Show there is a term t and formula $\theta$ such that $\mathbb{N} \models \forall x<t \theta(x)$ and $\mathbb{N} \models \forall x<t \lnot \theta(x)$

Question

I've been reading Richard Kaye's book, Models of Peano Arithmetic, and I saw this question as an exercise.

Let $\mathbb{N}$ be the standard model of the natural numbers. Give a closed term $t$ and a formula $\theta$, both in the language of arithmetic $\mathcal{L}_A$, where $\mathbb{N} \models \forall x<t \,\, \theta(x)$ and $\mathbb{N} \models \forall x<t \,\, \lnot \theta(x)$.

I've thought about this for a while now and am quite stuck on conjuring an example, even though it should be simple. A helpful nudge would be quite appreciated!

Answer 1

Equivalently, we want $\mathbb{N} \models \forall x < t (\theta(x) \land \neg \theta(x))$. Note that $\forall x < t (\theta(x) \lor \neg \theta(x))$ is logically equivalent to $\neg \exists x (x < t)$. In other words, $t$ would need to be the smallest element of $\mathbb{N}$, which is zero.

So we can define $t = 0$. Once we do this, we see any possible $\theta$ works.