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I'm a student of mathematics in Germany.


Dec
10
answered Let $R=M_n(D)$, $D$ is a division ring. Prove that every $R-$simple module is isomorphic to each other.
Dec
8
comment Why is Mobius group denoted by $Aut(\hat{\mathbb{C}})$
$\mathrm{Aut}(\hat ℂ) = \{f \colon \hat ℂ → \hat ℂ;~\text{$f$ is biholomorph}\}$, if I remember correctly. Is it this you are looking for?
Dec
8
awarded  Caucus
Dec
8
comment What is the best path to learn Category theory and Type thoery?
I wouldn’t know. I don’t know Awodey.
Dec
7
comment What is the best path to learn Category theory and Type thoery?
Have you already had a look at the catsters videos on category theory?
Dec
7
comment What is the best path to learn Category theory and Type thoery?
Learning category theory will be tough if you don’t have a reasonable background in abstract algebra. Perhaps one can compensate this with a good understanding of functional programming? I would start off by learning the basics of Haskell (if you haven’t done this before) and learn about fundamental algebraic structures like monoids, rings and groups. Once you are used to these concepts, moving on to category theory will be much less tedious. (Also, I woudn’t focus too much on the set-theoretic aspects of algebra because that could lead you astray.)
Dec
7
comment Why are all the interesting constants so small?
@user1708 “everything there is to say abot this number” How would you possibly know?
Dec
6
comment showing the natural numbers exist from axioms (help with making sense of book)
@AlecTeal Okay, well, good luck with this then. That’s all of the time I’m investing in this question. I’m tired and I actually don’t have much time this weekend.
Dec
6
revised showing the natural numbers exist from axioms (help with making sense of book)
corrected mistakes, removed typos
Dec
6
answered showing the natural numbers exist from axioms (help with making sense of book)
Dec
6
comment showing the natural numbers exist from axioms (help with making sense of book)
@AlecTeal What is your suggestion for the inductive set you get, something like $B = \{x; S(x) ∈ B\}$? This would be unrestricted comprehension (and I’m not sure this would be a ‘legal’ unrestricted comprehension either). You can only do restricted comprehension, that is something like $B = \{x; x ∈ A ∧ S(x) ∈ B\}$ for some set $A$ (and again I’m not sure if this is a legal comprehension – I know very little about axiomatic set theory). So what’s your set $A$? Or do you want to something entirely different? (Actually, I’m pretty certain these comprehensions are not legal.)
Dec
6
revised showing the natural numbers exist from axioms (help with making sense of book)
clarified lemma, corrected tex
Dec
6
comment showing the natural numbers exist from axioms (help with making sense of book)
The empty set and the axiom of union only give you finite sets, as you are only “allowed” to use the axioms finitely many times to create sets. Comprehensions only let you pick subsets of a set for a given property. None of this allows you to actually create an inductive set. Can you clarify the lemma – does it say that, if there is an inductive set $I$ such that $ℕ ⊂ I$, then $ℕ$ is inductive, too?
Dec
5
comment Proof about direct sum of vector spaces
If there is some $v ∈ V$ with $v = w_1 + … + w_n$ and $v = w_1' + … + w_n'$, what’s $v - v$?
Dec
5
comment Proof about direct sum of vector spaces
Yes, and furthermore $v = \sum_{i ∈ I} w_i$ for some $w_i ∈ W_i$ and $v = \sum_{j ∈ J} w_j$ for some $w_j ∈ W_j$. Therefore $0 = v - v = \sum_{i ∈ I} w_i - \sum_{j ∈ J} w_j$. You now use $I ∩ J = ∅$ to conclude that this is really a sum of the needed form. (You even need to know that there is a $w_i ≠ 0$ and a $w_j ≠ 0$ in the sum.)
Dec
5
revised Proof about direct sum of vector spaces
added 16 characters in body
Dec
5
answered Proof about direct sum of vector spaces
Dec
5
comment Proof about direct sum of vector spaces
@AlonAlon … such that …? Finish the following sentence: “For subspaces $W_1, …, W_n$ of $V$, we say $V$ is the internal direct sum of $W_1, …, W_n$ – written $V = W_1 \oplus … \oplus W_n$ –, if …”
Dec
5
comment Proof about direct sum of vector spaces
Also, what is your definition for $W_1, …, W_n$ yielding an internal direct sum of $V$?
Dec
4
comment Let $G$ be a group. If $H\leq G$ is a subgroup and $N\vartriangleleft G$, then $HN$ is a subgroup of $G$.
A better title might be “On proving that $HN$ is a subgroup of $G$, for $H ⊂ G$ and $N ⊂ G$ normal”.