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 Incoming links: First-order definition of exactly finitely many witnesses

Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof resit / 8 / a / Solution 2026-10-25

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Using first-order logic with equality, one suitable sentence is

\exists e (Eq (e) \land \exists u \exists v [u \neq = v \land S o l (e, u) \land S o l (e, v) \land \forall w (S o l (e, w) \to (w = u \lor w = v))]) .

(1)

The distinct witnesses give at least two solutions, and the universal clause excludes every third solution. All variables are bound, and the same equation variable

e

is used in every solution predicate. This is an instance of a first-order definition of exactly finitely many witnesses.

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