Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 16J (Logic & Set Theory)

A class is axiomatisable if it is exactly the class of models of some set of first-order sentences in the given language.

Compactness theorem says a set of first-order sentences has a model whenever each finite subset has a model.

If a theory axiomatized precisely the Archimedean fields, add a constant $c$ and sentences $t_n<c$ for every $n$. Every finite subset has an Archimedean model with $c$ sufficiently large, so compactness gives a model of the whole set. Its field reduct satisfies the proposed theory but is non-Archimedean, a contradiction.

Atomic one-variable formulas reduce to polynomial equalities and inequalities after clearing terms. Their solution sets are basic by the supplied fact. Basic sets are closed under finite union, intersection, and complement; structural induction through the Boolean connectives therefore proves the claim.

Quantifier elimination makes $S=\{x:\exists y(y^2=x)\}$ basic. It contains every square and no negative element. If some $a>0$ were not a square, $S$ would have a boundary inside $(0,\infty )$; but multiplication by positive squares and the ordered-field inequalities force membership to be locally constant along positive multiplicative intervals, contradicting the finite-endpoint form of a basic set. Hence every positive element, and also zero, is a square.