mu<N$ elements, restrict the same truth assignment to the atoms involving only that subset and relabel its elements. This realizes one of the $\mu$-forms involved in $A_N$. Every possible choice and ordering of the subset occurs among the assignments quantified by the sentence, so each such involved form must occur in the $\mu$-row of $P$. Thus $A_N$ is completely contained through all smaller sizes.

Sufficiency

Conversely, suppose $P$ contains $A_N$ and every form involved in it. Choose one complete alternative $q\in A_N$ and label an $N$-element universe by $y_1,\ldots,y_N$. Define each relation on every tuple of these elements by the truth value assigned to the corresponding atom in $q$. Equality consistency of $q$ makes this a well-defined interpretation. Its nullary literals, if any, give the shared truth values of the propositional constants.

The original complete alternative listed every relation atom formed from the $x_i$. Since each of the $N$ equality classes contains an $x_i$, it therefore specifies every relation tuple on this $N$-element universe, including tuples with repeated entries.

Consider any assignment of $x_1,\ldots,x_n$ in this structure. It uses some $\mu\leq N$ distinct elements. After the equality reduction, its complete truth alternative is obtained by restricting $q$ to those elements and then permuting their labels. If $\mu=N$, this is a representative of $A_N$ itself; if $\mu<N$, it belongs to a $\mu$-form involved in $A_N$. In either case the form occurs in $P$. Hence $F$ is true under every assignment, so the constructed structure is a model.

The condition is exact for each prescribed nonempty cardinality $N\leq n$. Empty universes are outside the source's convention.