Zermelo-Frankel Set Theory for Algebraists

Question

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\CRings{CRings}\DeclareMathOperator\Grp{Grp}\DeclareMathOperator\Sets{Sets}$I'm not a logician/set theorist, and I have some questions on set theory and references that may seem "trivial" for experts. Still I ask the question – if you have references this would be interesting.

In algebraic geometry (See Hartshorne's book, Appendix A) the following theorem is proved:

Let $\Var(k)$ be the "category of non-singular quasi-projective varieties over an algebraically closed field $k$ and morphisms of varieties over $k$. This category is defined in Hartshorne's book.

Theorem 1.1. There is a unique intersection theory $A^*(X)$ for algebraic cycles on $X\in \Var(k)$ modulo rational equivalence satisfying the axioms A1–A7.

The axioms A1–A7 are listed on page 426-427 in the book. For a variety $X\in \Var(k)$ one defines
a commutative unital ring $A^*(X)$ – the Chow ring – and this construction is unique. There is only one way to do this, meaning there is a unique functor

$A^*(-) : \Var(k) \rightarrow \CRings$

such that axioms A1–A7 hold. Here $\CRings$ is the category of commutative unital rings and maps of unital rings.

In algebra one defines a group $(G, \bullet)$ as a set $G$ with an operation $\bullet: G\times G \rightarrow G$ satisfying $3$ axioms: G1 Associativity, G2 existence of identity and G3 existence of inverse. One defines a morphism of groups and the "category of groups" $\Grp$. Clearly the category of groups $\Grp$ contains non-isomorphic groups, hence the axioms G1–G3 does not uniquely determine
one group. There are many different groups satisfying G1–G3.

In ZF set theory set theorists write down 9 axioms ZF1-ZF9, and these axioms determine $\Sets$ – the "category of sets". $\Sets$ is a category with "sets" as objects and "maps between sets" as morphisms. We would like the category $\Sets$ to be uniquely determined by the axioms ZF1–ZF9 similarly to what happens for the Chow ring. Is it? Is there a unique category $\Sets$ fulfilling the axioms ZF1–ZF9? If yes I ask for a reference.

For reference, Wikipedia has the page ZF set theory.

Answer 1

We say that a mathematical theory is categorical if it has exactly one model, up to isomorphism.

We intend some theories to be categorical, for instance the Peano axioms for natural numbers, Euclid's planar geometry, and set theory. Other theories are designed not to be categorical, i.e., the theory of a group, the theory of a ring, etc.

You are asking whether there are general theorems about categoricity, and whether in particular the Zermelo-Fraenkel set theory is categorical. First we have:

Theorem: If a theory expressed in first-order logic is categorical, then it axiomatizes a unique (up to isomorphism) finite structure.

Thus Zermelo-Fraenkel set theory and Peano arithmetic are not categorical. In fact, they both have many models:

Theorem: (Löwenheim-Skolem theorem) If a theory expressed in first-order logic has an infinite model, then it has an infinite model of every cardinality.

How should one react to these results? Perhaps we need not worry about it. So what if there are many models of Peano arithmetic and set theory? If we can accept the fact that there are many different groups, why not accept the fact that there are many different set theories? The mathematical universe just gets richer this way (but the search for "absolute truth" has to shift focus).

We could also "blame" first-order logic for these undesirable phenomena. For instance, whereas Peano axioms do not "pin down" the natural numbers, the category-theoretic notion of the natural numbers object does: all natural number objects in a category are isomorphic (because they are all the initial algebras for the functor $X \mapsto 1 + X$). This is possible because the category-theoretic description speaks about the entire category, not just the object of natural numbers.

We can in fact do the same for set theory: assuming a suitable "category of classes", the Zermelo-Fraenkel universe of sets can be characterized (uniquely up to isomorphism) as a certain initial "ZF-algebra" (this point of view has been studied in algebraic set theory). Note however that from a foundational point of view we have not achieved much, as we just shifted the problem from sets to classes.

As an algebraist, should you be worried that the category of sets is not "uniquely determined" by the Zermelo-Fraenkel axioms? I don't think so. Algebra is generally quite robust, and works equally well in all models of set theory. Of course, there are also parts of algebra that depend on the set-theoretic ambient, but is that not a source of interesting mathematics?