Let \(Prop^\mathbb{N}\) be the set of all mathematical statements one can make about a natural number.
Examples:
n is a prime
n = 2
\(n \geq 11\)
We say \(P \leq Q\) if for all \(n \in \mathbb{N}\), \(P(n) \implies Q(n)\)
Define a monoidal unit and product on \((Prop^\mathbb{N}, \leq)\).
Let the unit be \(\lambda x. true\) and product be \(\land\)
montonicity: \(P_1(x)\leq Q_1(x) \land P_2(x) \leq Q_2(x) \implies (P_1 \land P_2)(x) \leq (Q_1 \land Q_2)(x)\)
If the \(P\) properties hold for a given number, then each of the \(Q\) properties hold
unitality, associativity, symmetry: same as \(\mathbf{Bool}\)