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 :-)