# Isn't proving $A$ iff $B$ iff $C$ by showing $A\to B$ and $B\to C$ and $C\to A$ circular?

Suppose you have $$A$$ iff $$B$$ iff $$C$$.

If you assume $$A$$ to be true to prove $$B$$, $$B$$ to be true to prove $$C$$, and $$C$$ to be true to prove $$A$$, then doesn't that imply you've assumed $$A$$ to be true to prove $$A$$?

I ask because of the method of proof in https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction.

• You are not proving any of them. You are proving that they are equivalent. Commented Feb 17, 2019 at 20:10
• The real question I have, is why do you need A, B, and C in your question? Isn't just proving that A iff B circular enough for you? Commented Feb 18, 2019 at 8:25
• @Karagila because of the given context. Commented Feb 19, 2019 at 17:39

Yes, you are absolutely right the proof $$A \implies B \implies C \implies A$$ only shows that $$A,B,C$$ are equivalent. So if one of them is true, all the others are true and if one is false all the others are false. However, the proof you linked doesn't try to show that the principle of mathematical/complete induction is true or the principle of well ordering is true, it only shows that they are equivalent. In fact, in the usual framework of mathematics these are taken as axioms. The proof shows that you only need to assume one of them as an axiom and you get the other for free.
• Or that if you prove from other things one of $A$, $B$ or $C$ (usually the easiest one), the others follow immediately. Commented Feb 18, 2019 at 2:49