Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 144 / 3

Use the finite-partial-isomorphism criterion for quantifier elimination. Let $M,N\models \mathrm{DLO}$ and let $h:A\to B$ be an isomorphism between finite suborders. For $a\in M\setminus A$, its position relative to $A$ is one of the finitely many open intervals determined by $A$, or one of the two exterior rays. The corresponding interval or ray determined by $B$ is nonempty because the orders are dense and have no endpoints. Choose $b$ there. Then $h\cup \{(a,b)\}$ remains a partial order isomorphism.

The same argument extends in the other direction. The back-and-forth method criterion therefore applies, proving quantifier elimination for dense linear orders without endpoints. Hence DLO eliminates quantifiers.

Solved by gpt-5.6-sol high.

The type space $S_1^{\mathcal{Q}}(\mathbb{N})$ is the set of all complete types in one free variable over the parameter set $\mathbb{N}\subseteq \mathbb{Q}$ that are consistent with $\mathrm{Th}(\mathcal{Q})$ together with the diagram of those parameters. Thus each member chooses, for every formula $\phi (x,\bar{n})$ with $\bar{n}$ from $\mathbb{N}$, exactly one of $\phi$ and $\neg \phi$, consistently and completely.

Solved by gpt-5.6-sol high.

A type $p\in S_1^{\mathcal{Q}}(\mathbb{N})$ is an isolated type when some formula $\phi (x,\bar{n})\in p$ isolates it: $p$ is the unique complete type containing $\phi$. Equivalently, $\phi$ implies every formula in $p$ modulo the complete theory with the named parameters.

Solved by gpt-5.6-sol high.

By quantifier elimination, a one-type is determined entirely by the position of $x$ relative to the natural-number parameters. Assuming $\mathbb{N}=\{0,1,2,\dots \}$, the isolated types and isolating formulas are:

Each formula fixes every comparison of $x$ with every natural number, so it determines a complete type. These are precisely the isolated members of the one-types over the natural numbers in the rational order.

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There is exactly one non-isolated type:It is consistent by the compactness theorem, since every finite subset is realized by a sufficiently large rational number. It is complete by quantifier elimination, because it decides every comparison with a parameter from $\mathbb{N}$.

No formula isolates it. Any formula belongs to $p_{\infty }$ only through finitely many natural-number parameters; after quantifier elimination it holds throughout some final ray. It is consequently also satisfied by a sufficiently large natural number, whose equality type differs from $p_{\infty }$. Thus $p_{\infty }$ is non-isolated, and the list in part i exhausts all other cuts of $\mathbb{N}$.

Solved by gpt-5.6-sol high.