Q:

How can we define the "larger than" (or "smaller than") relationship between two natural numbers without taking into account set theory?

Since set theory is proven to be inconsistent by Russell, what is the method mathematicians use to define the equality, the "greater than" and "the smaller than"relationships between two natural numbers without taking the cardinalities of sets into account?
For instance, I could state that a set containing 2 elements is smaller than a set containing 3 of them because there is no surjective function that has the latter set as its domain and the former as its image. 
How can I accomplish a solid definition of these relationships without set theory?

A:

If you're speaking in the language of Peano arithmetic, you can define $a < b$ recursively as:

$0 < S(0)$
$\forall x, y : (x < y) \to (x < S(y))$
$\forall x, y : (x < y) \to (S(x) < S(y))$

where $S(x)$, the successor of $x$, can be read as "$x+1$".

