Answer 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 ArticleComment by Gro-Tsen on Decomposition of symmetric powers of the fundamental...
For completeness of MathOverflow, compare with: this answer which is about the alternating powers of the same standard (=first fundamental) representation $V$.
View ArticleComment by Gro-Tsen on Is every recursively axiomatizable and consistent...
In case a reference is useful, I believe the following is relevant here: Hájek & Pudlák, Metamathematics of First-Order Arithmetic (available here), chapter I, theorem 4.27 (“Low Arithmetized...
View ArticleComment by Gro-Tsen on Existence of a special ordering of the elements of a...
I'm not too familiar with this, but if you consider the product of the $1+g+\cdots+g^{\textrm{ord}(g)-1}$ not over all elements of $G$ (in some order) but only over some of them (but still want the...
View ArticleComment by Gro-Tsen on $\omega\times\omega$-Hadamard matrices
Saying that lim.sup equals lim.inf is the same as saying that lim exists with that value. So your condition just says that $f(k)\cdot g(k)$ is summable with sum $0$. Which can't happen because...
View ArticleComment by Gro-Tsen on Who wins the Scrambler-Solver game for infinitary...
I think this question has great potential, but as it is currently phrased it is very confusing (although I admit it is rigorously defined). IIUC, as per @JoelDavidHamkins's comment, this isn't at all...
View ArticleComment by Gro-Tsen on The orders of the exceptional Weyl groups
I can't do it now as I'm on my phone, but did you check whether there are historical notes to Bourbaki's volume on Lie groups? Did you search for the number 696729600 in Cartan's papers? Did you check...
View ArticleComment by Gro-Tsen on Efficiently computing $\prod\limits_{i=1}^{n} A_i$
The lower-right entry seems to be Fibonacci for $k=0$, number of involutions for $k=1$, $n!$ for $k=2$, A167449 for $k=3$ (where no formula is given), and unknown in the OEIS for $k=4$. This does not...
View ArticleComment by Gro-Tsen on Efficiently computing $\prod_{i=1}^{n} A_i$
The lower-right entry seems to be Fibonacci for $k=0$, number of involutions for $k=1$, $n!$ for $k=2$, A167449 for $k=3$ (where no formula is given), and unknown in the OEIS for $k=4$. This does not...
View ArticleComment by Gro-Tsen on Stone-Čech compactification of a Boolean subalgebra of...
Thanks! This was exactly the explanation I needed. Both papers are quite interesting, and, crucially, I had missed the fact that when $B$ is countable it can be partitioned into countably many nonempty...
View ArticleComment by Gro-Tsen on Understanding the quotient of a free...
Indeed, the tags to the question (both “commutative algebra” and “noncommutative algebra”), the notation (brackets are normally used for polynomial rings) and the previous comment by OP (about elements...
View ArticleComment by Gro-Tsen on What is the least $\alpha$ such that $L_\alpha$...
Just to be clear, you answered the question “what is the smallest $α$ such that $L_α$ contains a subset of $2^ω$ which is not measurable in the universe ($V=L$)?”; another possible interpretation of...
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 Can we describe open cover compactness of a space in...
Saying that the evaluation map $X \to [0,1]^{\operatorname{Hom}(X,[0,1])}$ (where Hom refers to the set of continuous functions) is a closed subspace embedding seems like it has the same flavor as your...
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...
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