Construction of the smallest nucleus above a prenucleus: what does this proof...
While reading Hyland's paper on the effective topos [retyped version here] in the L. E. J. Brouwer Centenary Symposium, specifically prop. 16.3, I realized that the following proposition is...
View ArticleComment by Gro-Tsen on Is every real number the limit of a sequence of...
@MohammadTahmasbizade The counterexample is easily modified to additionally satisfy $1\leq x\leq 2$, so it is not provable that a real in $[1,2]$ (or, a fortiori, a nonzero real) is the limit of a...
View ArticleAnswer by Gro-Tsen for Is every real number the limit of a sequence of...
This is more an extended comment than an answer (or at least it is a very partial answer, which I'm sure OP is already aware of), but because of the confusion such questions typically cause (as...
View ArticleAnswer by Gro-Tsen for Special rational numbers that appear as answers to...
According to this answer on this site, the area of the Pythagoras tree fractal (obtained by starting with a unit square, putting two smaller squares of size scaled down by $\sqrt{2}/2$ adjacent to one...
View ArticleWhy is it so hard to give examples of differentially closed fields?
The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability;...
View ArticleAnswer by Gro-Tsen for Open Sets in Hausdorff spaces
Here is a counterexample showing that it is not true in general that, for $X$ Hausdorff, every non-empty open set contains a closed set with non-empty interior.Let $X$ be the countable complement...
View ArticleAnswer by Gro-Tsen for Construction of the smallest nucleus above a...
Answering the first bullet point in my own question, I have found two references for this result, which provide at least some context. One is:Peter T. Johnstone, “Two Notes on Nuclei”, Order7 (1990)...
View ArticleSoftware for testing validity of propositional formulas in finite Kripke...
I havea formula $\varphi$ of propositional modal logic or propositional intuitionistic logic,a finite Kripke frame $W$,and I would like to test whether $\varphi$ is valid in $W$.This is an enumeration...
View ArticleComment by Gro-Tsen on Different forms of completeness of intuitionistic...
@EmilJeřábek Sanity check to be sure that I'm not even more confused than I thought I was: the argument for “it certainly holds … for arbitrary Heyting algebras” is that we take the set of all formulas...
View ArticleA generalization of extremal disconnectedness (closure of the intersection of...
For $n\geq 2$, consider the following property $(\mathbf{P}_n)$ on a topological space $X$:If $U_1,\ldots,U_n \subseteq X$ are $n$ arbitrary open sets, then$$...
View ArticleTonelli's theorem for Riemann integration
I asked this question on Math StackExchange last month, but received no answer there. I think it is worth reposting it on MathOverflow, so here we go.Recall that if $f\colon [0,1]^2 \to...
View ArticleAnswer by Gro-Tsen for Locally isomorphic graphs
Let $G_0$ be two pentagons ($5$-cycles) connected by an edge: say $V_0 = (\mathbb{Z}/5\mathbb{Z})\times\{0,1\}$ with $(i,p)$ connected to $(i-1,p)$ and $(i+1,p)$ except that $(0,p)$ is also connected...
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Answer by Gro-Tsen for Software for testing validity of propositional...
So I ended up writing my own program for the intuitionistic case. It can be found here on GitHub. As an example, I used it to generate this document tabulating the validity of a number of sample...
View ArticleComment by Gro-Tsen on Is there an “opposite” hypothesis to the (Generalized)...
@bof: Maybe we should leave the problem of the size of $2$ to even later generations, but I might point out, in relation with the present question, that in constructive mathematics, the powerset of a...
View ArticleComment by Gro-Tsen on Sampling from the uniform distribution on preorderings...
@JackEdwardTisdell I think this doesn't work: the distribution on the partitions defined by the equivalence classes of a random preorder is not uniform: larger classes have a smaller probability of...
View ArticleOn the history of the “bb” propositional formula that characterizes finite...
The intuitionistic propositional formula $\mathbf{bb}_n$ (in the $n+1$ variables $p_0,\ldots,p_n$) is:$$\bigwedge_{i=0}^n \Big ( \big (p_i \Rightarrow \bigvee_{j\neq i} p_j\big) \,\Rightarrow \,...
View ArticleAnswer by Gro-Tsen for When do globally generating sections generate the...
As pointed out in the comments, the correct definition of “the sections $s_1,\ldots,s_n$ (on $X$, say) generate $\mathcal{F}$ [as an $\mathcal{A}$-module]” is that the morphism of $\mathcal{A}$-modules...
View ArticleIn the internal language of the topos of sheaves on a topological space, can...
For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject...
View ArticleCondition for a “checkering” to be connected?
Let $A_0\cup B_0=A_1\cup B_1=\mathbb{R}$ and$A_0\cap B_0=A_1\cap B_1=\varnothing$. We define the associated “checkering” of $\mathbb{R}^2$ to be the partition of $\mathbb{R}^2$ as $\mathbb{R}^2 = P\cup...
View ArticleTopological spaces in which countable intersections of dense open sets have...
In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense.Now consider the following strengthening of the Baire...
