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What is the appropriate term to use for titling a mathematical statement which will be proven false? Note that I'm focusing on the context of labeling and organizing results within a paper or similiar, e.g., Theorem 1.3.5 is a consequence of Lemma 1.2.4 and Proposition 1.2.3. The following conventions seem typical in my experience of research papers:

  • a Lemma is a true statement, but is generally not of interest in its own right and is subservient to other grand results.
  • a Theorem is a true statement of the utmost importance in the exposition.
  • a Corollary is a true statement that is easily derived from other results.
  • a Conjecture is a statement that is not proven, but is often believed to be true by those working in the field.
  • a Proposition is a bit murkier, but my experience has been that it is generally used in papers to denote a true statement whose importance/interest is somewhere between that of a Lemma and a Theorem. Note that many textbooks use Proposition as a catch-all term for any statement whose truth is yet to be resolved. For example, "Problem #1. Prove or give a counter example. Proposition: ...."

However, there doesn't seem to be a good use for listing a result that is false, although the following would all achieve the result after jarring the reader to varying degrees.

Idea 1

Theorem: even plus even is odd

Counter-Example: 2+2 = 4 $\blacksquare$

My issue here is that the reader has to actually look at where the proof would go and not immediately run off to write a listicle titled "You won't believe these 10 counter-intuitive results in mathematics!"

This becomes even more problematic if your proof isn't simply a counter-example, but rather page upon page of complex manipulations. Even with guiding narration, it seems to ask a lot of the reader. Perhaps something like "Theorem:... Anti-proof:....QED" in this case?

Idea 2

Fallacy: even plus even is odd.

This seems the closest to what I'm after, but fallacy seems a bit too specific and yet quaint (depending on the field). I imagine that if I opened a paper and saw "Fallacy 1.3", I might immediately start looking for "Fable 2.4" and "Fuzzy Notion 6.2".

Idea 3

Proposition: even plus even is --NOT-- odd

This hammers home the point (depending on the amount of emphasis added to the negation), but feels...jarring? There is also a balance between the reader missing a single word and thinking you're an idiot vs. adding so much emphasis that it becomes an eyesore.

What is your approach for handling this situation in formal mathematics?

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  • $\begingroup$ “Consider the following (false) proposition:” — how about that? $\endgroup$ – rb612 Aug 18 at 8:47
  • $\begingroup$ My take on the word "Proposition" is that a proposition is a statement that you propose the reader to study. Usually, in an article, we usually have no space to waste with false statements, so most propositions that appear are true. $\endgroup$ – Taladris Aug 18 at 8:56
  • $\begingroup$ Conjecture should suffice. Then one provides some kind of arguments in support and then later exposes them as incorrect. Or simply statement. $\endgroup$ – Alvin Lepik Aug 18 at 9:06
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The simplest thing may be to just say what you mean. That is, if your theorem is that some statement is false, then just make that the theorem you're claiming. The only reason to display the false statement as you want is if there is going to be a protracted development before it is refuted or perhaps before you even begin to refute it. If you are going to immediately refute the statement and the proof is moderately short (say a page or less), then I would do this, i.e. state the actual theorem which is that the negation of the false statement holds.

You definitely should not label the false claim a theorem and then provide a counter-example instead of a proof.

"Proposition" would be fine as technically that doesn't imply any assertion of provability, though often it is taken to mean a true statement.

What I would recommend other than my first paragraph is "Claim", though you would still need to make it clear either immediately before or after that you are going to show that this claim is false, as "claim" doesn't have a connotation that you are going to refute it.1 This would be best if the fact that the claim is false is surprising. At that point, the structure of your text would be something like "A natural statement is . Surprisingly, we will show that it is false."

I've also seen "Non-Theorem" rarely which definitely would make it clear that you are asserting that it doesn't hold. When I've seen this, it's usually in more expository work that is pointing out statements that prior experience might naturally suggest hold but don't, e.g. when moving from vector spaces to modules. Typically these non-theorems are not the main point of the text but just warnings. If something like this is your intent, then this may be appropriate.

1 Well, except the fact that you said "Claim" rather than "Theorem" or "Conjecture" strongly suggests that you don't have much faith in it.

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In logic, the term "proposition" is a standard one that captures the meaning you want, namely a boolean-valued statement whose truth-value is not specified. But you are right that non-logicians often use that very term to mean something that they are claiming is true, and frequently there is little to no distinction between "lemma" and "proposition". One way to avoid this issue is to use the term "statement", which is not only consistent with established terminology but also is unlikely to be misinterpreted.

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