I believe that the pushback against *$\epsilon,\delta$ definitions* (which unfortunately spills over to pushback against *$\epsilon,\delta$ techniques*) is entirely justified because *$\epsilon,\delta$ definitions* arise from the (unfortunately widespread) confusion between a statement being *formal* and a statement being *rigorous*.

Consider the *formal "definition"* of continuity of a function $f$ at a point $a$:
$$\forall\epsilon\exists\delta\forall x(0<|x-a|<\delta\rightarrow |f(x)-f(a)|<\epsilon)$$ This is just an objuscated way of stating the informal, but rigorous:

For every ball $B_{f(a)}$ centered at $f(a)$, there is a ball $B_a$ centered at $a$ so that $f$ sends every point of $B_a$ into $B_{f(a)}$.

which is logically equivalent to the conceptually clearer, though still informal, though still rigorous:

Whenever the image $f(S)$ of a set $S$ is separated from the image $f(a)$ of a point $a$, the set $S$ was already separated from the point $a$.

which is the contrapositive of the, informal and rigorous, *intuitive definition* of continuity of $f$ at a point $a$:

Whenever a set $S$ of points are close to a point $a$, the set of images $f(S)$ of those points are close to image point $f(a)$.

I strongly believe that the equivalence of the blocked statements and the IDEA that equivalence expresses, which is that we CAN distill an intuitive notion into a rigorous definition, is much more interesting, important, and memorable, than the formal $\epsilon,\delta$ "definition". Furthermore, I can't even bring myself to calling the formal "definition" a definition, since what it expresses is not a description of what it *means* for a function to be continuous, but a technique (of $\epsilon,\delta$ proofs) for how to *check* that a function is continuous.

This, in my opinion, is *the reason* for the pushback against $\epsilon,\delta$ "definition" and arguments: instead of expressing the rigorous idea or concept of continuity, the $\epsilon,\delta$ "definition" only gives a technique for working with continuity, and, when presented as a definition, only obfuscates the meaning of the concept (in a very efficient way, I might add, since the path from the intuitive and meaningful definition to the $\epsilon,\delta$ definition involves taking a contrapositive...).

Finally, I do think that being aware of how to rigorously translate (as above) from the intuitive definition of continuity to the statement of the $\epsilon,\delta$ technique will certainly not hurt, and I suspect could actually help students in using the ($\epsilon,\delta$) technique, especially with the simple functions that arise in Calculus and basic analysis.

(Someone might criticize the above saying that the notion of a ball is confusing in single-variable Calculus. My perhaps controversial response is that there really isn't any good reason not to teach Calculus using $2$ or $3$ variables from day $1$ and that the narrow viewpoint offered by single-variable Calculus obscures more than it simplifies).

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.

## Best Answer

In this situation, it seems that a negative statement is a proposition that comes with a logic negation in front of it. Let me try to explain it with an example, that should be clearer:

Positive statement: “There exists a function $f:X \rightarrow Y$ such that $f(x)$ is constant on $X$“

Negative statement: “There does not exist a function $f:X \rightarrow Y$ such that $f(x)$ is constant on $X$“, that logically means “for every function $f:X \rightarrow Y$, $f(x)$ is not constant on $X$ (that is there exists at least an $a$ and $b$ both in $X$ such that $f(a) \neq f(b)$.