0

I would like to get a book on set theory that presents the construction of number systems so that one really has: $\Bbb N$ contained in $\Bbb Z$, $\Bbb Z$ contained in $\Bbb Q$, $\Bbb Q$ contained in $\Bbb R$, $\Bbb R$ contained in $\Bbb C$. (As usual, $\Bbb N$ denotes the set of natural numbers, $\Bbb Z$ the set of integers, $\Bbb Q$ the set of rational numbers, $\Bbb R$ the set of real numbers, $\Bbb C$ the set of complex numbers.) It seems to me that such a construction is presented in Solomon Feferman's Number Systems - Foundations of Algebra and Analysis. I ask for help.

Heptapod
  • 1,123
Paulo Argolo
  • 4,210
  • 3
    Um, if you already has the title of a book that does it in the way you prefer, then what is your question here? – hmakholm left over Monica Oct 14 '18 at 23:52
  • 2
    It seems to me that, rather than manipulating the definitions of number systems to get each to be literally a subset of the next, it would be more useful to prove a general theorem saying that, if $A$ is isomorphic to a substructure of $B$, then $A$ is literally a substructure of an isomorphic copy of $B$. Then apply that theorem iteratively to the number systems you listed. (Caution: The obvious "proof" of the theorem, namely to replace in $B$ each element of the copy of $A$ with the corresponding element of $A$, is not in general correct, but easy to repair.) – Andreas Blass Oct 15 '18 at 00:52
  • @Henning Makholm: I'm not sure if the book I mentioned brings the building I want to see. – Paulo Argolo Oct 15 '18 at 01:03
  • @Andreas Blass: Many thanks for the suggestion. – Paulo Argolo Oct 15 '18 at 01:05
  • 1
    FWIW, this answer of mine shows a bit more of how @Andreas's suggestion would go -- though not explicitly phrased as a general theorem. – hmakholm left over Monica Oct 15 '18 at 01:08
  • @Henning Makholm: Thank you very much! – Paulo Argolo Oct 15 '18 at 01:42

0 Answers0