compactness theorem confusion

Trying to think about a problem, I recently stumbled into this pit,
which I can't seem to get out of.

Suppose we have a FOL with |N| variables x_1,x_2,... and one unary
predicate P which isn't constantly true or constantly false. Let S be
the set of wffs
S = { exists x_1 P(x_1) } union { not P(x_i) | i = 1,2,3,... }

If S_0 is a finite subset of S, there's a j so big that "not P(x_j)" is
not in S_0, so we can choose an assignment such that P(x_j) is true but
such that P(x_k) is false for each k<>j. This assignment satisfies
S_0.

By the compactness theorem, S should be satisfiable. But that's
obviously absurd if the language has no function symbols.

.

Relevant Pages

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