What can be written instead of the supremum axioms to get same structure in the set of real numbers Question goes as follows : "What can be written instead of the supremum axioms, to get the same structure in $\mathbb{R}$."
Now for the supremum axioms I have :
If $K_a \neq Ø \Rightarrow \exists$ a smallest upper bound
$ \exists min$ $K_a \iff \exists \alpha \in K_a$ $\forall k \in K_a : \alpha \leq k$
$ \alpha = sup(A) $ ergo the least upper bound.


*

*$ \alpha$ is upper bound, $ \forall a \in A : a \leq \alpha$

*$ \alpha$ is the smallest/least upper bound, $ \forall \epsilon \gt 0 $ $\exists  a \in A : a \gt \alpha - \epsilon$


Now my question is what can I write instead of these axioms. I was thinking some kind of isomorphism to preserve the structure, but I am not sure.
 A: If you're hoping for an alternate axiom which is "purely algebraic" - that is, avoids talking about arbitrary sets of reals, and is closer to (say) the field axioms - then there is a sense in which no such axiom exists: specifically, any first-order axioms will fail to capture the real numbers, in the following sense. If $T$ is a set of first-order sentences which are true about the structure $(\mathbb{R}; +, \times, <)$, then


*

*There will be models of $T$ which are "too big", e.g. have cardinality much greater than that of $\mathbb{R}$. In particular, there will be models which are non-Archimedean (an ordered field is Archimedean if for any $x>0$, there is some natural number $n$ such that $1+1+...+1$ ($n$ times) is $>x$).

*There will be models of $T$ which are "too small," e.g. are countable. Some of these will be non-Archimedean, but we'll also have proper substructures of the reals satisfying $T$.
This is due to the compactness and Lowenheim-Skolem theorems of first-order logic, respectively.

That said, we can certainly avoid sups and infs! $\mathbb{R}$ is a maximal Archimedean ordered field; and any two maximal Archimedean ordered fields are isomorphic. Note, however, that this is non-first-order in two ways:


*

*We have to say that $\mathbb{R}$ is Archimedean.

*We have to say that any proper ordered field extension of $\mathbb{R}$ is not Archimedean.
There is a precise sense in which this second sentence is "even less first-order" than the first; but that's getting a bit off-track. If you're interested in these sorts of things, you might look at abstract model theory.
