Proving order in real number set

Question

Can we prove such a statement, or is it axiomatic ?

$$ \forall (x,y,z) \in \mathbb{R}^3 ∶(x \leq y)∧(y \leq z)⇒(x \leq z) $$

Answer 1

The relation $\leq$ on the set of real numbers is the paradigmatic relation defining a partial order, because it is reflexive, antisymmetric, and transitive. (Indeed, the relation $\leq$ on $\mathbb R$ is a total order.)

The property you post asserts that $\leq$ is transitive on $\mathbb R$.

See also: Real numbers, axiomatic approach.

Answer 2

That depends on which definitions and axioms you're working from.

In the most common way to develop real analysis, we work in a setting where the basic properties of real numbers are not proved, but just assumed. Among the things we assume is that there is an ordering on the reals that satisfies certain laws such as the one you quote. During this development of real analysis, these assumptions are axioms.

Later one may get to prove, using set theory, that there actually exists a set with operations that satisfy the axioms. Afterwards we say that the set we've constructed "is" the real numbers. As part of this existence proof, we of course need to prove from more basic principles that the particular relation on our set we claim works as the ordering satsifies the transitive law.

What the particular connection between the two paragraphs above is, is to a certain extent a matter of temperament and philosophy. It is common to say that what counts as "axioms" in elementary real analysis are effectively just promises that "we will prove this later, don't worry about it now". On the other hand, it may arguably be a better description of how (many) mathematicians actually think to say that the real numbers exist in and of themselves, and that the axioms of real analysis really are axioms; we accept them because they are "obvious". In this view, the point of constructing the real numbers using set theory is not to tell us anything about them, but just to construct a set-theoretic model of the real numbers, and thereby show that set theory can work as a basis for ordinary mathematical reasoning.