Algebraic definition or construction of real numbers Is there any algebraic definition or construction of real numbers ?
If not, why ?
 A: There is a strong sense in which the answer is no:
It's an informal concept, but we might reasonably say that an "algebraic" construction of the reals is one which only refers to algebraic operations - e.g., polynomials and their roots - and does not talk about more complicated things, such as arbitrary sets of rational numbers (e.g., Dedekind cuts and Cauchy sequences). In particular, we might ask for this construction to take place in first-order logic: https://en.wikipedia.org/wiki/First-order_logic.
This, we can prove, is impossible. Here's one way to say this:

There is a countable field $F$, with $\mathbb{Q}\subset F\subset\mathbb{R}$, such that $F$ satisfies every first-order sentence in the language of fields which is true in $\mathbb{R}$.

This is a direct consequence of the Lowenheim-Skolem Theorem, and it is very powerful and flexible; for instance, we can replace "the language of fields" with "the language of fields with exponentiation, $\sin$, $\cos$, and the Gamma function," and we could still find such an $F$. So no algebraic construction is guaranteed to get all of $\mathbb{R}$.
