 Solution

ID: past-exam-of-the-model-theory-course-of-the-university-of-oxford/2011/1/b/ii/beta/solution

Yes. One is the unique identity element of multiplication, defined by

\forall y (x \cdot y = y) .

The integer one satisfies the first-order formula, and testing at the actual integer one forces any satisfying

x

to equal one. Thus this is a parameter-free universal formula even though the language has no named identity.

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