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It is typical to find a corollary that following theorems, but is it right to use the word corollary for a statement following a conjecture, where the statement is true only if the unproven conjecture is true?

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Such statements are said to have "conditional proofs". . I'm voting to close as "not a real question" – Gjergji Zaimi Mar 11 '10 at 4:15
I think it's a reasonable question. – Noah Snyder Mar 11 '10 at 4:31
I don't like this business of voting down stuff, especially when I do like the question. – Anonymous Mar 11 '10 at 4:31
I didn't give it a vote down, the poster asked a clear question. The only debatable thing is whether this the right place to find the answer. Maybe this should be discussed at meta (or maybe there is a discussion already), but at least change the tag? :-) – Gjergji Zaimi Mar 11 '10 at 4:35
I retagged the question. – Kevin H. Lin Mar 11 '10 at 5:44
up vote 12 down vote accepted

I think it's generally bad form to have a corollary dependent on an earlier conjecture. I recommend one of the following:

Theorem: Assuming Conjecture A, properties X, Y and Z are true.


Theorem: Conjecture A implies X, Y and Z.

Most importantly, it should be crystal clear that the result is dependent on the conjecture.

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I would write "Proposition Z: If X holds, then Y is true." Even if the deduction of Y from X were trivial, I think labelling this a corollary would be confusing. (After all, what is the statement "X implies Y" a corollary of?) However, I wouldn't have a problem writing something like "as we saw above, Y would be a corollary of X" later on. (The subjunctive voice is important here!)

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+1 for mentioning the subjunctive (and to be serious, I agree with this and with Douglas' take). – Yemon Choi Mar 11 '10 at 5:13

I'm reminded of the following story that I posted on my personal web journal a couple years ago:

At the Topology seminar yesterday, the speaker presented a theorem, which he immediately followed with a refinement: a statement that directly and obviously implies the theorem. He labeled his refinement a "corollary". I turned to Noah Snyder, and said that it was more an "uncorollary, or an anticorollary", but as soon as I said as much, the two of us simultaneously correctly labeled the refinement as a "rollary".

There should be more rollaries in mathematical writing.

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I am not sure if I understand your joke correctly, but shouldn't it be contrarollary? – Willie Wong Mar 11 '10 at 12:51

Making a new Theorem environment that let you have the bolded part say "Corollary to Conjecture X" seems to me a good compromise of concise and unlikely to confuse anyone.

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I'm sure I've written stuff like "Corollary (of Conjecture X)" in class notes, but I think that in careful writing one should be even more clear about the conditional nature of a result than this. Strictly speaking it makes no sense for a conjecture to have a corollary, because the logical statuses are completely different. The more careful statement: "Proposition [or Theorem, or whatever]: Conjecture X implies Y" seems preferable. – Pete L. Clark Mar 11 '10 at 9:44
I also think that there is room for improvement in the way mathematicians state conditional results. In some subfields it seems to be almost forgotten that certain standard conjectures are not known to be true, so you see things like "Theorem: Something amazing (conditional on GRH)" (which, by the way, is not even one conjecture but a bundled together family of conjectures) Or, to hit closer to home: "We show something fantastic relating analytic, Mordell-Weil and Selmer ranks (assuming the finiteness of Sha)". I say boo: put your assumptions first! – Pete L. Clark Mar 11 '10 at 9:48
I actually think this is much clearer than "Prop: X implies Y." If you're skimming through you can't miss that "Corollary to conjecture" means it hasn't been proven, whereas if you have a proposition and you phrase it a little bit poorly a skimmer might think the conclusion had been proven. The clearest thing is to have the fact that it's unproven in bold. – Noah Snyder Mar 11 '10 at 17:00
Here's an example of when this construction is used: – Douglas S. Stones Mar 13 '10 at 22:32

The correct term for such an item is CONJOLLARY. ;)

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Isn't that a relative of the Bandersnatch? – Yemon Choi Mar 11 '10 at 18:48

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