Here is an example from set theory.
Set theorists commonly study not only the theory $\newcommand\ZFC{\text{ZFC}}\ZFC$ and its models, but also various fragments of this theory, such as the theory often denoted $\ZFC-{\rm P}$ or simply $\ZFC^-$, which does not include the power set axiom. One can find numerous instances in the literature where authors simply define $\ZFC-{\rm P}$ or $\ZFC^-$ as "$\ZFC$ without the power set axiom."
The notation itself suggests the idea that one is subtracting the axiom from the theory, and for this reason, I find it to be instance of incorrect notation, in the sense of the question. The problem, you see, is that the process of removing axioms from a theory is not well defined, since different axiomizations of the same theory may no longer be equivalent when one drops a common axiom.
And indeed, that is exactly the situation with $\ZFC^-$, which was eventually realized. Namely, the theory $\ZFC$ can be equivalently axiomatized using either the replacement axiom or the collection axiom plus separation, and these different approaches to the axiomatization are quite commonly found in practice. But Zarach proved that without the power set axiom, replacement and collection are no longer equivalent.
- Zarach, Andrzej M., Replacement $\nrightarrow$ collection, Hájek, Petr (ed.), Gödel ’96. Logical foundations of mathematics, computer science and physics -- Kurt Gödel’s legacy. Proceedings of a conference, Brno, Czech Republic, August 1996. Berlin: Springer-Verlag. Lect. Notes Log. 6, 307-322 (1996). ZBL0854.03047.
He also proved that various equivalent formulations of the axiom of choice are no longer equivalent without the power set axiom. For example, the well-order principle is strictly stronger than the choice set principle over $\text{ZF}^-$.
My co-authors and I discuss this at length and extend the analysis further in:
We found particular instances in the previous literature where researchers, including some prominent researchers (and also some of our own prior published work), described their theory in a way that leads actually to the wrong version of the theory. (Nevertheless, all these instances were easily fixable, simply by defining the theory correctly, or by verifying collection rather than merely replacement; so in this sense, it was ultimately no cause for worry.)
An interesting example is the enumeration of 2,3-trees here. The number $a_n$ of such trees with $n$ vertices satisfies
$$ a_n\sim \frac{\phi^n}{n}u(\log n), $$
where $\phi=(1+\sqrt{5})/2$ and $u(x)$ is a positive nonconstant continuous function satisfying $u(x)=u(x+\log(4-\phi))$. The average value of $u(x)$ is $(\phi\log(4-\phi))^{-1}$.
Best Answer
In logic, this relation is called almost less than or equal, and is denoted with an asterisks on the relation symbol, like this: $f \leq^* g$.
For example, the bounding number is the size of the smallest family of functions from N to N that is not bounded with respect to this relation. Under CH, the bounding number is the continuum, but it is consistent with the failure of CH that the bounding number is another intermediate value.