Free Abelian Groups .... Aluffi Proposition 5.6

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In summary: A## is a set, and let $$j: A \longrightarrow \mathbb{Z}$$ be the natural function given by sending elements of ##A## to the function ##j_a: A \longrightarrow \mathbb{Z}##.Then, for any two elements ##a, b## of ##A##, the function ##j## associates to ##a## the element ##j_a(a)##, which is the element obtained by sending ##a## to the function ##j_a##; and likewise for ##b##.In summary, fresh_42's first definition states that the function ##j## associates to each element of ##
  • #36
mathwonk said:
mike is emil's son. (artin)
... and a prof of mine once said: A mathematician's talent is transmitted to his son-in-law :wink:
 
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  • #37
fresh_42 said:
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.
 
  • #38
lavinia said:
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).
 
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  • #39
@fresh. well John Tate was Emil Artin's son in law, having married Karin Artin, so maybe.
 
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  • #40
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}##
 
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  • #41
micromass said:
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.
 
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