Free Abelian Groups .... Aluffi Proposition 5.6

[fresh_42]

mike is emil's son. (artin)

... and a prof of mine once said: A mathematician's talent is transmitted to his son-in-law

 

[lavinia]

Let $C = ℝ$ be the continuum and basis of the free abelian group $F$. Further let $ι : C → F$ be the embedding (of the alphabet) and $id: C → ℝ_+$ the identity. Then there is a unique group homomorphism $π: F → ℝ_+$ which extends $id$, i.e. $id = π \, ι$ because $F$ is free. $π$ is onto. The question is what kernel $π$ has or what makes $4 + 5 = 9$ since $4 + 5$ and $9$ are different elements in $F$. The Dedekind cuts were what occurs to me. Maybe there is a more algebraic way to define the equivalence classes.

It seems to me that Dedekind cuts assume that you already know what addition is.

 

[fresh_42]

It seems to me that Dedekind cuts assume that you already know what addition is.

I thought one needs the order in ℝ and the embedding of ℚ. But you are right, this breaks the definition requirements only down on ℚ.
So we are left with the interesting question: What is addition? Why are 4+5 and 9 equivalent? I have to think about it. I never really lost a thought on it. (PF is an ever lasting fount of challenges ...)

Edit: How about defining $(a,0) \sim a$ for all $a \in ℝ$ and then $(a,b) \sim c ⇔ \nexists d : ( (a,b) \sim d)$ proceeding by transfinite induction on the well-ordering of $ℝ$? I'm almost certain that it cant't be done without the axiom and the ordering. But I have to admit that I'm no logician.

Edit2: Perhaps some more care with the definition is needed which I consider a technical issue (well definition, commutativity, inverse).

 

[mathwonk]

@fresh. well John Tate was Emil Artin's son in law, having married Karin Artin, so maybe.

 

[micromass]

As for the addition structure on $\mathbb{R}$. It might help to look at $\mathbb{R}$ as a $\mathbb{Q}$-vector space of dimension $2^{\aleph_0}$. So we could see it as something like $\bigoplus_{2^{\aleph_0}~\text{factors}} \mathbb{Q}$

 

[lavinia]

As for the addition structure on $\mathbb{R}$. It might help to look at $\mathbb{R}$ as a $\mathbb{Q}$-vector space of dimension $2^{\aleph_0}$. So we could see it as something like $\bigoplus_{2^{\aleph_0}~\text{factors}} \mathbb{Q}$

Right. Then one needs the structure of the rationals.