# Formal statement of the well-ordering theorem

Out of interest, how would you write the well-ordering theorem in pure set-theoretic language?

• What? Just $\forall, \exists, \to, \land, \lor, \neg, \in$ and variables? I doubt someone is willing to do that. – Git Gud Apr 10 '14 at 0:20
• I agree with @Git Gud. This is not an exercise in set theory, this is a horrible exercise in logic, that you give to people who committed serious crimes against society. – Asaf Karagila Apr 10 '14 at 0:30
• @AsafKaragila OTOH, if you start out with set theory, it can't hurt to convince yourself that these things really can be translated, so doing a few relatively easy translations isn't the worst idea one could have... – fgp Apr 10 '14 at 1:54
• @fgp: Of course, I'm not disagreeing with you. This is not one of these easy translations, though. – Asaf Karagila Apr 10 '14 at 1:55
• I would do this, with statements more complex, because I'm an obsessive, anal weirdo. But not for someone else :p – Malice Vidrine Apr 10 '14 at 19:07

For every $X$ there is a relation $W \subset X\times X$ on $X$ such that $W$ is a antisymmetric, reflexive, transitive and total, and such that for every $U \subset X$ there's a $x \in U$ with $(x,y) \in W$ for all $y \in U$.
• Only that "relation", "Cartesian product", "ordered pairs", are all not in the language of set theory. Also, $U\neq\varnothing$. Oh yeah, $\varnothing$ is also not in the language of set theory. – Asaf Karagila Apr 10 '14 at 0:28
• I agree with you that "once in my life" it is an exercise worth to be done. The trick is to work "bottom-up". In your example, start with "$W$ is a relation on $X$" and call it $Stat_1$; then "$W$ is antisymmetric" and call it $Stat_2$... and so on, building up all the "catalogue" of the "basic" sentences $Stat_i$ that you need. – Mauro ALLEGRANZA Apr 10 '14 at 5:32