Comment 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 Topological spaces in which countable intersections of...
I understand that the property “the intersection of a sequence of essential ideals is an essential ideal” is an algebraic reformulation of the topological condition I wrote down. But is it a standard...
View ArticleComment by Gro-Tsen on When are two proofs of the same theorem really...
@Jon23 I mean inclusive or ($\lor$).
View ArticleComment by Gro-Tsen on Isomorphism classes of finite $\mathbb{N}$-groups
I think there are two main ways to see such objects: either as a group together with a chosen endomorphism (which is something you said in an earlier revision, and I think you might have kept that part...
View ArticleComment by Gro-Tsen on Countably compact Boolean algebras versus distributivity
Just to be sure, “the canonical forcing which adds a Cohen subset to $ω_1$” is the complete Boolean algebra of regular open subsets of the topological space $\{0,1\}^{ω_1}$ with the product topology —...
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 How to define Dedekind reals and Eudoxus reals such...
There are at least two other definitions I can think of: namely the smallest subset of the Dedekind reals which contains the rationals and is closed under limits of unmodulated Cauchy sequences (resp....
View ArticleComment by Gro-Tsen on Dense subsets of the affine space
Even with the added assumption $V\neq\varnothing$, the field $\mathbb{R}$ still fails this: take $f(x_1,x_2) = -x_1^2-x_2^2$: this is a square only for $(x_1,x_2)=(0,0)$. But maybe you might want to...
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 ArticleComment by Gro-Tsen on When a null uncountable set can be image of some...
Just as an illustrative example of what we can do, consider the Cantor staircase function $f$ and let $g$ be its “inverse” where for each dyadic $r$ (i.e., every “step” of the staircase) we let $g(r)$...
View ArticleComment by Gro-Tsen on Why are extremally disconnected spaces so hard to give...
@JosephVanName I added at the end of your answer a number of remarks clarifying a few minor points and a proof of the proposition you stated: could you review this before I approve your answer?
View ArticleComment by Gro-Tsen on Dividing a square into 5 equal squares
This is the same answer as Douglas Zare's.
View ArticleComment by Gro-Tsen on Real matrix rings and associative hypercomplex numbers
So your question is “does any finite-dimensional (unitary, associative) $\mathbb{R}$-algebra admit a basis of elements whose squares are in $\{-1,0,1\}$?”, is it? Because if this is it, then it would...
View ArticleComment by Gro-Tsen on Real matrix rings and associative hypercomplex numbers
This example is nice because it even refutes the weaker form of the question, “does any finite-dimensional (unitary, associative) $\mathbb{R}$-algebra admit a generating set (as an algebra) whose...
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}...
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