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Let me wrote a quick introduction to this idea: 1) Locales I do not know if you are already familiar with the notion of locale that Andrej is referring to in his talk: They are a small variation on th …

Yes there is: the formal locale of p-adic integer is simply defined as the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as a pro-finite locale). So internally in any topos a continuous function …

A first big difference between Brauer & Hansen's result and the one you are talking about is that CZF is a predicative theory (it doesn't have power set/power object) so consistency with CZF doesn't …

As the title said, I would like to know if constructive measure theory has been developed somewhere ? I am more precisely interested in the (constructive) theory of completely continuous valuation on …

No the problem isn't quite choosing an element from an unordered pair, even if I agree with you that it somehow feel like it is. The map you are talking about is indeed always an epimorphism. One way …

I'm looking for references presenting a constructive treatment of the theory of ordinals. By constructive I mean valid in the internal logic of a topos (so no axiom of choice and no law of excluded mi …

Classically, if a locally small category $C$ has all limits of shape $K$ (for some small diagram $K$), then its free co-completion also has $K$-shapped limits. But all proof I know of that result reli …

For this type of question the first reference that comes to my mind is P.T.Johnstone Sketches of an elephant, part C. Most of the results in this book are constructively valid: If a result is proved …

Here is a very brief sketches of the connection between this and forcing. I'll describe you how I understand forcing, this is quite different from how it is generally described by logician, but this h …

There is on the nLab page "Grothendieck topos" a part about the theory of Grothendieck toposes in weak foundation. It claims that the equivalence for a category between the Giraud's axioms and being …

I have thought about this recently, and here is I think the best constructively valid statement one can extract from Brouwer fixe point theorem (framework : internal logic of an elementary topos, real …

Barr's covering theorem assert that any Gorthendieck topos can be covered by a Grothendieck topos (even a locale) satisfying the axiom of choice (and hence also the law of excluded middle). Its corrol …

Here is a method which is very efficient in the case were "constructive" is interpreted as "no axiom of choice at all, not even countable and no law of excluded middle", i.e. essentially "topos logic" …

I'm looking for references for the following closely related facts: Given a Boolean algebra $B$, I denote by $\mathbb{Z}[B]$ the free ring generated by symbols $e_b$ such that $e_b e_{b'} = e_{b \cap …

You can always rewrite a subobject $V \subseteq \mathbb{Q}$ as a function $\mathbb{Q} \to \Omega$, but you'll need to includes all the axiom that are in the definition. Even if you only look at defini …