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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 …

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 …

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 …

There are several (equivalent) way to go about it: You can start form the fields operation on $\mathbb{Q}$ and use that they are "locally uniformly continuous" to extend them by continuity to the loca …

Let me first clarify some confusion in the comments to the original question. To be clear : I'm not at all saying the persons making them were confused, as far as I can tell all the comments were corr …

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 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 …

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 …

AS I said, it depends way to much on your framework to give a definitive answer ! here are some exemples that works in some cases: Take $E$ to be the free $\mathbb{Q}$-vector space on a set $S$, and …

I claim that the following result have constructive* proof: 1) Let $f : [0,1] \rightarrow \mathbb{R}$ be a uniformly continuous function such that $f(0)\leqslant 0$ and $f(1) \geqslant 0$ then (as a …

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 …

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 …

The following works constructively over an arbitrary local ring $R$ (constructively, $\mathbb{R}$ is a local ring). Assume that you matrix $M$ is canceled by a polynomial $Q$, of degree $m$ (with lea …

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" …

The notion of "Cauchy real" is always a bit ambiguous: it depends on what you call a Cauchy sequence. For the argument that follow I need a notion of Cauchy sequence that is geometric (is classified b …