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.

You have to start somewhere. Say we have the integers and we know how to add, subtract, and multiply. This distinguishes the multiplicative identity element $1$. We say that $1$ is "positive," and recursively define positive numbers by declaring that if $p$ is positive, then so is $p+1$. This gives us the usual set of positive numbers.
Define $a<b$ if $b-a$ is positive. Then if $a\neq b$, then either $a-b$ is positive or $-(a-b)=b-a$ is positive, so either $a<b$ or $b<a$, which is what we wanted to show.
What about $52$ and $48$? Well, $52-48=4\ldots$
