Tao Analysis I - 3.2.3

Exercise 3.2.3

Show (assuming the other axioms of set theory) that the universal specification axiom, Axiom 3.9, is equivalent to an axiom postulating the existence of a “universal set” $\Omega$ consisting of all objects (i.e., for all objects $x$ , we have $x \in \Omega$). In other words, if Axiom 3.9 is true, then a universal set exists, and conversely, if a universal set exists, then Axiom 3.9 is true.

For equivalence we need to show the implication in both directions, as stated by the question.

Let's remind ourselves of Axiom 3.9.

Axiom 3.9 (Universal specification, or axiom of comprehension). (Dangerous!) Suppose for every object $x$ we have a property $P(x)$ pertaining to $x$. Then there exists a set $\{x : P(x)\}$ such that for every object $y$,

$$y \in \{x : P(x)\} \iff P(y)$$

Let's formulate an axiom that postulates the existence of a universal set $\Omega$ consisting of all objects.

Axiom 3.10 (Universal set). There exists a universal set $\Omega$ consisting of all objects.

$$\forall x (x \in \Omega)$$

Let's start with Axiom 3.9. If we take $P(x)$ to be true for all objects $x$ then there exists a set which contains every object $x$. This is the definition of the universal set. So Axiom 3.9 implies Axiom 3.10.

Now let's start with Axiom 3.10. It tells us there exists a universal set $\Omega$ consisting of every object $x$. Let's use Axiom 3.6, the axiom of (not universal) specification on the set $\Omega$, with a property $P(x)$ which is true for all objects $x$. This Axiom 3.6 tells us there exists a set

$$\{x \in \Omega: P(x) \}$$

An object $y$ is in this set if $P(y)$ is true, and if $y \in \Omega$. Since every object $y$ is in $\Omega$, we are no longer bound by the set $\Omega$ (this is the key), and so we have universal specification for $x$.

$$ y \in \{x: P(x)\} \iff P(y)$$

So Axiom 3.10 implies 3.9.

We have shown Axiom 3.9 implies 3.10, and Axiom 3.10 implies 3.9, so we can conclude they are equivalent. $\square$