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I encountered a post on locales and geometric theory here https://grossack.site/2022/05/22/locale-basics.html

In about the middle of this blog, the author gives a geometric theory defining a function from $\Bbb N$ to $\Bbb R$, and asked: Do you see why this is a nontrivial theory with no classical models?

I am not confident that I understand it. It seems like the third clause given there indicates there is a surjection to the real numbers. But we actually do not have a surjection. So where can we obtain a model of such a theory, and how does it imply that there is no frame morphism from $L[{\Bbb T}]$ to $2$?

Moreover, is there any more typical example of geometric theory without classical model?

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Indeed, the third axiom says that $f\colon\mathbb{N}\to\mathbb{R}$ is surjective. Therefore, a classical model for this theory is an actual function $f\colon\mathbb{N}\to\mathbb{R}$ of sets that is surjective, which doesn't exist, so there is no classical model. However, a frame morphism $\mathcal{L}[\mathbb{T}]\to\{\top,\bot\}$ is exactly the same data as such a classical model (it assigns to each geometric formula a truth value, and this assignment respects the usual logical operations in the expected way), so such a frame morphism does not exist either.

You can use this idea to come up with more geometric theories without classical models: just come up with something basic that cannot happen in set theory but is not an ''obvious logical contradiction'' (as Alex Kruckman pointed out below, you want to deal with infinite sets in a way to get a non-trivial theory). Axiomatize the situation, and in good cases you have a geometric theory, which automatically cannot have classical models.

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  • $\begingroup$ Thanks a lot for such a swift answer! It is not obvious to me if there is anything useful to say about such theories. What does it indicate when the theory, say the theory in the post, has a model in some frame? Is there any interesting properties arise from here? $\endgroup$
    – Y.X.
    Apr 24 at 23:11
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    $\begingroup$ I think your finitary examples will result in trivial geometric theories, in which $\top\vdash\bot$. The important thing about infinitary examples like the one in the question is that they generate non-trivial frames, and hence have models in non-trivial toposes (and @Y.X. one thing you can do with this is forcing!) $\endgroup$ Apr 24 at 23:49
  • $\begingroup$ @AlexKruckman Thanks for pointing that out! $\endgroup$ Apr 25 at 0:01
  • $\begingroup$ @AlexKruckman Ah, thanks! I have not systematically learnt about forcing, but I remembered in one step you want a map to a nontrivial Boolean algebra. I would appreciate in the case that you might take another few lines to elaborate a bit on this. $\endgroup$
    – Y.X.
    Apr 25 at 1:00
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    $\begingroup$ @Y.X. In very short, by various topos-theoretic constructions one can pass from the classifying topos of a geometric theory, which contains by definition a model of that theory, to a topos very similar to that of sets (for instance, Boolean and two-valued, and in many cases also satisfying the axiom of choice) containing a model of the same theory--such as a surjection $\mathbb N\to\mathbb R.$ For details I'd look at Chapter VI of Mac Lane-Moerdijk's book Sheaves in Geometry and Logic. $\endgroup$ Apr 25 at 4:20

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