Exercise 7.66

Exercise 7.66#

In the topos \(\textbf{Set}\), where \(Ξ© = 𝔹\), consider the predicate \(p: β„•Γ—β„€\to 𝔹\) given by

\[\begin{split} \begin{align*} p(n, z) &= \begin{cases} \text{true} & \text{if } n \leq |z| \\ \text{false} & \text{if } n > |z| \end{cases} \end{align*} \end{split}\]

  1. What is the set of \(n \in β„•\) for which the predicate \(\forall(z: β„€). p(n, z)\) holds?

  2. What is the set of \(n \in β„•\) for which the predicate \(\exists(z: β„€). p(n, z)\) holds?

  3. What is the set of \(z \in β„€\) for which the predicate \(\forall(n: β„•). p(n, z)\) holds?

  4. What is the set of \(z \in β„€\) for which the predicate \(\exists(n: β„•). p(n, z)\) holds?

Author’s solution#

We have the predicate \(𝑝 : β„• Γ— β„€ β†’ 𝔹\) given by \(𝑝(𝑛, 𝑧)\) iff \(𝑛 ≀ |𝑧|\).

  1. The predicate \(βˆ€(𝑧 : β„€). 𝑝(𝑛, 𝑧)\) holds for \(\{0\} βŠ† β„•\).

  2. The predicate \(βˆƒ(𝑧 : β„€). 𝑝(𝑛, 𝑧)\) holds for \(β„• βŠ† β„•\).

  3. The predicate \(βˆ€(𝑛 : β„•). 𝑝(𝑛, 𝑧)\) holds for \(βˆ… βŠ† β„€\).

  4. The predicate \(βˆƒ(𝑛 : β„•). 𝑝(𝑛, 𝑧)\) holds for \(β„€ βŠ† β„€\).

Alternative answer#

  1. {0}

  2. β„•

  3. {∞} (but see Does the set of natural numbers contain infinity? - MSE)

  4. β„€