I'm trying to tie some loose ends here. My lecturer didn't bother to go into details, so I have to work it out myself. I usually hate to be pedantic, but these questions have been bugging me for a while.
First, what's the proper definition of the natural numbers? The one given to me is: "the smallest set such that $\emptyset\in N$ and $x\in N\implies x\cup\lbrace x\rbrace\in N$". This definition does not seem rigorous to me. What does "smallest" mean here? Does it mean that no proper subset have the same property? And how to prove the existence of such set (under ZFC)? Lastly, how to prove this set satisfies Peano's axioms, especially the last axiom (about induction)?
Next, how to prove that the relation $<$ is a total order? The definition is $x
Lastly, how to prove the well ordering principle? I suspect this would be easy after proving the previous statement though.
Edit: Why didn't the curly bracket appear? I'll use the square bracket instead.