 Solution

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

Yes. Zero is the unique absorbing element of the multiplicative first-order structure. It is defined by

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

Zero satisfies this universal formula. Conversely, substituting the actual integer zero for

y

in the semantic verification forces

x = 0

. This use of zero in the verification does not insert a constant symbol into the first-order formula.

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