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. Feb 17 '19 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? Feb 18 '19 at 8:25
• @Karagila because of the given context. Feb 19 '19 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. Feb 18 '19 at 2:49