Is $2x\geq x$ an axiom?

Question

From intuition we know $2x\geq x$ is true.

But can we really prove this or is it more of an axiom? If so, does it have a name or more general form?

Answer 1

Note that

$$2x\geq x\iff2x-x \ge 0\iff x\ge0$$

More in general for an ordered field $\mathbb{F}$ the following order axioms hold

• $\forall x,y\in \mathbb{F}$ exactly one is true $x>y, x=y, x<y$

• $\forall x,y,z\in \mathbb{F} \quad x<y \quad y<z\implies x<z$

• $\forall x,y,z\in \mathbb{F}\quad x>y \implies x+z>y+z$

• $\forall x,y,z\in \mathbb{F}\quad x>y \quad z>0 \implies xz>yz$

Answer 2

For $x\geq 0$ you have $2x-x=x\geq 0$ hence $2x\geq x$.

Answer 3

When the real numbers are constructed or specified by a set of axioms the assertion $$ 2x > x \text{ for positive } x $$ will be a theorem, easily proved from some more general and more useful axioms or theorems.

When you deal with numbers informally, starting in elementary school, it is indeed intuitive, and little more needs to be said,