How can be proven that any number X is greater,lesser or equal to any other number Y? I have looked for it on the internet, really, but all I have found are particular cases like 1 > 0, or such. Is there an algebraic proof for proving that x > y or, x = y, or x < y? I thought of using Euclid's Fundamental theorem of Arithmetic as a tool, and comparing the primes that produce a compound number, but I still would have to prove that those primes (and primes in general) are ones greater (or lesser) than others.

Different text may do these differently but they are all equivalent.
x < y;x=y; or y < x are mutually exclusive and exhaustive by definition.
Addition and multiplication (and the inverses subtraction and division) are given by fiat so that a+0=a 1xa =a, a+(-a)=0.  If a ne 0 then ax1/a =1 and a (b+c)=ab+bc.  These are axioms.
From these we can prove basic this like (-(-a))=a and (-a)(-b)=ab and 0xa=0 and so on.
We are then given two essential axioms of order.
x+y < x+z if y < z
And xy > 0 if x >0 and y > 0.
From there we can prove everything.
We can prove if x > 0 then - x < 0
If x >0 and y < z then xy < xz.
That $x^2 > 0$ is $x \ne 0$ and in particular $1^2 = 1 > 0$.
That if 0 < x < y then 0 < 1/y < 1/x
So that's how we can compare any two rational numbers.  
But what of irrational.  Well when we define the reals it is as the limits of sequences of rationals and the laws of order are extended.
