What 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 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 ArticleAnswer by Gro-Tsen for Does the first fundamental representation of...
(Copied from my own comments.) Yes. In fact, if $V_k$ (for $1\leq k\leq n$) denotes the $k$-th fundamental representation in the order of the nodes of the Dynkin diagram, and $V_0$ the trivial...
View ArticleComment by Gro-Tsen on What are the Nash equilibria of the “aim for the...
@MichaelGreinecker Right: sorry, I should have excluded that case.
View ArticleExamples of concrete games to apply Borel determinacy to
I'm teaching a course on various mathematical aspects of games, and I'd like to find some examples to illustrate Borel determinacy. Open or closed determinacy is easy to motivate because it proves the...
View ArticleAnswer by Gro-Tsen for Alternate descriptions of finite fields
Instead of having a single polynomial-quotient (aka “rupture”) step $\mathbb{F}_p[X]/(P)$ with $P \in \mathbb{F}_p[X]$ irreducible, you can also construct finite fields in several steps, i.e.,...
View ArticleWhat are the possible symmetry groups of n-point constructions in the...
Let $k$ be an infinite field, perhaps take $k = \mathbb{C}$ if it simplifies matters.I will be asking a question about $\mathbb{P}^2$ for definiteness and to simplify definitions/notations, but feel...
View ArticleFormulas for the line joining two points in the projective plane over a...
Let $K$ be a[n associative] division algebra (= skew field). By the “projective plane”$\mathbb{P}^2(K)$ over $K$ I mean, as usual, the set of triples $(x,y,z)$ of elements of $K$, not all zero, up to...
View ArticleAnswer by Gro-Tsen for Hilbert's Satz 90 for real simply-connected groups?
[Copied and extended from comments.]First, it needs to be clarified in relation to the question that $\mathbb{R}$ does not satisfy the cohomological condition (viꝫ., having cohomological dimension...
View ArticleComment by Gro-Tsen on Condition to guarantee that an inhabited and bounded...
@JeanAbouSamra No this is not stupid at all: constructively, “least upper bound” (l.u.b.) and “supremum” need not coincide: “supremum” is the notion I defined, and “l.u.b.” is (as name suggests) the...
View ArticleClosed sets versus closed sublocales in general topology in constructive math
This question is set in constructive mathematics (without Choice), such as in the internal logic of a topos with natural numbers object, or in IZF.Short version of the question: if $X$ is a sober...
View ArticleAnswer by Gro-Tsen for Equivalence of omniscience principles for natural...
Let me try to tackle the LPO case. I will even try to show that it doesn't matter whether we assume our Cauchy sequences to have a modulus or not. (Please check me carefully because I know I've made...
View ArticleAnswer by Gro-Tsen for Completing half of Hilbert's program: Foundations that...
One possible such theory is described by Solomon Feferman in chapter 13 (“Weyl vindicated: Das Kontinuuum seventy years later”) of his book In the Light of Logic (Oxford University Press 1998);...
View ArticleComment by Gro-Tsen on Terminology: A "corollary" to a proof?
@EmilJeřábek Of course the reader will not really forget everything about the proof as soon as they reach the “end of proof” symbol. What will really happen, however, is that many will read the...
View ArticleComment by Gro-Tsen on How should we picture the set of monomial orders (=...
@SamHopkins Thanks! To spell it out explicitly, they explain (prop. 1.7, prop. 3.1 and remark following cor. 4.1) that the set of monomial orders is homeomorphic to the Cantor set.
View ArticleComment by Gro-Tsen on Topological spaces in which countable intersections of...
@NotMike Indeed, I should have mentioned the P-space property. I added a remark to the question on this subject.
View ArticleComment by Gro-Tsen on When are two proofs of the same theorem really...
@Jon23 I mean inclusive or ($\lor$).
View ArticleAnswer by Gro-Tsen for What is a "scholium"?
Bourbaki defines a "scholie" in the preface of the Éléments de mathématiques as follows:Sous le nom de « scholie », on trouvera quelquefois un commentaire d'un théorème particulièrement important.I.e.,...
View ArticleCountably compact Boolean algebras versus distributivity
Let us say that a complete Boolean algebra $B$ is:countably distributive when for any sequence $(I_n)_{n\in\mathbb{N}}$ of sets and any elements $(u_{n,i})_{n\in\mathbb{N},i\in I_n}$ of $B$ we...
View ArticleComputing the truncations (“ancestors”) of a surreal number from its Hahn...
If $x$ is a surreal number and $\alpha$ an ordinal, let us denote $T_\alpha(x)$ and call $\alpha$-truncation of $x$ the surreal number whose sign sequence is obtained by truncating the sign sequence of...
View ArticleComment by Gro-Tsen on What makes the surreals special among other...
@JoelDavidHamkins Indeed, but all the “surreal-like” fields I mention in this question are isomorphic as ordered fields, so they all satisfy this property. The extra datum I am trying to grasp is the...
View ArticleComment by Gro-Tsen on Why are extremally disconnected spaces so hard to give...
@godelian This seems promising, and your example interests me. I can (sort of) form a mental picture of an $η_1$ totally ordered set, like the set of functions $ω_1 \to \{-1,0,+1\}$ that are eventually...
