Theorems true but wrong. [closed]

Question

Many theorems have the form : Premise(es) implies Conclusion(s)

Example A of wrongness:
There are many examples in which a theorem is stated without mentioning that part of the premise is not necessary to reach the conclusion.
Usually it is simple (and much better) to add a remark stating that the result is not sharp (ideally providing an example of weaker premise holding with the solution).

But there is another type of bias :

Added Note: Below composition means the AND of two relations ( for classical composition the transitivity does not compose! ( thanks to HenrikRüping remark).

Example B of wrongness:
Theorem 1 : The composition of 2 equivalence relations is an equivalence relation.
Or in fewer words : Equivalence relations are stable under composition.

Actually there is a much finer version of B :

Theorem A: For relations each of the following properties are stable under composition : Reflexive , Transitive , Symmetric.
By conjunction of the above we obtain:
Corollary B: Equivalence relations are stable under composition

Note: The second form is not only more precise but it also makes the mention "left as an easy exercise" more acceptable.

The "WRONG" notion:
I called theorem 1 (or its statement) wrong as it induced the reader to think that the conjunction of the 3 properties plays a role in proving the conclusion.

Of course only true theorems may be qualified as wrong.

Taking an absolute stance you may call wrong any theorem that is not a tautology.
A less absolute stance would call wrong any theorem that is not a tautology and in which you forget to mention non-sharpness.

Question 1: is there a better / more adequate term than wrong ( the subtext is: do you think it is a good notion?) .

Question 2: Do you know examples that follow a pattern like B or some variation in lack of tautology?

ADDED TO BE MORE SPECIFIC:

Question 3: More specifically : Are there other types of patterns showing a distance between premise and conclusion. The types need to be common in the mathematical literature, not purely logical types ( of course those are more countable).

Answer 1

A classic example for B is the theorem (proved using Gauss' lemma) usually stated as: if $R$ is a unique factorization domain, then so is the ring of polynomials $R[x]$. Now, $R$ is a UFD iff it is a GCD-domain with no infinite descending chain of proper divisors (ACCP), and

• if $R$ is a GCD-domain, then so is $R[x]$,

• if $R$ satisfies ACCP, then so does $R[x]$.

Answer 2

I remember years ago sitting in Leo Harrington's office in Berkeley explaining my dissertation to him (he was on my committee), and he spent some time just scanning through the dissertation seeking out any theorem of the form If P, then Q. At such a theorem, he would stop, smile with glee, and then turn to me and ask: Is the converse true? And I would have to explain why or give a counterexample. This little exercise definitely made a better dissertation.

His point, of course, was that such theorems could be seen as flawed in a way very similar to the sense of your question. If the converse was true, then this fact might become part of the theorem, which could be stated as the full if and only if version. And if the converse was not true, then the hypothesis was wastefully strong, and might be improved by weakening it and finding a better theorem. So the exercise guides one to what might be better theorems lurking just nearby your existing results. Since that time, I have often found this perspective illuminating---it has helped my own mathematical writing and understanding in many instances---and so now I find myself carrying out that little exercise with my own students...

At the same time, I recognize that one should not take a dogmatic view on it. There are numerous instances where one wants to draw attention to a surprising or illuminating implication, even though it isn't optimal, because one wants to focus attention on a particular aspect of the mathematics at hand. The choice after all of how to present a mathemetical result is also a choice about non-mathematical issues, such as style or emphasis, and surely many of us have wished in certain cases that the author of a text had given more attention to such presentational aspects of a mathematical text. Perhaps the best way to communicate the mathematical idea you want to communicate is to focus only on the implication P implies Q, even in cases when the hypothesis can be weakened or when the converse is also true, since those other aspects might be a distraction from the construction you want to present or the example you want to explore. Perhaps part of the point is that the implication is easy when P holds, while the optimal implication may be difficult. And so we should relax, and in such circumstances allow such flawed theorems into our papers.

(But still, you should nevertheless try to know the answer to the Harrington exercise for your theorems, even if you decide ultimately not to include those more exact results for the reasons I mentioned.)

---

But you seemed particularly interested in phenomenon B, so let me offer a specific example, as you requested:

Theorem. Every forcing extension of a model of ZFC is a model of ZFC.

This theorem breaks apart in a manner similar to your equivalence relation example, since for most of the stronger axioms, to verify the axiom in the extension V[G] one appeals to the axiom in the ground model V.

But I definitely don't call this a wrong theorem in any sense, and I wouldn't see it as a necessary improvement to deliniate exactly which ground model axioms are needed to get the particular axioms in the forcing extension, unless the focus of the work was specifically on models that did not satisfy all of ZFC. If one is interested just in ZFC models, then this theorem expresses exactly the desired implication, and the broken-apart version in the style of your Theorem A could be seen as an irrelevant technical distraction.

Almost any theorem about ZFC models would exhibit a very similar phenomenon to this.

Answer 3

I found the blog piece by Professor Tao that I commented above seems to be relevant to this question:

http://terrytao.wordpress.com/advice-on-writing-papers/dont-overoptimise/

I hope the OP will tell me whether this is, in fact, relevant to the issues he wishes to consider.

If this is on the right track, perhaps "suboptimal", "not sharp" or "unnecessarily weak" give a better description of the phenomenon than "wrong"?

Answer 4

Another way of understating theorems seems quite common: The theorem merely asserts that two sorts of things are equivalent (or in bijection), for example, equivalence relations on a set and partitions of that set. The proof gives more information, namely the usual ways of transforming any object of either sort into an object of the other sort. That these two transformations are inverse to each other is sometimes not mentioned at all.

There is a conflict here between style and mathematical content. On the one hand, the explicit transformations and the fact that they're inverse to each other are important pieces of information, often needed (though not so often mentioned, since they're so elementary). On the other hand, incorporating them explicitly into the statement of the theorem makes the statement unpleasantly long, and it may obscure the central fact.

When I teach this sort of material, in courses where students are just beginning to do rigorous proofs, I usually first state the mere fact of equivalence, but eventually I write down the whole story (including that the transformations are inverses). And I emphasize that, behind every theorem of the form "these two sorts of things are equivalent," there should ordinarily be a more explicit statement (inverses and all).