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 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$$...
View ArticleTonelli's theorem for Riemann integration
I asked this question on Math StackExchange last month, but received no answer there. I think it is worth reposting it on MathOverflow, so here we go.Recall that if $f\colon [0,1]^2 \to...
View ArticleAnswer by Gro-Tsen for Locally isomorphic graphs
Let $G_0$ be two pentagons ($5$-cycles) connected by an edge: say $V_0 = (\mathbb{Z}/5\mathbb{Z})\times\{0,1\}$ with $(i,p)$ connected to $(i-1,p)$ and $(i+1,p)$ except that $(0,p)$ is also connected...
View ArticleClik here to view.
Answer by Gro-Tsen for Software for testing validity of propositional...
So I ended up writing my own program for the intuitionistic case. It can be found here on GitHub. As an example, I used it to generate this document tabulating the validity of a number of sample...
View ArticleComment by Gro-Tsen on Is there an “opposite” hypothesis to the (Generalized)...
@bof: Maybe we should leave the problem of the size of $2$ to even later generations, but I might point out, in relation with the present question, that in constructive mathematics, the powerset of a...
View ArticleComment by Gro-Tsen on Sampling from the uniform distribution on preorderings...
@JackEdwardTisdell I think this doesn't work: the distribution on the partitions defined by the equivalence classes of a random preorder is not uniform: larger classes have a smaller probability of...
View ArticleOn the history of the “bb” propositional formula that characterizes finite...
The intuitionistic propositional formula $\mathbf{bb}_n$ (in the $n+1$ variables $p_0,\ldots,p_n$) is:$$\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 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 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 ArticleClik here to view.
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 continuity of the gradient norm imply continuity...
This was also asked on Reddit
View ArticleComment by Gro-Tsen on Is the set of theorems of a PA +“PA is inconsistent”...
@JeanAbouSamra No. (But I very much suspect that the answer to my first question is positive, so I gambled by asking the stronger question in that direction.)
View ArticleComment by Gro-Tsen on With what probability does an inscribed/circumscribed...
Do you have a reference for the computation with the inscribed triangle?
View ArticleAnswer by Gro-Tsen for Does continuity of the gradient norm imply continuity...
A partial answer: in dimension 1, the answer is “yes” (this is easy, but too long to fit in a comment):Assume $f:\mathbb{R}\to\mathbb{R}$ is such that $|f'|$ is continuous. We wish to prove that $f'$...
View ArticleAnswer by Gro-Tsen for Is the set of theorems of a PA +“PA is inconsistent”...
My question already has two answers which are correct and well-written, but now that I understand what is going on I thought it'd be instructive to add one of my own. What follows is a retelling of...
View ArticleConvergence spaces, pseudotopologial spaces, etc.: what's the big picture?...
I've often wanted to learn more about convergence spaces, but I've found myself lost in a maze of definitions (sometimes with conflicting names across sources) with no intuition about what each one is...
View ArticleAnswer by Gro-Tsen for Convergence spaces, pseudotopologial spaces, etc.:...
I will attempt to at least partially answer my own question by saying something of what, after reading several papers, I understand about how pre- and pseudotopological spaces shed light on quotients...
View ArticleComment by Gro-Tsen on What is the consistency strength of Russell &...
This may be more a history of math than an actual math question, but I'm a bit confused about what part of PM's presentation makes it not a fully specified formal system so that Church had to later...
View ArticleWhat is the consistency strength of Russell & Whitehead's ‘Principia...
Russell and Whitehead's Principia Mathematica is of mostly historical interest (e.g., in that Gödel's incompleteness theorem was originally formulated against it), and I must admit never having read...
View Article
