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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 :)

Shaun
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    I don't know what role you want $t$ to have in the topos. You can certainly find a topos whose truth values are a totally ordered set with $n+2$ elements but I don't know what "$m$ steps until truth" is supposed to mean in a topos. – Eric Wofsey Dec 06 '20 at 15:41
  • Here $t$ is intended to play an analogous role to that of $p$ in this answer, @EricWofsey. – Shaun Dec 06 '20 at 15:48
  • What's the first thing about $t$ that is not clear? Have I misled people with the wrong choice of symbol for the set currently denoted by $\Omega$? It seems like you understand, @EricWofsey, what I'm aiming for with your example of a topos with a totally ordered set of $n+2$ truth values. Perhaps you could elaborate on that in an answer. I hope this post is not a silly question. – Shaun Dec 06 '20 at 16:19
  • I've edited the "$m$ steps until truth" to "$m$ steps (through $t$) until truth", @EricWofsey. – Shaun Dec 06 '20 at 17:16
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    As Eric says, you can certainly find a topos whose truth values are a totally ordered set with $n + 2$ elements, and like Eric I have no idea what "$m$ steps until truth" is supposed to mean. – Qiaochu Yuan Dec 06 '20 at 18:33
  • Let me jump on that train: I don't get the role of $t$ and "$m$ steps until truth" either. – Z. A. K. Dec 06 '20 at 18:36
  • I don't understand what the difficulty is, since "sets through time" as described here and here are clear enough, where $t$ in the question corresponds to $p$, @QiaochuYuan. – Shaun Dec 06 '20 at 18:40
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    @Shaun: that question is about a specific topos of functors $\omega \to \mathbb{N}$. Your question is about an arbitrary topos; what is $t$ supposed to be in that context? Do you want the topos to be equipped with an endomorphism? What kind of endomorphism? – Qiaochu Yuan Dec 06 '20 at 18:57
  • Consider the white king in chess, moving forward at each step when it has a clear path from e1 to e8 the other side of the board. Then at first: the truth of "the white king is at a1" is false, so corresponds to $\infty$, or "never true", and the truth of "the white king is at e8" is $7$, so $t$ corresponds to making the white king move. Do you see what I mean, @QiaochuYuan? (I'm sorry if I appear impatient; I really want to understand this.) – Shaun Dec 06 '20 at 19:11
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    @Shaun: I do not understand what would constitute an answer to your question. If I gave you an example of a topos $T$ and a logical functor $F : T \to T$ whose set of truth values $\text{Hom}(1, \Omega)$ is isomorphic to your poset $N$ and such that the induced map $F : \text{Hom}(1, \Omega) \to \text{Hom}(1, \Omega)$ is $t$, or proved that no such pair $(T, F)$ existed, would that answer your question? Or are you asking something else? – Qiaochu Yuan Dec 06 '20 at 19:19
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    Yes, I believe that would answer my question, @QiaochuYuan. I'm sorry about any ambiguity. – Shaun Dec 06 '20 at 19:22
  • (When I say I believe it would answer my question, @QiaochuYuan, I mean that, as far as I can tell, it makes sense in that the words look arranged in an appropriate manner, but, as yet, I do not understand it; I will leave the job of explaining it to you (or whoever wants to answer that way).) – Shaun Dec 06 '20 at 19:36

1 Answers1

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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.

Josh Wrigley
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  • Thank you for this answer. I have a paperback copy of SGL. The exercises are much more challenging to me than those of Topoi: A Categorial Analysis of Logic, at least to start with; much more category theory is assumed in SGL. I have no formal training in category theory, let alone topos theory. – Shaun Jan 09 '21 at 17:22
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    Yes, category theory is necessary. I can also suggest Eugenia Cheng's videos for an intro to category theory: https://www.youtube.com/user/TheCatsters/featured – Josh Wrigley Jan 10 '21 at 11:59