How to Define Order Relations in Irrational Numbers Compared to $\Bbb N,~ \Bbb Z$, and $\Bbb Q$

Question

I'm curious about how order relations are defined among irrational numbers. While the ordering of integers and rational numbers is straightforward, in the sense:

After constructing natural numbers $\{0,1,2,3,...\}$ using Peano's axioms, we define the order '$\leq$' using successor function. Once the order defined over $\Bbb N$, we can easily extend the order in to $\Bbb Z$ in such a way that $a \leq b$ if and only if $b-a \in \Bbb N$. Similarly, we can define the same over $\Bbb Q$ as $$\frac{a}{b} \leq \frac{c}{d} \Leftrightarrow ad \leq bc.$$

But I stuck at the definition of '$\leq$' over irrational numbers. I'm really interested in understanding the criteria for comparing irrational numbers and consequently in the set $\Bbb R$.

Answer 1

[...] we define the order $\leq$ using successor function. Once the order defined over $\mathbb{N}$ [...]

I suppose you mean

$x \lt_{\mathbb{N}} y :\iff \exists z(x+succ(z) = y)$

which gives us the c(l)ue ('add something of a positive magnitude') to define the usual ordering on the reals as/by

$x \leq_{\mathbb{R}} y :\iff \exists z(x+z^2 = y)$