TITLE: Construction of $\Bbb R$ from $\Bbb Q$
QUESTION [6 upvotes]: As it is true that we can construct all rational numbers $\Bbb Q$ from the set of integers $\Bbb Z$, is it possible to construct the set of real numbers $\Bbb R$ from $\Bbb Q$? If yes, how? Is there any procedure? And if no, is there any proof that we can't ? Thanks!

REPLY [3 votes]: To complete the list from Spivak, we can also define the reals 'naively' as infinite decimal sequences, but this is essentially picking a representative element from an equivalence class of Cauchy sequences.
(eg $\pi$ should be the limit of the sequence 3, 3.1, 3.14, 3.141, ...)