 Solution

ID: 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/8/b/solution

Express that the order is a dense order:

ψ = \forall x \forall y (x < y \to \exists z (x < z \land z < y)) .

In either real interval,

(x + y) /2

lies strictly between any two distinct ordered elements and remains in the domain. In the integers,

x = 0, y = 1

have no intervening integer, so this first-order sentence is false. The endpoints of a closed interval do not prevent a dense order.

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