A 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$$...
View ArticleComment by Gro-Tsen on Is there an “opposite” hypothesis to the (Generalized)...
@bof: Maybe we should leave the problem of the size of $2$ to even later generations, but I might point out, in relation with the present question, that in constructive mathematics, the powerset of a...
View ArticleComment by Gro-Tsen on Sampling from the uniform distribution on preorderings...
@JackEdwardTisdell I think this doesn't work: the distribution on the partitions defined by the equivalence classes of a random preorder is not uniform: larger classes have a smaller probability of...
View ArticleOn the history of the “bb” propositional formula that characterizes finite...
The intuitionistic propositional formula $\mathbf{bb}_n$ (in the $n+1$ variables $p_0,\ldots,p_n$) is:$$\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 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 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 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 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 ArticleCan the set of points in $\mathbb{R}^3$ with exactly one coordinate in $A$ be...
Context: This is a followup-question to this one, which asks whether the set of elements of $\mathbb{R}^3$ with exactly one coordinate in $\mathbb{Q}$ is connected: the answer to this is negative, but...
View ArticleAnswer by Gro-Tsen for Improving readability of proofs
✱ Try to add some redundancy as a kind of error-correcting code, especially in definitions:No matter how careful you are, there will always be parts of the proof that will be ambiguous or risk being...
View ArticleComment by Gro-Tsen on Improving readability of proofs
The first point is indeed underappreciated. When writing computer code, one does not simply call a function, one calls it on specific arguments (which may not have the same name as the ones used...
View ArticleComment by Gro-Tsen on The Bring quintic and the Baby Monster?
What exactly does “it” in “it is in fact true” refer to? Could you please write down a precise mathematical statement, with quantifiers clearly spelled out, whose truth you are asking about? All I see...
View ArticleComment by Gro-Tsen on How to prove $\sup_{n,k}...
@SamHopkins Are you saying that (as far as you know) this particular answer was AI-generated, or that this user posted some other answer that was AI-generated and that, as a result, all of their...
View ArticleComment by Gro-Tsen on The Bring quintic and the Baby Monster?
Just to be clear: the Baby Monster is a red herring in all of this, correct?
View ArticleComment by Gro-Tsen on Interpreting $1/f$ as a distribution when $f$ is only...
Concerning your second question, doesn't $x^2 + \exp(-\frac{1}{x^2})$ provide an easy answer? Is this really what you want to ask?
View ArticleComment by Gro-Tsen on How big can a multiprojective variety be for which...
I think there is no good answer here other than “go ahead and try it” (starting with simpler cases if you can). As for anything based on Gröbner bases computations, the worst case complexity is pretty...
View ArticleComment by Gro-Tsen on “Who dies first?” Probability distributions such that...
@FedorPetrov Maybe. But it is unclear: the available data is extremely poor at this point. From the table, I have, for an age gap of 8, the probability that the younger survives the older starts at...
View ArticleAnswer by Gro-Tsen for Is the infimum of the gradient norm equal to its...
This is too long for a comment, but I think the following perspective will be useful:The key fact alluded to in both answers is that if $f$ is differentiable everywhere and $f'\geq 0$almost everywhere...
View ArticleComment by Gro-Tsen on Cardinals with ${∣\{\lambda \leq \kappa\}∣} = \kappa$...
@JoelDavidHamkins I'm confused by your last comment: in ZFC, $\omega_1$ has $\aleph_0$ smaller-or-equal cardinalities (namely, the finite cardinals, $\aleph_0$ and $\aleph_1$), doesn't it? Do you mean...
View ArticleComment by Gro-Tsen on Embedding of the Lobachevsky plane
Is pseudosphere even an embedding of part of the hyperbolic plane? Don't you need to slice it vertically for that? (By “slice” I mean, remove one tractrix if the pseudosphere is defined as the rotation...
View ArticleComment by Gro-Tsen on What is the probability of coming out ahead in the...
Let me check I understood the question: you define a probability distribution for a random variable $X$ by $\mathbb{P}(X=-2^m) = \frac{1}{2^m}$ for $m\geq 1$ (and no other value is possible), then for...
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 Implications of $g^6 = 1$ for Cayley-numbers $g$
It is a basic fact about octonions that the $\mathbb{R}$-algebra generated by a single octonion¹ is isomorphic to $\mathbb{R}$ or $\mathbb{C}$.More descriptively, if $z$ is an octonion and $u$ the...
View Article