Comment by Gro-Tsen on Preimage of a sublocale by a morphism of locales:...
I should note that I'm not necessarily looking for a formula that is a finite expression in the alphabet $\land,\lor,\Rrightarrow,j,f^*,f_*$. For example, the formulas given in this answer to another...
View ArticleComment by Gro-Tsen on Does the support of a smooth compactly supported...
Regarding the claim that every closed set is the zero-set of a smooth function, the answers to this question on MSE (or alternatively this one) may be simpler to understand than the very general answer...
View ArticleAnswer by Gro-Tsen for How slow can an uncomputable function from...
Here's one possible interpretation of your question, I don't know if this is what you're after:Proposition. For every nondecreasing $b\colon\mathbb{N}\to\mathbb{N}$ such that $b(n) \to +\infty$ as $n...
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...
View ArticleComment by Gro-Tsen on Mathematical strength of the statement "Heyting...
Slightly related question about the strength of the statement “Heyting arithmetic has the existence property”.
View ArticleComment by Gro-Tsen on How much of mathematical General Relativity depends on...
Cross-referencing this question (+the comments below the question) which explains why any $\Pi^1_4$ statement of second-order arithmetic that is provable from ZFC is provable from ZF alone. (I.e., ZFC...
View ArticleComment by Gro-Tsen on Is there an irrational number which in every base, has...
@AnuragSahay One major difference is that the question you link to requires that every digit in base $3$ be in $\{0,2\}$ and every digit in base $4$ be in $\{0,3\}$ (and apparently there are even very...
View ArticleAnswer by Gro-Tsen for Where does one learn about the weather?
I have myself only glanced through it, but I believe the book An Introduction to Dynamic Meteorology by James R. Holton (4th edition 2004) is a standard reference in the domain. It's more on the...
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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 ArticleComment by Gro-Tsen on Poincaré and the principle of induction
@JeanAbouSamra You might want to check out Petr Hájek & Pavel Pudlák, Metamathematics of First-Order Arithmetic (1998) (freely available online), chapter I, theorem 4.27 (“low arithmetized...
View ArticleComment by Gro-Tsen on Is it possible to discover a "Universal formula" that...
This answers reminds me of the following — probably apocryphal — story. When the MRDP theorem was proved, some great Soviet mathematician (maybe Kolmogorov?) was told about it in the form “there is a...
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 curious square relating models of hyperbolic space
We can do something similar for the 3-sphere $\mathbb{S}^3$: taking a point in $\mathbb{S}^3$, we take its gnomonic projection to $\mathbb{R}^3$, and we interpret it as the stereographic projection of...
View ArticleComment by Gro-Tsen on What was Gill and Ladner’s joke about P=NP?
Maybe the point is that if you know the joke you can check very fast that it's funny, but if you don't know it, it might take a really long time to find. 🙃
View ArticleComment by Gro-Tsen on Could we allow the zero ring to be local?
The geometrical motivation is that a local ring describes what happens “locally around a point” (where “locally around” is to be understood in the sense of the Zariski topology). The zero ring...
View ArticleComment by Gro-Tsen on Diaconescu's theorem in type theory without...
Hyland's paper “Variations on realizability: realizing the propositional axiom of choice”, Math. Struct. in Comp. Science12 (2002) 295–317, which tries to construct various models of constructive math...
View ArticleComment by Gro-Tsen on Why are quotients so much trickier in constructive...
I'm not sure at what level of formality you want an answer, but an informal and handwavy one would be something like this: an equivalence class of $S$ under $E$ is a subset $C\subseteq S$ such that any...
View ArticleComment by Gro-Tsen on On Rational $\bar{\mathbb{Q}}$-variety
The following trick is folklore: choosing embeddings, a birational map, and whatever else might be necessary to witness that $X_{\mathbb{C}}$ is rational, the conditions expressing this fact (for these...
View ArticleComment by Gro-Tsen on What's the deal with predicative foundations?
Meanwhile, this question which is what about predicativism even means (and how it relates to constructivism) remains unanswered. ❧ See also this answer about the system proposed by Hermann Weyl and...
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...
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