View ArticleExamples of statements that are valid in every spatial topos
I am looking for statements¹ that, when interpreted in the internal language of a topos, are valid in all spatial toposes (i.e., the topos of sheaves of any topological space) that are not valid in all...
View ArticleAnswer by Gro-Tsen for Where can I find a reference table for ordinal...
I'm not sure what you mean or what you expect from a “reference table”, but your question prompted me to dig up this Haskell code I had written a dozen years ago: it's an implementation of the ordinals...
View ArticleComment by Gro-Tsen on Does this specific 5-state Turing machine halt?
I might as well add some gratuitous stats about this TM: the final tape contains $4098$ symbols $1$; during the $47176870$ execution steps, the machine head visited $12289$ different positions, from...
View ArticleComment by Gro-Tsen on Are lower semi-continuous images of compact sets Borel?
Do you know the answer to your question in the special case where $X$ is a compact subset of $\mathbb{R}$, or even in the even more special case when $X=[0,1]$?
View ArticleAnswer by Gro-Tsen for Simple explanation and/or detailed examples for...
It is not entirely clear to me what your actual question is, but if I correctly understand that at least part of it is how we can (algorithmically) compute the sequence counting the number of reduced...
View ArticleStandard term for a kind of compatibility of two partitions
Is there a standard term for the following compatibility condition between two partitions $P,Q$ of the same set $X$?For any two parts $A_1,A_2 \in P$ of the first and any two parts $B_1,B_2 \in Q$ of...
View ArticleComment by Gro-Tsen on When does the property of projective variety $V$...
A counterexample for the general claim is that, say, the twisted cubic curve $C$ in $\mathbb{P}^3$ is not complete intersection, but its intersection $C\cap L$ with a general plane $L \subseteq...
View ArticleAnswer by Gro-Tsen for Are lower semi-continuous images of compact sets Borel?
Claim: The image of a lower semicontinuous function $f\colon [0,1] \to \mathbb{R}$ can fail to be Borel.Proof. I will use the fact stated in this answer that every analytic subset of $\mathbb{R}$ is...
View ArticleComment by Gro-Tsen on Arithmetic of constructive Dedekind reals
One might also mention the following two references, even if they are not really satisfactory: ① P. Johnstone, Sketches of an Elephant (2002), section D.4.7, just before proposition 4.7.8 (a definition...
View ArticleComment by Gro-Tsen on Regarding the realizability topos on the computable...
Ah yes! This is in fact pretty much exactly the same argument as for the extensional realizability topos $\mathbf{Ext}$ [van Oosten, Extensional Realizability (1997)], which also satisfies continuous...
View ArticleComment by Gro-Tsen on Singular cohomology of varieties over function fields
Trivial remark: if you are satisfied with torsion coefficients, you can, of course, use étale cohomology, which over $\mathbb{C}$ agrees with singular cohomology but works over any base.
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Answer by Gro-Tsen for Riemann mapping to a specific curvilinear triangle
This is not exactly a full answer because I don't know enough about hypergeometric functions to say more, but I can at least say this:Seeing the unit disk as the hyperbolic plane under the Poincaré...
View ArticleRegarding the realizability topos on the computable part of Kleene's second...
Let:$\mathcal{K}_1$ be the first Kleene algebra, meaning $\mathbb{N}$ endowed with the partial operation $(p,n) \mapsto p\bullet n := \varphi_p(n)$ where $\varphi$ is the $p$-th partial computable...
View ArticleComment by Gro-Tsen on Conditions on the unit and coünit of a geometric morphism
@KevinCarlson I will henceforth definitely be using the word “highfalutin” to describe myself. For what it's worth, I got the diaeresis convention from Eisenhart's book on Riemannian Geometry, in which...
View ArticleComment by Gro-Tsen on Lie algebras with "trivial" Killing form
A remark for those who, like me, feel a little rusty on the topic: even in characteristic $0$, a vanishing Killing for does not imply nilpotence of the Lie algebra (although the converse is true): the...
View ArticleComment by Gro-Tsen on Proof-theoretic strength of subsystems of...
Do you know a subsystem of KP that has proof theoretic ordinal $\Gamma_0 = \varphi(1,0,0)$?
View ArticleComment by Gro-Tsen on On the set of differentiability of a fat Cantor staircase
Remark: the set of non-differentiability of $f$ is included in the boundary $\partial C$ for the density topology (i.e., $C$ minus its density interior $E$, the latter consisting of points where $C$...
View ArticleAnswer by Gro-Tsen for On the set of differentiability of a fat Cantor staircase
Let me answer a different question that I think is relevant in connection to the one above. Let us ask:Can we find a function $g\colon[0,1]\to\mathbb{R}$ that is differentiable everywhere (so, in...
View ArticleComment by Gro-Tsen on Does mutual independence yield independence for...
I would say “pairwise independent” for what you call “mutually independent”. Anyway. Consider two independent uniform random variables $X_1,X_2$ on $\mathbb{R}/\mathbb{Z}$ and let $X_3=X_1+X_2$ (mod...
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