The universal set $U$ can be defined as $U = \{x : x=x\}$, which is equivalent to $\forall x[x \in U \leftrightarrow x = x]$. Since $x=x$ is a tautology, then we infer $\forall x \in U$. This definition tells us that every free variable in the scope of the universal quantifier is an element of $U$. Since $x \in U$ means $\{x\} \subseteq U$, then $\forall x \in U$ could be rewritten as $\forall x[\{x\} \subseteq U ]$ or $\forall X[X \subseteq U]$. Therefore, every set $X$ is a subset of the universal set $U$.
However, I've seen in some proofs and commentaries cases which define the universal set as a finite collection of objects, for example $U = \{ x_0, x_1\}$. This formula is equivalent to $\forall x[x \in U \leftrightarrow x \in \{x_0, x_1\}]$ and to $\forall x[x \in U \leftrightarrow x = x_0 \lor x=x_1\}]$. Compatibilization problems with the classical definition $\forall x [x \in U \leftrightarrow x=x]$ arise when we substitute $x$ by a free variable that is not $x_0$ and $x_1$. For example, $x_2 \in U \leftrightarrow x_2=x_0 \lor x_2=x_1$ returns $x_2 \notin U$, while $x_2 \in U \leftrightarrow x_2=x_2$ is equivalent to $x_2 \in U$. Thus, we have an apparent contradiction between the standard definition of universal set and its definition as a finite collection of objects.
I was inclined to think that my mistake would lie on the impossibility of universally instantiating the formula $\forall x[x \in U \leftrightarrow x = x_0 \lor x=x_1\}]$ with a variable $x_2$ that is not in the universal set $\{ x_0, x_1\}$. To formalize such notion, we would have to explicit that the domain of this formula itself is "limited" to the defined universal set, thus considering it equivalent to $\forall x \in U[x \in U \leftrightarrow x = x_0 \lor x=x_1\}]$. Therefore, we would have by universal instantiation $x_2 \in U \rightarrow [x_2 \in U \leftrightarrow x_2=x_0 \lor x_2=x_1]$, which is a tautology, since $x_2 \notin U$. But then again, from the formula $\forall x[x \in U \leftrightarrow x=x]$, we would conclude $x_2 \in U$, so the contradiction remains.
Is the universal set not definable as a collection of finite objects?
Best Answer
Long comment
Consider formulas in the first-order language of arithmetic, with $0,+,\times, \le$.
The "natural" domain of interpretation is the set $\mathbb N = \{ 0,1,2,\ldots \}$ of natural numbers.
We can prove e.g. that $\forall x (x \le 1 \to P(x)) \leftrightarrow (P(0) \land P(1))$, but we cannot prove that $\forall x (x \ge 0 \to P(x)) \leftrightarrow (P(0) \land P(1) \land \ldots)$ because the infinite-long conjunction to the right is not a syntactically correct formula: in FOL, formulas are expression of finite length.
We can prove instead: $\forall x (x \ge 0 \to P(x)) \rightarrow P(n)$, for every $n$.
But you can consider so-called Infinitary Logic, i.e. logic based on
Regarding the "universal set" $U$, we can mimic it in predicate logic with an universal predicate $U(x) \leftrightarrow (x=x)$, in which case the formula $∀x[x=x ↔ U(x)]$ becomes a tautology.