Is division by zero a not well-formed expression(formula)?

Question

Suppose I have the following definition: "$a^{-1}$ is such a real number that $a^{-1}\cdot a = 1$". Will the folowing expression be well-formed ? $$1\cdot 0^{-1} = 2$$ It seems to me that we cannot reason about truth of the following expression so it is not. If it is well-formed, what is the exact technical mistake about the expression above ? And if it is not well-formed, what is the reason by that ?
Note: I am not trying to argue that division by zero must be accepted, but rather trying to understand rigorous explaining of its "imposibility".
And also, I know that we can prove that there exists no real number $x$ such that:$x\cdot0=1$. So you do not need to prove it to me, if you need it in the answer.
EDIT: I`m talking about real numbers only.

Answer 1

For a well-formed formula to even make sense, you need to first specify the language. I'll assume that you mean the language of fields: $0,1$ are constant symbols, $+,\cdot$ are the binary function symbols.

Now we add another symbol $\cdot^{-1}$, and we wish to postulate that it denotes the multiplicative inverse. And therein lies the rub.

You want to treat this as a function symbol, but a function symbol needs to be defined on the whole domain of the structure. So $0^{-1}$ needs to be specified. We can say that $0^{-1}=0$, and in the added axioms that postulate that $a\cdot a^{-1}=1$, require that $a\neq 0$. This way, we generally get to ignore the value of $0^{-1}$ for all practical purposes.

Another way of doing this, is to take $\cdot^{-1}$ as a binary relation symbol, and postulate that it defines a function on the non-zero reals. This has the obvious drawback of not being a function symbol, meaning you cannot use it in terms, but you can work around this, of course, by saying that there exists some $z$ which satisfies the $\cdot^{-1}$ with $a$ (namely, $z=a^{-1}$) and then treat $z$ as $a^{-1}$ for all practical purposes of your formula.

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In either case, the problem in your question comes from not paying enough attention to the details of what it means to be a well-formed formula, in the sense of not paying attention to the specifics of the language and its interaction with the pre-existing structure.