What does universal quantification mean?

In ZFC, for example, there is no universal set, so what does it mean to write $\forall x (\cdots)$, i.e., for all sets something is true? Does it avoid the problem by quantifying over all elements but not allowing those elements to form a set?

A related question I have is about quantifying over infinite sets. If $X$ is a finite set with $n$ elements $x_1, x_2, x_3, \ldots, x_n$, Then it is reasonable to define $\forall x\in X :P(x)$ to be the statement $P(x_1)\wedge P(x_2)\wedge\cdots\wedge P(x_n)$ (which can be formalized using a recursive definition). But what about infinite sets? It seems that if $Y$ is a countably infinite set with elements $y_1,y_2,\ldots$, then $\forall y\in Y: P(y)$ would mean $P(y_1)\wedge P(y_2)\wedge\cdots$, which we couldn't finish stating. For uncountable sets, we would miss an element in between every time we count upwards.

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...which is why universal quantification is not the same as a conjunction, and why lattices are not always complete lattices. Universal quantification is part of the underlying first-order logic of set theory, so there is no need to have a universal set to quantify over. – Arturo Magidin Apr 15 '12 at 21:54
@russell11 For each cardinal $\kappa$ the language $L_{\kappa,\omega}$ allow you to use conjunctions of length $\kappa$ i.e., $\bigwedge_{\xi<\kappa} P(x_\xi)$. – azarel Apr 15 '12 at 21:59

While it is useful motivation to think of the universally quantified statement as an "infinite" conjunction, this is not literally true and cannot be formalized "using a recursive definition". If you want to quantify "universally" over all elements of a set, the natural way to do this is simply $\forall x (x \in X \implies P(x))$.