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

Question

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. — Tobias Kildetoft, 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? — Asaf Karagila, Feb 18 '19 at 08:25

Jannik Pitt · Accepted Answer · 2020-01-13T08:15:16.570

19

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.

edited Jan 13 '20 at 08:15

answered Feb 17 '19 at 20:11

Jannik Pitt

1,980

Or that if you prove from other things one of $A$, $B$ or $C$ (usually the easiest one), the others follow immediately. – Remellion Feb 18 '19 at 02:49
@Jannik Pitt Suppose we have A iff B and A iff C. Does this mean that B iff C? – user599310 Jan 12 '20 at 21:58
@user599310 Yes. – Jannik Pitt Jan 13 '20 at 08:15

score 0 · Answer 2 · answered Feb 23 '19 at 19:34

Yes. But, this helps math, it gives a bunch of statements, that we either can all dismiss at once with a disproof of one, or accept if we accept or prove the others. Goldbach's conjecture, is sufficient to proves Bertrand's postulate. But, Bertrand's postulate is not sufficient to prove Goldbach's conjecture. In this case, we can't use the proof of Bertrand's postulate to prove Goldbach's conjecture. But we could use ANY proof of Goldbach's conjecture, to prove Bertrand's postulate again possibly in a new way. IFF means we could prove any statement in the chain and the other's follow.

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

2 Answers2