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I have seen how the natural, integer and rational numbers are constructed. These relate it through injective functions with which $\mathbb N\subset \mathbb Z \subset \mathbb Q$ can be considered, from a practical point of view it has many advantages (it simplifies things for us). But if we are rigorous, it is not true that $\mathbb N\subset \mathbb Z \subset \mathbb Q$ (speaking in a set-wise way). Is there any downside to not considering "$\mathbb N\subset \mathbb Z \subset \mathbb Q$" in a practical way (besides tedious, obviously)?

Jose Avilez
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    The downside is that people will look at you funny if you don't consider ${\bf N}\subset{\bf Z}\subset{\bf Q}$. – Gerry Myerson Jan 22 '22 at 22:09
  • @VeryForgetfulFunctor I refer to the construction from the "Peano axioms" (natural), then from the integer domains (integers), and finally from the field of quotients (rational). – ops Jan 22 '22 at 22:11
  • @Very, I suppose OP means that if you define, say, the rationals to be the set of ordered pairs of integers under a certain equivalence relation, then technically the integers are not a subset of the rationals (though they can be identified with an appropriate subset of the rationals). – Gerry Myerson Jan 22 '22 at 22:11
  • To summarize the previous comments, there doesn't seem to be anything you lose by associating $\mathbb N$ with a subset of $\mathbb Z$, or $\mathbb Z$ with a subset of $\mathbb Q$. – Rushabh Mehta Jan 22 '22 at 22:15

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