Comment 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 ArticleConstruction of the smallest nucleus above a prenucleus: what does this proof...
While reading Hyland's paper on the effective topos [retyped version here] in the L. E. J. Brouwer Centenary Symposium, specifically prop. 16.3, I realized that the following proposition is...
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Answer by Gro-Tsen for Is there a natural bijection from $\mathbb{N}$ to...
Since the Calkin–Wilf tree has been mentioned in other replies to this question, I think the Stern-Brocot tree needs to be mentioned as well, because (although it is very much related, basically up to...
View ArticleComputing the Heyting operation on the frame of nuclei
(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound...
View ArticleComment by Gro-Tsen on A topos for realizability under a variable oracle
@FrançoisG.Dorais Yes, I believe it's very different: the topos I'm trying to describe is supposed to “collect” the realizability topoi over $\mathcal{K}_1^x$, where $x$ ranges over Baire space, but...
View ArticleComment by Gro-Tsen on Removing disks with maximal radius in a disk: must the...
@JeanAbouSamra Isn't it a classic saying that the best part of emmenthal is the holes?
View ArticleEffective (algorithmic) computation of the moduli space of algebraic curves...
Much literature (see, e.g., references here) has been written about the existence and “construction” of moduli spaces of algebraic spaces of a given genus — coarse or fine, smooth or stable, possibly...
View ArticleDo these nonlocal gadgets define a strict hierarchy?
Definition: For $0\leq p\leq 1$, a $p$-trigadget is a pair of boxes, each with three input buttons (say, $X,Y,Z$, only one of which can be pressed) and an light which can flash one of two colors (blue...
View ArticleReference request: a real-valued semicontinuous function on a Baire space is...
I thought the following result was well-known, but I can't seem to find it in any standard textbook on real analysis or general topology:Theorem (★). Let $X$ be a Baire topological space and $f\colon...
View ArticleHow can we characterize cardinals of the form $\lambda^{\aleph_0}$?
This question is a followup to another one concerning the Hamel basis cardinalities (i.e., dimension qua vector space) of Hilbert spaces. Nik Weaver proved in an answer that they are exactly the...
View ArticleWhat does “the” mean in “the first Kleene algebra”? (In what sense is it...
Definition:“The” first Kleene algebra $\mathcal{K}_1$ is the set $\mathbb{N}$ of natural numbers endowed with the partial operation $(p,n) \mapsto p\bullet n := \varphi_p(n)$ where $\varphi_p$ is the...
View ArticleTopological spaces in which positive and negative values of a real function...
(This question is about classical mathematics, despite the appearance of constructive mathematics terms for reasons explained below.)Definition: Say that a topological space $X$satisfies LLPO when, for...
View ArticleDoes analytic WLLPO together with sequential LLPO imply analytic LLPO?
This question is about constructive mathematics, without any form of Choice except Unique Choice, such as in the internal logic of a topos with natural numbers object, or in IZF. The “reals” (and the...
View ArticleAnswer by Gro-Tsen for What to do if paper was rejected with good report?
This happened to me once, and the editor of the editor whose editorial board took the decision to reject the paper offered to disclose the referee's identity to another journal if I chose to resubmit...
View ArticleComment by Gro-Tsen on Is every real number the limit of a sequence of...
@MohammadTahmasbizade The counterexample is easily modified to additionally satisfy $1\leq x\leq 2$, so it is not provable that a real in $[1,2]$ (or, a fortiori, a nonzero real) is the limit of a...
View ArticleAnswer by Gro-Tsen for Is every real number the limit of a sequence of...
This is more an extended comment than an answer (or at least it is a very partial answer, which I'm sure OP is already aware of), but because of the confusion such questions typically cause (as...
View ArticleAnswer by Gro-Tsen for Special rational numbers that appear as answers to...
According to this answer on this site, the area of the Pythagoras tree fractal (obtained by starting with a unit square, putting two smaller squares of size scaled down by $\sqrt{2}/2$ adjacent to one...
