TITLE: What does a period in between quantifiers mean?
QUESTION [5 upvotes]: I'm currently reading the notes (rather a book) of an MIT preliminary math course for discrete mathematics.
In section on page 39, some ZFC axioms are written and roughly explained.
For example, the "Union" axiom states the following:
$$\forall z. \exists u \forall x. (\exists y. x \in y \space \land y \in z) \iff x \in u.$$
And the "Pairing" axiom states the following:
$$\forall x, y. \exists u. \forall z. [z \in u \iff (z = x \space \lor z = y)]$$
So the "Union" axiom contains something like $\exists u \forall x$, whereas the "Pairing" axiom has a period between both quantifiers: $\exists u. \forall z.$
So what is the point (pun intended) of the period? As far as I can tell, a period indicates that the statement before it relates to the one after it, that is, $\forall x. \exists y$ means "For all $x$, there exists a $y$." OTOH, $\forall x \exists y$ simply means "For all $x$ and for some $y$."
Is that correct? Or what is the meaning of inserting a period?
Besides, why is there no $.$ at the end of the "Pairing" axiom?

REPLY [2 votes]: It has been traditional in mathematical logic to take the quantifiers to have higher operator precedence than the propositional connectives: so $\exists y\,x \in y \land y \in z$ would mean $(\exists y\,x \in y) \land y \in z$. Russell and Whitehead had a convention of using dots to modify precedence and a few authors followed this convention, but it is rather complicated and unintuitive (compared with brackets, which are familiar from algebra) and it fell into disuse.
Nowadays, many authors (particularly computer scientists, like the authors of the notes in question) adopt the convention that quantifiers have lower operator precedence than the propositional connectives and use a dot as a clue to readers familiar with the older conventions. So they write $\exists y.x \in y \land y \in z$ rather than $\exists y(x \in y \land y \in z)$. This agrees with the traditional notation for $\lambda$-calculus, where $\lambda x. x + y$ is not read as $(\lambda x.x) + y$, and is (in my opinion) much more convenient in practice.
In the examples you cite, there is a dot missing after $\exists u$ in the union axiom. This must be a typo. The authors are also using commas to abbreviate nested quantification: $\forall x, y.\cdots$ abbreviates $\forall x. \forall y. \cdots$