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So, oftentimes it is said that $\mathbb{Q} \subset \mathbb{R}$, but how is this the case when $\mathbb{R}$ is constructed out of subsets of $\mathbb{Q}$? Would it not be better said, "the representations of the elements of $\mathbb{Q}$ in $\mathbb{R}$ are a subset of $\mathbb{R}$," which is obvious?

mineben256
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    If you prefer ,you can just say that there is a natural injection of $\mathbb Q$ into $\mathbb R$, which is compatible with the arithmetic sructures. – lulu Oct 09 '21 at 17:37
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    Such things happen all the time in mathematics. Similarly, you can say that $\mathbb{Z}$ is not a subset of $\mathbb{Q}$, just that $\mathbb{Q}$ contains a subset which is basically a copy of $\mathbb{Z}$. Thing is, the sets are the same up to naturally changing the names of the elements, and have the same properties. So we just identify them. – Mark Oct 09 '21 at 17:40
  • You are correct. The same is true when we say $\mathbb R\subset \mathbb C$. – John Douma Oct 09 '21 at 17:44
  • The right way to think about $\mathbb{R}$ is as the completion of $\mathbb{Q}$. Informally, this is akin to filling in the gaps between rational numbers, and this is why we usually say that $\mathbb{Q}\subset\mathbb{R}$, but what we mean is that we can identify $\mathbb{Q}$ with a subset of $\mathbb{R}$. – Saegusa Oct 09 '21 at 17:48
  • Technically yes, but I think the point is that algebraic structures like $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$, etc are more usefully identified by their algebraic souls than by the specific appearance of their elements. So saying $\mathbb{Q} \subset \mathbb{R}$ really just means that the algebraic soul of $\mathbb{Q}$ is canonically reflected in that of $\mathbb{R}$. However, at some point it saves more time to stop interpreting $\mathbb{Q} \subset \mathbb{R}$ as a suggestion and start interpreting it literally. – joeb Oct 09 '21 at 18:02
  • I dunno, are you a platonist or a formalist? :-) – Brian Tung Oct 09 '21 at 18:20

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I recommend regarding the construction of the real numbers from the natural numbers a proof of the existence of a complete ordered field (based on the axioms of set theory) - not a definition of the real numbers. The real numbers are defined up to a natural isomorphism by the characterization as a complete ordered field, see this answer.

Filippo
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