View ArticleWhy is it so hard to give examples of differentially closed fields?
The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability;...
View ArticleAnswer by Gro-Tsen for Open Sets in Hausdorff spaces
Here is a counterexample showing that it is not true in general that, for $X$ Hausdorff, every non-empty open set contains a closed set with non-empty interior.Let $X$ be the countable complement...
View ArticleAnswer by Gro-Tsen for Construction of the smallest nucleus above a...
Answering the first bullet point in my own question, I have found two references for this result, which provide at least some context. One is:Peter T. Johnstone, “Two Notes on Nuclei”, Order7 (1990)...
View ArticleSoftware for testing validity of propositional formulas in finite Kripke...
I havea formula $\varphi$ of propositional modal logic or propositional intuitionistic logic,a finite Kripke frame $W$,and I would like to test whether $\varphi$ is valid in $W$.This is an enumeration...
View ArticleComment by Gro-Tsen on Different forms of completeness of intuitionistic...
@EmilJeřábek Sanity check to be sure that I'm not even more confused than I thought I was: the argument for “it certainly holds … for arbitrary Heyting algebras” is that we take the set of all formulas...
View ArticleA 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 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 ArticleComment by Gro-Tsen on On the strength of ZFC with Replacement removed and an...
So your answer proves that $\Sigma^1$-Replacement does not imply axiom $A$ over ZC; but what about the other way around: does ZC + $A$ imply $\Sigma^1$-Replacement?
View ArticleAnswer by Gro-Tsen for Locally isomorphic graphs
Let $G_0$ be two pentagons ($5$-cycles) connected by an edge: say $V_0 = (\mathbb{Z}/5\mathbb{Z})\times\{0,1\}$ with $(i,p)$ connected to $(i-1,p)$ and $(i+1,p)$ except that $(0,p)$ is also connected...
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Answer by Gro-Tsen for Software for testing validity of propositional...
So I ended up writing my own program for the intuitionistic case. It can be found here on GitHub. As an example, I used it to generate this document tabulating the validity of a number of sample...
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 Is there an “opposite” hypothesis to the (Generalized)...
The MPccc scheme is indeed very natural and appealing, and the fact that it makes $2^{\aleph_0}$ very large without increasing the consistency strength of ZFC makes it an excellent answer to my...
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 ArticleComment by Gro-Tsen on Symmetry of the Riemannian logarithm
I don't even understand what the statement means: $\log_x y$ is an element of $T_x M$ (the tangent space at $x$) and $\log_y x$ is an element of $T_y M$. Do you mean to imply that you parallel...
View ArticleComment by Gro-Tsen on Why is it so difficult to define constructive...
Answer to a related question on MSE about the cardinality of the set $\Omega := \mathcal{P}(1)$ of truth values. (Beware that I use slightly different definition of “finite”, “subfinite”, etc., than in...
View ArticleComment by Gro-Tsen on Why is it so difficult to define constructive...
@WillSawin Even without CSB, “$X$ injects into $Y$” defines a preordering, and one might imagine redefining “cardinality” as equivalence classes not under “$X$ is in bijection with $Y$” as per Cantor,...
View ArticleComment by Gro-Tsen on Equivalence of definitions of locally finitely...
That $\mathcal{F}$ is “generated by the $s_i$” means precisely that the morphism $\mathcal{A}^{\oplus n} \to \mathcal{F}$ is surjective (see, e.g., EGA 0.5.1.1). This is often expressed, instead, as a...
View ArticleComment by Gro-Tsen on Configurations of injections and surjections (without...
Empty set aside, if there's an injection $f:A\to B$ then there's a surjection $B\to A$ by taking elements of the image of $f$ to their antecedent, and other elements to some fixed element of $A$. (Of...
View ArticleAnswer by Gro-Tsen for Equivalence of definitions of locally finitely...
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 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 \,...
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