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

* $x=n$, for each $n\in\mathbb N$;
* $x<0$;
* $n<x<n+1$, for each $n\in\mathbb N$.

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>.

Solution

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

{title2=$S_1^{(\mathbb Q,<)}(\mathbb N)$}

Quantifier elimination for dense linear orders shows that the one-types over $\mathbb N\subset\mathbb Q$ are determined by cuts: equality to a natural number, one of the intervals between consecutive natural numbers, the ray below zero, or the cut above every natural number. Only the last type is non-isolated.

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 Incoming links: One-types over the natural numbers in the rational order