This question already has an answer here:

I want to prove that:

Well ordering principle ⟺ Complete Induction.

I am interested in both directions of the implication. That is, that if every non-empty set contains a least element then if a property P holds for all naturals less than n-1 it must hold for n. And viceversa.

I want the proof to be classical (I suspect this is quite difficult to prove constructively)


marked as duplicate by Pragabhava, Daniel W. Farlow, Frunobulax, Jack's wasted life, quid Feb 14 '16 at 19:09

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ I see, didn't realize this is being asked for. The proof by contradiction seems to be quite simple though, any reason why seeking "non contradiction" proof? $\endgroup$ – Sil Feb 14 '16 at 15:26
  • $\begingroup$ Wait, now I'm confused. I thought "classical" refers to "classical logic" - thus including tertium non datur. $\endgroup$ – Stefan Mesken Feb 14 '16 at 15:43
  • $\begingroup$ @Sil Simply for the interest of an intuitionistc proof. Which is what the OP seems to be looking for. The proof in the link you suggested was classical. The OP is using contradictory concepts in his question. $\endgroup$ – Git Gud Feb 14 '16 at 16:58
  • $\begingroup$ @Stefan I made a mistake, I meant intuitionistic instead of classical. $\endgroup$ – Git Gud Feb 14 '16 at 16:59