2 Added example of phenomenon B

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

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

This theorem breaks apart in a manner similar to yourequivalence relation example, since for most of thestronger 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.

1

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.)