What exactly is in universe in category theory?

Question

From Mac Lane's Category theory:

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 $b \in U$

He then states that $\omega \in U$ implies that all sets of real numbers and related infinite sets are in $U$.

My questions are:

1. How is ANY set of real numbers in $U$?

Since $U$ is just defined as a set that obeys those properties and contains $\omega$, how do we get $\{0.5\}$ from these properties? And what are the "related infinite sets"?

2. Are there multiple U's?

Say you generate $U_1$ from $\{a,b,c, \omega\}$ and $U_2$ from $\{X,Y,Z, \omega\}$, are they both viable universes or are they equal?

3. What universe $U$ is used in the definition of the category Set?

Mac Lane says:

Fix $U$. Then we define the category Set of all small sets to be the category in which $U$ is the set of objects. Does this contain every small set I can think of or just the ones in $U$?

Answer 1

1. The properties given for a universe are sufficient to run the entire construction of $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{R}^n$ (for each $n \in \mathbb{N}$), etc. within. How exactly you do this depends on your particular constructions.

For example, suppose we wish to show that $\mathbb{Z} \in U$ for any universe $U$. A typical approach for defining $\mathbb{Z}$ is as a quotient $(\mathbb{N}\times\mathbb{N})/{\sim}$, where $(x,y)\sim(z,w)$ if $x+z = w+y$. First we may notice that each pair $(x,y)$ of natural numbers is in $U$ by hypothesis (2). Each equivalence class is countable and contains only elements of $U$, so $\omega$ being in $U$ by hypothesis (4) along with hypothesis (5) shows that each equivalence class is in $U$. A second application of hypotheses (4) & (5) (and the fact that there are only countably many equivalence classes) shows then that $\mathbb{Z} \in U$. Likewise, we can show $\mathbb{Q} \in U$.

For $\mathbb{R}$, we may take the Dedekind cut construction, letting each real number be defined as a subset $A$ of $\mathbb{Q}$ which is downward closed (i.e., $p < q$ and $q \in A$ implies $p \in A$) and has no maximum element. Each element of such a Dedekind cut $A$ is an element of $\mathbb{Q}$ and hence an element of $U$ (either by the proof that $\mathbb{Q} \in U$ above or via hypothesis (1)), and each element is countable. Thus, another application of hypotheses (4) & (5) show that each Dedekind cut is in $U$. Hypothesis (3) shows that $\mathcal{P}(\omega) \in U$, so coupling that with hypothesis (5) shows that $\mathbb{R}$ (the set of all Dedekind cuts) is in $U$.

Now, to see that, e.g., $\{1/2\}$ or $\{\sqrt{2}\}$ are elements of $U$, we simply note that $1/2,\sqrt{2} \in \mathbb{R}$ and hypothesis (1) hence shows $1/2,\sqrt{2} \in U$. Hypothesis (2) then shows that $\{1/2\} = \{1/2,1/2\}$ and $\{\sqrt{2}\} = \{\sqrt{2},\sqrt{2}\}$ are elements of $U$.

2. There could be multiple universes, but it is also possible for there to only be a single one.

The arguments here show that if $U$ is a universe, then there is an associated strongly inaccessible cardinal $\kappa$ (in particular, the existence of a universe is independent of $\mathsf{ZFC}$ since it implies the consistency of that theory), and vice-versa. To get only a single universe, you could start with a model of $\mathsf{ZFC}$ plus the existence exactly one strongly inaccessible cardinal. On the other hand, if you have a model of $\mathsf{ZFC}$ plus the existence of more than one strongly inaccessible cardinal, then that would be a model of $\mathsf{ZFC}$ with more than one universe. It is not atypical to accept the Tarski-Grothendieck Axiom when using them in the context of category theory: For every set $x$, there exists a (Grothendieck) universe $U$ containing $x$. This is equivalent (over $\mathsf{ZFC}$) to the existence of arbitrarily large strongly inaccessible cardinals.

3. The point is that it doesn't really matter what universe you take. They all contain all 'standard' mathematical objects. The point of the Tarski-Grothendieck axiom above is to ensure that given any sets you care about, they're guaranteed to be in some universe, so that you can simply go back to dealing with a single, fixed one.

In particular, if you have any set $S$ of mathematical objects you care about, then the Tarski-Grothendieck axiom guarantees that there is a universe $U$ such that $S \in U$. By hypothesis (1), each element of $S$ is an element of $U$, so $U$ should, in principle, encompass anything you want. You're correct, however, in that your instantiation of $\mathbf{Set}$ does depend on the choice of universe.

On the other hand, you can not do the same with a proper class. In that case, you might consider a different set theoretic foundation. That being said, if you're already comfortable accepting large cardinal hypotheses, then your need for proper classes might be asking for far more than you actually want (since the existence of large cardinals imply the existence of standard (set) models of $\mathsf{ZFC}$).

Answer 2

You are getting a lot of learned and correct answers above but I think the issue may be that you are not familiar with how the usual mathematical number systems are constructed within a universe of pure sets. Forgive me if I am wrong.

Given a universe U of sets satisfying the above properties within a sufficiently powerful set theory like ZFC or MK one can construct all the familiar number systems, or at least structures that behave just like them.

You ask about a half. Here is the set corresponding to 0.5 regarded as a real number within just one of the many ways of constructing the reals.

0.5 is the left cut L of the set of rationals such that for all rationals $a/b$ in L we have $a<2b$.

A rational is an equivalence class of pairs of integers with nonzero second ordinate under the equivalence $<a,b>\sim<c,d>$ iff $ad=bc$. We usually denote $<a,b>$ by $a/b$

An integer is an equivalence class of pairs of whole numbers under the equivalence $<a,b>\sim<c,d>$ iff $a+d=b+c$.

The whole numbers correspond to $\omega$.

This hierarchy of constructions starts from $\omega$ in U and stays within U every step of the way. Note the usual operations and relations on the number systems are also defined for each level and stay within U.

Once you have $\mathbb R$ you can construct $\mathbb C$ and $\mathbb R ^n$ and $\mathbb C ^n$ and the rest within U.

Now you have all this you can define every mathematical object you can think of and still stay within U because you cannot leave U by doing any of the standard set operations like $\wp$ or $\bigcup$, U is closed by definition.

The only times a mathematician needs to step outside U is to consider proper classes, these being collections that are too large to handle relative to the axioms of the set theory being used.

McLean wants to talk about categories and these involve proper classes relative to ZFC so he added a universe to get that obstacle out of the way.

Regardless of what universe you start with the mathematical objects constructed as above are the same so to a mathematician it doesn't matter which universe you fix. If you want you can intersect them all and get a new universe, so just fix this smallest universe.