 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/a/solution

Express existence of a minimum using only the given strict-order relation:

φ = \exists m \forall x \neg (x < m) .

In the closed interval

[0, 1]

, the witness is

m = 0

. In the open interval

(0, 1)

, every possible witness

m

has the smaller member

m /2

, so the first-order sentence is false. These witnesses are explanations in the ambient real numbers, not additional symbols in the first-order signature.

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