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I am aware of three different constructions of the field of real numbers :

  1. The Cauchy sequence construction : in this case, we see the field $\mathbb{Q}$ as a metric space and $\mathbb{R}$ is the completion of $\mathbb{Q}$ for the standard metric on $\mathbb{Q}$.

  2. The Dedekind cut construction : in this case, the field $\mathbb{Q}$ is seen as a partially ordered set (with the standard order), and $\mathbb{R}$ is the completion of $\mathbb{Q}$ in the sense of the smallest complete lattice containing $\mathbb{Q}$.

  3. The "Eudoxus" reals : in this case, we don't start from $\mathbb{Q}$ but directly from $\mathbb{Z}$. Reals numbers are identified as equivalent classes of "almost-homomorphisms" from $\mathbb{Z}$ into $\mathbb{Z}$ (functions $f$ from $\mathbb{Z}$ into $\mathbb{Z}$ such that $\{f(m+n)-f(m)-f(n): m, n\in\mathbb{Z}\}$ is finite and two almost-homomorphisms $f$, $g$ are equivalent iff the set $\{f(m)-g(m): m\in\mathbb{Z}\}$ is finite. As far as I understand, only the additive group structure on $\mathbb{Z}$ is used.

My question is this : We see that the first two constructions are actually just specific examples of application of a general process (metric completion, order completion). What is the "completion" process associated to the Eudoxus real construction (if there is one)? Could it be applied to other abelian groups, was is already explored ?

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    $\begingroup$ (I originally accidentally posted this as an answer:) I believe the appendix to arxiv.org/pdf/math/0405454v1.pdf is relevant: given two abelian groups $G$ and $H$, there is a natural "Eudoxus construction" $\mathbb{E}(G, H)$ coming from the set of quasihomomorphisms from $G$ to $H$. For example, $\mathbb{E}(\mathbb{Z}^n, \mathbb{Z}^n)$ is just $M_n(\mathbb{R})$. So this confirms your guess that the construction can be substantially generalized. $\endgroup$ May 27, 2016 at 18:18
  • $\begingroup$ Doesn't Gromov have some construction that associates a space to a finitely generated group (maybe with presentation), in some sense completing its Cayley graph? I am no expert here, just hoping to jog other people's memories. $\endgroup$ May 27, 2016 at 18:32
  • $\begingroup$ You can also define real numbers as infinite decimal expansions (that do not contain infinitely many consecutive 9's). (This is not directly related to the question.) $\endgroup$ May 29, 2016 at 3:56
  • $\begingroup$ Behrend's paper A Contribution to the Theory of Magnitudes and the Foundations of Analysis. Mathematische Zeitschrift, 63:345–362, 1956 is particularly relevant to Gabriel's comment. Behrend gives the lie to the traditional objection that you can't define how to add infinite decimal expansions explicilty. There are also many other much more wacky constructions of the reals (I am guilty of one such, which starts with an analysis of the order structure of rings of quadratic integers.) $\endgroup$
    – Rob Arthan
    Jul 23, 2020 at 23:10

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This method can be used to construct the fields $\mathbb{Q}_p$ and the ring $\mathbb{A}_{\mathbb{Q}}$ of adeles over $\mathbb{Q}$. See T.D.J. Hermans' Bachelor's thesis: https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/bachelor/2017-2018/hermans-bscthesis.pdf. In particular, see Proposition 2.4 and Theorem 2.11.

In her Master's thesis (not available online, unfortunately), entitled "The abelian category of abelian groups and quasi-homomorphisms," Hermans undertakes a study of the category in the title of the thesis.

I learned of this fascinating approach to completions when Hendrik Lenstra gave a talk on it at a conference honoring Alice Silverberg: https://www.math.uci.edu/node/30659

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