Comment by Gro-Tsen on Hilbert's Satz 90 for real simply-connected groups?
I'm not sure whether I misunderstood your question, but ⓐ to be clear, $\mathbb{R}$ does not satisfy the condition of having cohomological dimension $≤2$ (since its Galois group is finite cyclic...
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 Journal name abbreviations
I am flabbergasted to learn that there is an ISO standard for abbreviating journal titles. (On the other hand, there is an ISO standard for brewing tea, so maybe I shouldn't be so surprised.) But given...
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 ArticleComment by Gro-Tsen on Smooth analogue of Cartan's Theorem B
My memory may serve me incorrectly, but if I recall correctly, the real-valued $C^\infty$ functions on a (paracompact) $C^\infty$ manifold $X$ form a “fine” sheaf $\mathscr{F}$ (“fine” means any...
View ArticleComment by Gro-Tsen on Condition to guarantee that an inhabited and bounded...
I'm not sure whether this is supposed to be a comment or whether you are claiming to answer my question. In the latter case, could you make a mathematically precise statement as to what you are claiming?
View ArticleComment by Gro-Tsen on Linear independence in $\mathbb{Z}_q^n$
For $n=1$, there are $\varphi(q)$ elements of $\mathbb{Z}/q\mathbb{Z}$ that are (each, separately) “linearly independent”, so to find a single vector you need $q-\varphi(q)+1$ vectors. This does not...
View ArticleComment by Gro-Tsen on What do we know about the computable surreal numbers?
Is it not clear that if you call $(r_n)$ the dyadic rational whose $k$-th binary digit is $1$ when $k≤n$ and the $k$-th Turing machine halts in less than $n$ steps, and $0$ otherwise, thus defining a...
View ArticleComment by Gro-Tsen on Condition to guarantee that an inhabited and bounded...
This is, indeed, a fairly natural property that is probably the best we can get, and it is very strong. As you point out, for $S = \mathbb{N}$, it implies LPO. (Proof: consider $(b_n)$ a binary...
View ArticleComment by Gro-Tsen on Field extensions over which algebraic varieties cannot...
@EricTowers The spelling “viꝫ”has an entry in Wiktionary; the character ‘ꝫ’ (Unicode U+A76B LATIN SMALL LETTER ET, see here) is a medieval abbreviation mark for “something ending in -et or -est”: so...
View ArticleComment by Gro-Tsen on Is the Tarski–Seidenberg theorem constructively provable?
May I suggest that before we even attempt to think about the (likely very delicate) question of constructive quantifier elimination for the reals and/or for real-closed field (and to what extent...
View ArticleComment by Gro-Tsen on Which are the hereditarily computably enumerable sets?
If the program $e$ enumerates $e$ (and only $e$, say), do you consider $x_e$ to be undefined, or do you allow this to represent a set such that $x=\{x\}$ depending on if there is one (such as under...
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 ArticleComment by Gro-Tsen on How can I calculate $\chi(\mathscr{O}(P))$
To clarify how to derive the result given in Jason Starr's comment, write down the additivity of $\chi$ for the short exact sequence $0 \to \mathcal{O}_{\bar X}(P) \to \mathcal{O}_{\bar X} \to...
View ArticleComment by Gro-Tsen on Sets of algebraic integers whose differences are units
For those, like me, who forgot why this is true: writing $\zeta_n$ for a primitive $n$-th root of unity, $1-\zeta_n$ is a unit in $\mathbb{Z}[\zeta_n]$ when $n$ has at least two distinct prime factors...
View ArticleComment by Gro-Tsen on Is there an infinite subset of $\Bbb{R}$ not...
I'm confused about the claim “then $f_α$ continuously extends to an injection on $\mathbb{R}\setminus Y_α$”: your assumption is that for each $x \not\in X_α \cup Y_α$ separately, $f_α$ continuously and...
View ArticleComment by Gro-Tsen on Can exist a positive integer number $x$ such that...
I understand the proof, but I'm confused as to what “makes it work”. How difficult is it to find $n$ such that we can cover the residue classes mod $n$ by arithmetic sequences with difference $r_i$...
View ArticleComment by Gro-Tsen on Equivalence of omniscience principles for natural...
By a “Cauchy sequence” $(u_n)$, do you mean a Cauchy-with-modulus sequence $∃μ:\mathbb{N}\to\mathbb{N}.∀k.∀p,q≥μ(k).|u_p-u_q|<2^{-k}$ or a Cauchy-without-modulus sequence...
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);...
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