This question is based on the answers to this question.
The Question:
Let $n\in\Bbb N$. Let $N$ be a set with $n+2$ elements, labelled $0$ to $n$, and the $(n+2)$th element labelled $\infty$. Suppose we have a function
$$\begin{align} t:N &\to N,\\ 0 &\mapsto 0,\\ m &\mapsto m-1,\quad\text{(for } m\in \overline{1,n}\text{)}\\ \infty &\mapsto \infty. \end{align}$$
Does there exist a topos whose internal logic corresponds to $\infty$ being "false", $0$ being "true", and each $m\in \overline{1,n}$ being "$m$ steps (through $t$) until truth"?
Motivation:
An answer to this question, hopefully, will kill two birds with one stone: objections to: (1) systems of logic with more than two truth-values and (2) whether infinity has a rigorous place in mathematics.
Thoughts:
It is my belief that such a topos can be created; however, I don't know how.
I'm aware that $(N, t)$ is a dynamical system. I don't have much experience with them.
For an idea of my abilities in topos theory, see this: The legitimacy of topos theory and intuitionism.
Please help :)
I recommend you read SGL by MacLane and Moerdijk (available here) since I'm going to be quoting results from there and generally it's a great introduction to Topos theory.
In particular, in I.4 we are given the following characterisation for the subobject classifier of a presheaf topos ${\bf Sets}^{\mathcal{C}^{op}}$: $$ \Omega\colon \mathcal{C}^{op} \to {\bf Sets}, \ \ \Omega(C) = \{\,S \mid S \text{ is a sieve on $C$}\,\}.$$ In the text, they offer one motivating description of $\Omega(C)$ as "the set of "paths to truth"". The map ${\rm true} \colon 1 \to \Omega$ has as components $\ast \mapsto t_C$ (where $t_C$ is the maximal sieve of all arrows to $C$).
We might want to restrict to when $\mathcal{C}$ is a preorder category, in which case sieves correspond to down-sets in the order. If $\alpha$ is an ordinal, the nonempty down-sets of $\alpha^{op}$ are in bijection with the elements of $\alpha$. In some sense these are the "truth values". For a subfunctor $Q \rightarrowtail P$ in ${\bf Sets}^{\alpha^{op}}$, the characteristic map $\varphi^Q \colon P \to \Omega$ has components $\varphi^Q_\beta(x) = \gamma$, the least ordinal $\gamma \in \alpha$ such that $x\cdot f \in Q(\gamma)$ (where $f$ is the map induced by $\beta \leqslant \gamma$), or $\varphi^Q_\beta(x) = \bot$ should such a $\gamma$ not exist.
It's not hard to see that, taking $\alpha = n$, we get the topos you seek.