Dissolution of a topos
The dissolution of the locale associated with a frame $F$ is the locale associated with the frame $N(F)$ of nuclei of $F$ (see, e.g., Johnstone, “Stone Spaces” (1982), §2.5). Note that there is a frame...
View ArticleAnswer by Gro-Tsen for How to get a ball in the nonvanishing locus of a...
I'm interpreting your question as follows: given$f \in \mathbb{Z}_p[t_1,\ldots,t_n]$ and given$\underline{x} := (x_1,\ldots,x_n) \in \mathbb{Z}_p^n$ such that $f(x_1,\ldots,x_n) \neq 0$, you want to...
View ArticleKleene realizability in Peano arithmetic
For completeness of MathOverflow and for clarity of the question, I will first recall a few things, including the the definition of Kleene realizability: experts can jump directly to the question...
View ArticleAnswer by Gro-Tsen for Difference between constructive Dedekind and Cauchy...
(This is more of a comment than an answer, but I think the following caveat is indispensable whenever Cauchy reals are mentioned in constructive mathematics without Countable Choice.)I'm puzzled by the...
View ArticleNeed help in trying to understand an argument by V. A. Yankov on the...
(This is really long because I give a lot of context, but you can skip right to the end where the excerpt I'm trying to make sense of is copied and translated.)Background: I'm trying to understand the...
View ArticleAnswer by Gro-Tsen for Need help in trying to understand an argument by V. A....
So, I didn't understand the exact details of what Yankov was trying to do with his $K$ and $Q,P,R$ (the definition of $K$ probably depends on the details of the coding he uses for realizability, which...
View ArticleAnswer by Gro-Tsen for Did Gödel prove that the Ramified Theory of Types...
I realize this is an old question, but in case this is still of interest: if I correctly understand that you are talking about the ramified analytical hierarchy, namely the one in which each next level...
View ArticleLogical properties of realizability (topoi or McCarty models) defined by...
Setup: Let $\alpha$ be an admissible ordinal (viꝫ., one such that $L_\alpha$ is a model of Kripke-Platek set theory), identified as usual with the set of ordinals $<\alpha$. Then there is a standard...
View ArticleAnswer by Gro-Tsen for Intuition behind Kleene's “second algebra”...
I'm adding an answer to my own question to point out one thing about Kleene's $\mathcal{K}_2$ which I had previously failed to understand and which is an important difference with $\mathcal{K}_1$ that...
View ArticleIntuition behind Kleene's “second algebra” $\mathcal{K}_2$
The “second Kleene algebra”$\mathcal{K}_2$ is defined, e.g. here on nLab, or in section 1.4.3 of van Oosten's book Realizability: an Introduction to its Categorical Side (2008), or as example 3.4 of...
View ArticleAnswer by Gro-Tsen for Why is $\operatorname{Spec}(\mathbb Z)$ supposed to...
(I'm not sure this qualifies as an answer or an extended comment, because I don't have any deep knowledge in the matter, but I hope this can at least help clear some possible confusion.)It seems to me...
View ArticleÉtendue measure of the set of lines between two Euclidean balls
Let $d>0$ and $r_1,r_2>0$ such that $r_1+r_2 < d$. Consider two (say, closed) balls $B_1,B_2$ in $\mathbb{R}^m$ having radii $r_1,r_2$ and whose centers are at distance $d$. Let $C$ be the set...
View ArticleWhat is known about propositional realizability for the second Kleene algebra...
Short version: Various things are known about realizability of propositional formulas for Kleene's “first algebra” (i.e., $\mathbb{N}$), like examples of realizable but unprovable formulas, and some...
View ArticleWhat are the measure of the volume and boundary (and other quermaß measures)...
Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q)...
View ArticleAnswer by Gro-Tsen for Guaranteed correct digits of elementary expressions
There is a certain confusion in the answers, so let me try to dispel this confusion.There are two different issues here. One is “computing an approximation with arbitrary precision” and one is...
View ArticleAnswer by Gro-Tsen for When can a function defined on $[a, b] \cup [b, c]$ be...
Here is a positive result that's pretty obvious but still worth mentioning since you're just supposing that $S$ is a set: if $S=\Omega$ is the power object of a singleton, i.e., the set of truth...
View ArticleResults with a flavor “every automorphism of automorphisms is inner”
It seems that there are a number of results which take more or less the following form: let $X$ be some (specific) kind of structure, let $Y$ be the group of automorphisms of $X$ or perhaps ring of...
View ArticleLabeling binary trees so that adjacent vertices differ by a power of two
Let $T$ be a finite rooted binary tree (where "binary tree" means that each node has at most two children, possibly less) with $n$ nodes in total. Is there a labeling of the nodes of $T$ with the...
View ArticleAnswer by Gro-Tsen for Examples of conjectures that were widely believed to...
The Connes embedding conjecture, formulated in 1976, asserts that every type II₁ von Neumann factor embeds into an ultrapower of the hyperfinite II₁ factor. Even though it is sometimes known as the...
View ArticleAnswer by Gro-Tsen for Zero set of prime ideal
Let $k = \mathbb{R}$ and let $P$ be the prime ideal of $\mathbb{R}[x,y]$ generated by $h := y^4 + y^2 + (x^2-1)^2$. According to Sage¹ (though I'm sure this is easy to check by other means), $h$ is...
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