View ArticleAnswer by Gro-Tsen for Stone-Čech boundary is not extremally disconnected
The question has already been answered satisfactorily, but I think the following presentation, which is implicit in the second part of YCor's answer, will clarify things. The fact that $\beta\mathbb{N}...
View ArticleAnswer by Gro-Tsen for Clarification on proof of the algebraic completeness...
[Converted from a comment into an answer, and expanded with a copy of the statement from Siegel's book.]You may find clearer the proof given in Siegel, Combinatorial Game Theory (2013, AMS Graduate...
View ArticleStone-Čech compactification of a Boolean subalgebra of $\{0,1\}^S$
Setup: Let $S$ be a set. Let $B$ be a Boolean subalgebra of $\{0,1\}^S$; i.e., just to be clear $B$ contains the constant $0$ and $1$ functions, and is stable under binary pointwise $\land$, $\lor$ and...
View ArticleComment by Gro-Tsen on Is there a mathematical theory of negotiation games?
@StevenLandsburg Thanks! Formula 7.4.13 in the chapter you cite gives $x=(c+d_1-d_2)/2$ when $F(x)=(x-d_1)(c-x-d_2)$, confirming that Nash's result subsumes the particular case I discussed in ¶5. If...
View ArticleComment by Gro-Tsen on Topological rigidity of cartesian product with...
Let me just comment in passing that this isn't true without the “compact” hypothesis: if $V=\mathbb{R}^4$ and $W$ is an exotic $\mathbb{R}^4$, then $V\times\mathbb{R}$ and $W\times\mathbb{R}$ are both...
View ArticleComment by Gro-Tsen on Does the decomposability of $\mathbb{R}$ imply...
It's not important in this question because you clearly stated what you mean, but in general it seems to me that it's a really bad idea to use the notation “BISH” to refer to constructive mathematics...
View ArticleComment by Gro-Tsen on Extending polynomial hierarchy above $\omega$
Just to point out the obvious: in the arithmetic hierarchy, there is a universal/uniform oracle which lets you jump from one level to the next (viꝫ. the halting problem relativized to the previous...
View ArticleComment by Gro-Tsen on Is there a statement in Presburger arithmetic about...
I have a meta-question, which is whether the second paragraph in the conjecture's statement is algorithmically decidable (i.e., whether given $s$ we can algorithmically decide whether for all $M$ the...
View ArticleAnswer by Gro-Tsen for Does the decomposability of $\mathbb{R}$ imply...
(To lighten notations, let me simply write $\mathbb{R}$, instead of $\mathbb{R}^d$, for the set of Dedekind reals in what follows, since it is the only one that will appear. For the sake of notational...
View ArticleComment by Gro-Tsen on Non-trivial subfield of ${\bf Q}(\sqrt[3]{a+\sqrt{b}})$
@coudy In this case, put an edit in boldface at the top saying something like “this question is badly worded and shouldn't have been posted”, and briefly summarize the problem or the issue making it...
View ArticleWhat topos-theoretic construction lies behind the “symmetric model”...
Suppose we want to prove that (classical!) $\mathsf{ZF}$ does not prove, say, “for every infinite set $A \subseteq \{0,1\}^{\mathbb{N}}$ there exists an injection $\mathbb{N} \to A$” (I take this...
View ArticleComment by Gro-Tsen on Multiplicative cancellation for trivial vector bundles
Since I had “forgotten” why the isomorphism $I⊕J\cong(I⊗J)⊕O_F$ holds, here is a reference: Keith Conrad, “Ideal classes and relative integers”, lemma 8.
View ArticleComment by Gro-Tsen on Beyond the Bring Radical: What is known about...
Hilbert's 13th problem (of which there are conflicting interpretation, so caveat) and the notion of essential dimension seem relevant to mention here, but I don't feel competent enough to say more, and...
View ArticleComment by Gro-Tsen on When can we add choice to a model of ZF
Probably stupid question: what happens in your question if we replace “countable transitive model” by just “transitive model”? Is there a statement $\tau$ such that a transitive model $M$ of ZF...
View ArticleComment by Gro-Tsen on Delta distribution on manifolds
It seems to be that you've convincingly argued that the answer to your own question is no. If there were a “natural” way to define a distribution $δ_p$ for every $p$, then comparing (chartwise) it to...
View ArticleComment by Gro-Tsen on Is there a continuous partition of space into circles?
Wait, does the word “circle” here refer to a submanifold diffeomorphic to $S^1$ or to a bona fide Euclidean circle (set of points on a plane at equal distance from a center point)? Because the question...
View ArticleComment by Gro-Tsen on Internal logic of the topos of simplicial sets
@HDB: Regarding your comment of 2023-04-03 13:13:59Z, it seems to me that an abstract simplicial complex on $[n] := \{0,\ldots,n\}$ is precisely the same thing as a simplicial subset of $\Delta^n$...
View ArticleEvery real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.Or so I'm told, but this leaves me...
View ArticleSimplicial complexes on $[n] := \{0,\ldots,n\}$ that are identical under any...
For $n\in\mathbb{N}$, let us denote by $\Omega(n)$ the set of all (possibly empty) “abstract” simplicial complexes on $[n] := \{0,\ldots,n\}$ (“on $[n]$” means “labeled by the elements of $[n]$”). To...
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