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Here's a proof in constructive set theory (probably just a rephrasing of the topos theoretic proof but you might find it useful). Let $h : A \twoheadrightarrow B$ be an epimorphism. Define $$ C := \{ …

For reference, here is a more detailed version of Riehl's argument. Definition 1 (Garner, Understanding the small object argument, Proposition 3.8) Let $J$ be a category and $D \colon J \to \mathcal{ …

Here is an alternative proof based on Russell's paradox rather than cardinality that doesn't require sets cover, although I do need to assume that hom sets are 0 truncated. The rough outline is to mod …

I'm not sure if this example counts as natural, but one possibility is to adjust the definition of category of assemblies to use only finite existence predicates. An assembly is a set $X$, together w …

I don't know much about class forcing, so I'll just sketch a Frankel Mostowski style model that I think gives a model of $\mathbf{ZFA}$ where local smallness fails. We take the domain to be the discr …

The way it is usually presented, certainly yes. As you point out it usually refers to possibly uncountable regular ordinals, which would usually mean von Neumann ordinals. Once you have the regular or …

There are probably easier ways to see this, but my favourite example is to look at the filtered colimit over all finite coproducts of $1$ with inclusions. We can also view this as a countable sequence …

I don't know if this is what the authors of the paper had in mind, but here's one way to do it. Also, I'm not sure if there's a constructive way to do this, so I'll give an answer assuming the axiom o …

Following Mike's suggestion, I post my comment as an answer. Brouwer's theorem that all functions $\mathbb{R} \rightarrow \mathbb{R}$ are continuous holds in the effective topos. For example, this ap …

I think there are enough representable $\Sigma$-universes in any regular locally cartesian closed category with disjoint coproducts and $W$-types. One can show the category of sets has $W$-types in $\ …

I don't have a complete answer, but here's a couple of related observations. If $T$ is the algebraically free monad on an endofunctor $P$ in a category with binary coproducts, then for all objects $X$ …

Regarding the bullet point list of questions at the end, I believe that every true $\Pi^0_1$-sentence holds in the internal logic of any Grothendieck topos, so in that case $T_{\mathbf{GrTop}}$ is def …

As pointed out by Noah and Hanul in the comments, there are known models of set theory making use of class sized pcas. The earliest relevant to this question is probably Tharp, A quasi-intuitionistic …