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>.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-144/3/c/i/solution