TITLE: Is there a definition for the"size" of a number?
QUESTION [6 upvotes]: To clarify my question, here is an example:
We know that $1 < 2$  or $2 > 1$ (one is less than two, or, two is greater than one), but is this only based on our intuition or natural understanding of "size"?
To ask it in a different way, what does it mean for a number to be greater/less than another? Is there a way or method for establishing this, or is it, as mentioned above, just common sense (based on the definition of a number)? 
It may be silly to ask such a question, but it is worth the clarification nonetheless.

REPLY [2 votes]: Is there a way or method for establishing this, or is it, as mentioned above, just common sense (based on the definition of a number)?

From mathematics point of view there is very formal and precise method for establishing this. First of all you need to know what Zermelo-Fraenkel set theory is (and indeed ZF axioms are sort of arbitrary but they play well with our intuition) and then you can precisely define natural numbers:
$$0:=\emptyset$$
$$1:=\{\emptyset\}$$
$$2:=\{\emptyset, \{\emptyset\}\}$$
$$\vdots$$
$$n+1:=n\cup\{n\}$$
Now there is a natural ordering of sets: $X< Y$ if and only if there exists and injection $X\to Y$ but there is no injection $Y\to X$. This ordering restricted to naturals is what you are refering to. It captures the essence of "size".
For example if $X=\{1\}$ and $Y=\{1,2\}$ then there exists an injection $X\to Y$, $1\mapsto 1$. You can easily check that the only function $Y\to X$ is the constant one which is not injective. Hence $X<Y$.