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I would like to know if anyone has a good and rigorous reference for the following constructions:

  • construction of $\Bbb R$ as a simply ordered field and as a linear continuum (Dedekind Property or Least Upper Bound Property);
  • construction of $\Bbb R$ as the completion of $\Bbb Q$;
  • construction of real-exponent powers (i.e.: $x^{\alpha}, x > 0, \alpha \in \Bbb R$), preferably without using the exponential function.

The motivation behind this request is that I have familiarized with most of these constructions because they were quoted or partly left as exercises on books I have read, however I have never felt satisfied with them.

As always any comment or answer is well-accepted and let me know if I can explain myself clearer!

Matteo Menghini
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  • At the end of the first chapter of Walter Rudin's Principles of Mathematical Analysis there is an appendix that constructs $\mathbb{R}$ from $\mathbb{Q}$ – Torrente Aug 25 '22 at 10:28
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    For some textbooks books that are primarily devoted to the construction of the real numbers, see my answer to Is the real number structure unique? For references (papers and books) that deal with the history of this topic and/or various methods for carrying out the construction, see my answer to History of the construction of $\mathbb{R}$. – Dave L. Renfro Aug 25 '22 at 10:33
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    https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/anal1v.pdf this book constructs $\mathbb{R}$ from $\mathbb{Q}$ using Cauchy sequences. It also defines real powers, though using the exponential function. It seems that constructing a continuous power function and constructing the exponential function are equivalent tasks, and since the exponential works in greater generality, I guess it is presented. – Mason Aug 26 '22 at 01:47

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