It's been a while since my last post and now it's time for a mathematical post:

I'm currently preparing for a math exam (calculus) and I'm thinking it would be nice if there was a way to avoid much of the "for every $\epsilon > 0$ there is an $N \in \mathbb{N}$, so that for all $n \ge N$ some property ... holds" stuff you find in textbooks. Some textbooks actually shorten it to "for every $\epsilon > 0$ some property ... holds for almost all $n$".
However, I haven't found a quantifier to express this anywhere yet, so I'm proposing to introduce a new one:

$\tilde{\forall} x \in M: P(x)$ should mean "for almost all $x \in M$ $P(x)$ holds", which suggests that there are only finitely many elements for which it does not hold.
One can formalize this as:
$\tilde{\forall} x \in M: P(x) \equiv \exists n \in \mathbb{N}: \left | \left \{ x \in M \mid \neg P(x) \right \} \right | \le n$

Of course, a new existential quantifier is required then, too (for negation):
$\tilde{\exists} x \in M: P(x)$ stands for "there exist infinitely many $x \in M$, for which $P(x)$ holds".
And this can be formalized as:
$\tilde{\exists} x \in M: P(x) \equiv \forall n \in \mathbb{N}: \left | \left \{ x \in M \mid P(x) \right \} \right | > n$

It's easy to see that $\neg \left ( \tilde{\forall} x \in M: P(x) \right ) \equiv \tilde{\exists} x \in M: \neg P(x)$

Thus one can use the two quantifiers just as one would use $\forall$ and $\exists$ usually. Note however than $\forall$ and $\tilde{\forall}$ don't interchange and neither do $\exists$ and $\tilde{\exists}$.

One last note: it might be worth using a different notation, for example: $\exists ^ \infty$ might be easier to understand than $\tilde{\exists}$, and $\forall ^ \approx$ might be better than $\tilde{\forall}$.

I'll try to formalize these quantifiers some more when I find some spare time.

Stay tuned for coding related updates soon
Cheers,
Andreas