Wait, I also have an issue with definitions.

First, I am used to the definition that is complete if or : see PlanetMath and MathWorld. (I was surprised that Wikipedia defines completeness as maximal consistency.) That's why saying that Peano Arithmetic is incomplete immediately implies that there is an independent sentence, i.e., a sentence that is neither proved nor disproved.

Second, the phrase "maximal consistent" is not idiomatic. For any strict order <, saying "x is maximal" means that there is no y such that x < y. (As an aside, note that this is different from saying that x is the greatest, which means that y <= x for all y.) Here the order is set inclusion, so a set is maximal (in a family of sets) if it does not have a proper superset. Correspondingly, " is maximal consistent" means that every proper superset of is inconsistent, though itself is consistent.

Now, of course, when is consistent, these things:

(1) or

(2) , implies that is inconsistent

are equivalent, and any of them implies

(3) or

So one may say that this is arguing about definitions, though I think that distinguishing properties that hold by definition and those that need to be proved is helpful here.

Let's assume that is consistent and prove (1) => (2) => (1) and (1) => (3). (Below denotes a contradictory formula and I assume that is equivalent to .)

(1) => (2): Suppose . Take . Since , by (1). Therefore, both and are in , and so is inconsistent.

(2) => (1): Assume the contrary, i.e., and for some formula . Then both and are inconsistent by (2), i.e., and . By Deduction Theorem (or by Implication Introduction rule), and , i.e., and . This means that is inconsistent, a contradiction.

(1) => (3): This is obvious: for every , either or by (3). Therefore, or .

I am not sure what advice to give about how best to appropriate these proofs. Usually, I don't remember every detail, but I have an intuition about a maximal consistent set as something that knows everything, so one cannot add new knowledge to it.