Comment 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 “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 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 ArticleComment by Gro-Tsen on Have you seen this variation of Medvedev reducibility?
@BjørnKjos-Hanssen It does have some resemblance to that as well, of course, but Weihrauch reducibility has the important limitation that, informally speaking, “you can only use the oracle once”, which...
View ArticleAnswer by Gro-Tsen for Is there an algorithm to write down the 27 lines of a...
(I'm aware that I'm answering an old question, but I think this is worth explaining anyway.)Here is a general method to compute the lines contained in a projective surface $S$ defined in $\mathbb{P}^3$...
View ArticleAnswer by Gro-Tsen for Formula for the volume of hyper-spherical simplices
I once tried to look up what was known regarding this question (because it relates to the problem, that I was interested in, of finding the expected value and higher moments of the max of $n$...
View ArticleComment by Gro-Tsen on Why did Bourbaki's Élements omit the theory of...
@TimCampion Serre claims that flat cohomology can (might? could? he's not entirely clear) on the choice of the universe (unlike étale cohomology), so “even Deligne” (says Serre) is afraid to use it....
View ArticleNote rejected from arXiv: what to do next?
Short version: A note of mine was rejected by the arXiv moderation (something I didn't even know was possible) on account of being “unrefereeable”. The moderation process provides absolutely no...
View ArticleAnswer by Gro-Tsen for Does $\neg(j^{**}(\bot))=\neg\neg(j^{*}(\bot))$ hold...
No, this does not hold in general. Here is a counterexample.Let $L := \mathcal{O}(\mathbb{R})$ be the frame of open sets in $\mathbb{R}$. Let $D$ be a dense subset of $\mathbb{R}$ whose complement $E...
View ArticleComment by Gro-Tsen on For which sets do Arthur and Nimue have a winning...
@user14111 Indeed, if A+N have a winning strategy for $(P',Q')$ and $P\subseteq P'$ and $Q\subseteq Q'$, then A+N have a winning strategy for $(P,Q)$. I should probably have spelled this out explicitly.
View ArticleComment by Gro-Tsen on Can we “encrypt” in the Turing degrees?
@JeanAbouSamra I don't think this is true, however: for example, it is known that any $2$-generic and any $2$-random Turing degree form a minimal pair (that is, their inf is $\textbf{0}$), but there is...
View ArticleComment by Gro-Tsen on Does the airplane Julia set contain true circles?
Experimentally (zooming in with a graphical Julia set computing program that I wrote) ages ago, the right edge of the central “circle” is at $\sim 0.109650026$ to within $10^{-9}$, whereas the top edge...
View ArticleComment by Gro-Tsen on Is van Dalen’s “open problem” about $\bf{CT}$ and...
Regarding your question of the relation between CT and omniscience principles, I think this needs to be a separate question (and you will need to make clear exactly what you mean by CT because there...
View ArticleComment by Gro-Tsen on If $X'$ computes $\mathcal{O}^{Y}$ must $X$ compute $Y$?
@PeterGerdes Could you provide a reference or statement to what you call “Spector's jump inversion”? I can't find anything by that name anywhere. (Or is “Spector” meant to be “Cooper”?)
View ArticleComment by Gro-Tsen on Why does the Fourier transform of $μ(n)/n$ look like...
@DanielWeber Excellent idea! I generated 4 random “control” curves with “fake” $\mu$ functions that coincide with $\mu(n)$ for $n\leq 12$ (I think $100$ is too much, so I went with $12$) and otherwise...
View ArticleWhy and how do (classical) reverse mathematics and intuitionistic reverse...
Broadly speaking, the idea of “reverse mathematics” is to find equivalents to various standard mathematical statements over a weak base theory, in order to gauge the strength of theories (sets of...
View ArticleAnswer by Gro-Tsen for Algorithm for selecting a fixed-point free permutation...
This is not an answer to the mathematical question. But it is an answer to the practical problem, which, I think, is nice enough to be shared, and it doesn't fit in the comments.Many years ago, a group...
View ArticleWhat is the Jarlskog invariant, conceptually?
Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity:$$J_{ij,k\ell} :=...
View ArticleComment by Gro-Tsen on Essential results for a chapter on $\alpha$-recursion
For what it's worth, there is a short introductory course by Simpson (“Short Course on Admissible Recursion Theory”, pp. 355–390 in Generalized Recursion Theory II (Oslo 1977) by Fenstad, Gandy &...
View ArticleAnswer by Gro-Tsen for Deciding whether rational periodic external rays of...
I'm afraid I've forgotten what little I knew of the subject, but you may wish to consult the (partially written? in preparation? abandoned?) book Symbolic Dynamic of Quadratic Polynomials by Henk...
View ArticleAnswer by Gro-Tsen for Does a quartic have four rational roots if its...
First of all, there are multiple definitions of the “resolvent cubic”, and the Wikipedia article even lists 5 different ones, so you should have clarified which one you meant. I will assume you mean...
View ArticleAnswer by Gro-Tsen for Is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated...
No. The following counterexamples are part of the folklore:Even if $X$ is an affine variety over a field $k$ (so $\Gamma(X,\mathscr{O}_X)$ is certainly of finite type over $k$) and $U\subseteq X$ open,...
View ArticleComment by Gro-Tsen on Precise meaning of "picking a basis"?
Re “unique choice”, it might be relevant to point out, as I like to do, that the axiom of choice over ZF is equivalent to the assertion that for any set $X$ and equivalence relation $\sim$, the obvious...
View ArticleComment by Gro-Tsen on Theory of polynomial multiplication modulo 4
For another start, we understand the multiplicative monoid of $\mathbb{Z}_p[X]$ (where $\mathbb{Z}_p$ are the $p$-adics, not integers mod $p$) because the ring is a UFD (because $\mathbb{Z}_p$ is one)...
View ArticleComment by Gro-Tsen on “Uniformly” minimal pairs of Turing degrees
Yes, sorry $\langle,\rangle$ is a standard computable Gödel coding of tuples by natural numbers.
View ArticleComment by Gro-Tsen on “Uniformly” minimal pairs of Turing degrees
Very nice and simple argument! (I sort of expected something of the sort, but more complicated and probably involving Kleene's recursion theorem.) ❧ If I happen to use this in a paper, by what name...
View ArticleTopology of the set of Nash equilibria of a normal form game
Consider a normal form game with $n$ players (and finitely many options per player) defined by finite option sets $A_1,\ldots,A_n$ and payoff matrices $u_1,\ldots,u_n: \prod_{j=1}^n A_j \to...
View ArticleExamples of “usual” topological properties that can be expressed internally...
Any topological space $X$ defines a topos $\newcommand{\Sh}{\operatorname{Sh}}\Sh(X)$ of sheaves (of sets) on $X$, which has an internal language/logic: every such “internal” statement $\phi$ defines a...
View ArticleComment by Gro-Tsen on How do we incentivize breadth in mathematics? Should we?
I feel it is necessary to post this quote here: ”A leader in the theory of pseudo-parabolic partial differential equations in quasi-convex domains will not stoop to being understood by specialists in...
View ArticleComment by Gro-Tsen on Intuition of unique roles behind flow and diffusion...
This video titled “But how do AI images and videos actually work?” by 3Blue1Brown and Welch Labs (two remarkable science video creators on YouTube) tries to answer this exact question, so I would...
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