Which one is correct? Don’t we do as necessary as rigorous math?

Question

The mathematical statement $$\exists h\in \mathbb{R} \quad \dfrac{1}{h}=5 $$ seems a mathematical statement that is true.

However, its negation $$\forall h\in \mathbb{R} \quad \dfrac{1}{h}\neq 5$$ is not a mathematical statement, because it claims that $\displaystyle\frac10\ne5,$ which is nonsense since $0$ does not have a multiplicative inverse.

Consequently, the two statements either are not negations of each other or are mathematically nonsense. Which is it?

And how do I formalise “there exists a real number whose multiplicative inverse is $5$” ?

Answer 1

Assuming we interpret division to mean the usual notion of division in the real numbers, you're correct that the statement $(\exists h \in \mathbb{R})(1/h = 5)$ doesn't quite make sense, for the reason you stated. A more formal way of putting it is that $1/h = 5$ is not, in fact, a predicate with domain $\mathbb{R}$, so we can't quantify it over $\mathbb{R}$.

There are a few ways to correct this:

1. Rewrite the statement as $(\exists h \in \mathbb{R} \setminus \{0\})(1/h = 5)$ instead, so the domain matches the domain of division of real numbers.
2. Rewrite the statement as $(\exists h \in \mathbb{R})(1 = 5h)$, noting that $1/h = 5$ is equivalent to $1 = 5h$ for all $h \neq 0$, but the latter also makes sense (though it is false) for $h = 0$. I'd say this is the most straightforward way to formalize "there exists a real number whose multiplicative inverse is $5$".
3. Interpret the division symbol to mean division in the real projective line $\hat{\mathbb{R}}$, in which $1/0$ is a meaningful expression and is equal to $\infty$, a "point at infinity" that we add to the standard real numbers. With the division symbol interpreted in this way, $1/h = 5$ now makes sense for all $h \in \mathbb{R}$ (in fact, for all $h \in \hat{\mathbb{R}}$).

In practice, issues like this are often somewhat informally glossed over, because the intended meaning of the statement is clear and there's no ambiguity, as all these ways of resolving the formal syntactical issue lead to the same conclusion. But it's good to understand how a slightly informal notation like that can be correctly rewritten/reinterpreted to resolve any formal syntactical problems.