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

By quantifier elimination, a one-type is determined entirely by the position of

x

relative to the natural-number parameters. Assuming

N = {0, 1, 2, \dots}

, the isolated types and isolating formulas are:

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

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.

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