How does one define a value of a number? What is the value of the number 4? Asked differently, how does one show that a certain number is greater than another number?
After this, one might ask how do you quantify a real number?
This is a bit philosophical, but questions I wondered how math gives a rigorous answer to
On
Brief outline:
For natural numbers, we have operation "next", often written as $'$: $1' = 2, 3''=5$ and so on (or, rather, symbol $5$ is short for $1''''$). From that, we have well-known operation $+$, and for natural numbers, $a \ge b$ is, by definition, $\exists c: a=b+c$.
Rational numbers are defined as ordered pairs $(p,q)$, where $(p_1,q_1) \ge (p_2,q_2)$ is $(p_1*q_2 \ge p_2*q_1)$ ($p_i$ are integral, not natural, but order of integer numbers is rather obvious).
Real numbers can be defined differently (those different definitions produce essentially the same object). One way is this: for rational numbers $\mathbb{Q}$, let $A \subset \mathbb{Q}, B=\mathbb{Q} \setminus A, A \neq \emptyset, B \neq \emptyset$ and $\forall a \in A, b \in B, a \ge b$ (note that this is "rational" $\ge$ which we have defined). We can define real numbers as such pairs $(A,B)$, and $(A_1,B_1) \ge (A_2,B_2)$ is $(A_1 \subseteq A_2)$.
This depends entirely on the number itself and is primarely done through number construction, we construct all the numbers and through it we decide what constitute an equality and inequality. This is an excellent document showing for natural numbers to rational numbers, the real numbers is a bit lacking but it gives a good view of how it is done.