why universe is a set?

Question

Recently, I study algebraic geometry, and the first time I doubt the axiom. There exists a universe defined to be a set $U$ with these properties:

1. $x \in u \in U \Rightarrow x \in U$

2. $u \in U$ and $v \in U$ $\Rightarrow \{u,v\}, \langle u,v \rangle, \text{ and } u \times v$ are in $U$.

3. $x \in U \Rightarrow$ the power set of $x$ is in $U$ and the union of $x$ is in $U$.

4. $\omega$, the set of all finite ordinals, is in $U$

5. $f : a \rightarrow b$ is a surjective function with $a \in U$ and $b \subset U$, then $b \in U$

If there is an example, then I will be fine. However, many book let the existence be true, and I was wonder why this is natural like the other axiom in the Euclidean space. I mean, how can I show that universe is "a set". I think it is large enough ($\Bbb N \in U $ and so $\{\Bbb N\}\in U$), in this manner recursive. Why many book decide consider universe is actually a set, not a proper class?

Answer 1

Why many book decide consider universe is actually a set, not a proper set?

I guess you mean "proper class" instead of set?

In algebraic geometry, in particular when you work a lot with categories, you may have to face annoying foundational issues, such as the universe of all sets is not a set, and we're forced to have all the technical definitions such as small category and locally small category, etc. To get rid of the problem once and for all, Grothendieck suggested to work within a Grothendieck universe $\mathcal U$ instead: That is, all the usual operations of ZFC can be carried out within this set.

We know $\mathcal U$ as a set cannot be the proper class of all sets by either Russell's paradox or Cantor's theorem that $|2^{\mathcal U}|>|\mathcal U|$, or even the singleton $\{\mathcal U\}$ is not in $\mathcal U$, due to well-foundedness. And as pointed out by Ennar in the comment, we cannot prove there exists such a $\mathcal U$ using ZFC, because it will contradict Godel's incompleteness theorem.

This led to a new axiom due to Grothendieck and Verdier: any set is contained in a Grothendieck universe $\mathcal U$, which is related to the existence of inaccessible cardinals. Whether this axiom is consistent with ZFC is up for debate.

However, occasionally this indeed causes some unease. For example, can Fermat's last theorem be proved within ZFC? This turned out to be true but not completely trivial, as the original proof of Wiles had used modern algebraic geometry a lot. (More interestingly, can FLT be proved in PA?)

Most texts would just ignore the problem. Categories and Sheaves worked with $\mathcal U$ systematically. For beginners, it's better to ignore the abstract nonsense and focus on the more meaningful content.

Here is some fairly recent discussion by very serious people on whether Grothendieck universe is necessary, and how set theory is related to (infinite) category theory, etc.