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I can only answer some of your questions. Yes, the Zariski locale is extensively studied. It's one of the ways of setting up scheme theory in a constructive context: Don't define schemes as locally r …

Recall the well-known proof that a unique factorization domain is a GCD domain: Let $x, y \in R \setminus \{ 0 \}$. Factor $x$ and $y$ into pairwise non-associated irreducible elements: $$\begin{a …

I'm not precisely sure what you're looking for. Here is an example for the external interpretation of an apartness relation: Recall that the object of Dedekind reals $\mathbb{R}$ in a sheaf topos $\m …

This answer provides a scheme how to construct a constructive proof, though I'm still working to actually explicitly extract the constructive proof, so please don't accept the answer just yet. (Update …

The well-known reflection principle for classical Zermelo–Fraenkel states: For any formula $\varphi(x_1,\ldots,x_n)$ of the language of ZFC with free variables $x_1,\ldots,x_n$, ZFC proves $$ \f …

Recently I got interested in predicative foundations, mostly because of Laura Crosilla's work and because Agda employs a predicative type theory. From the point of view of a predicative foundation to …

Yes! Here is a proof which is slightly different from both your proof and the proof in Mines–Richman–Ruitenberg. First define the similarity relation on $\mathrm{List}(R \times X)$ as in Mines–Richma …

The spectrum of a commutative ring, defined as the classifying locale of its prime filters, is overt if and only if any element is nilpotent or not nilpotent (Proposition 12.51 in these notes of mine) …

Any subclass $\mathcal{C}$ of an abelian category determines a smallest Serre class containing it, by iteratively adding (the zero object and) the object $Y$ for any exact sequence $X \to Y \to Z$ whe …

This is not an answer to the question, but rather a comment which benefits from proper formatting. For me personally, Markov's principle is specifically associated with the natural numbers and not wi …

Last week, I learned from Ulrik Buchholtz that there are proof-theoretic reductions from certain impredicative systems to certain predicative ones. While these fall short of predicatively explaining t …