What is the deal with the bizarre philosophy in historical and current axiomatic set theory?

Question

Many respectable mathematicians have written about "true axioms" or similar concerns about whether all mathematical theorems are in fact "real" or "true". This seems to make a great deal of non-sense to me, to the point where I suspect I must be missing something. Historically (and to a lesser extent recently) mathematicians made a great fuss about Choice, and recently Continuum and large cardinal axioms are debated. This would be at least somewhat more understandable if the debate was about whether it is more interesting or useful to study the consequences of different mathematical systems, but instead the debate frequently consists of justifications about "what is mathematical reality as opposed to a falsehood" or "what is intuitively obvious and therefore true", etc. This entire situation seems ridiculous to me. I don't understand how it isn't the case that all axiom sets are equally "true", or more like the entire framing is bizarre to me. We don't speak of fields being more "true" than groups, or lattices, or quandles, or what have you. We also don't really argue about it being "better" to study groups than fields, but it at least does not seem so much like nonsense. Why is it not the case that working in ZF or NBG or whatever set theory simply other branches of mathematics studying theorems on different objects? How does the concept of "mathematical reality" make any sense at all? Being confronted with brilliant minds frankly worrying about such things makes me assume there must be something I, a newcomer to logic and set theory, am simply yet to grasp about the matter. Can someone explain this to me?

Answer 1

There are two sorts of axiom systems. Axioms of the first sort are intended to describe the important properties of some particular mathematical structure. Examples include Euclid's axioms for geometry (intended to describe the space in which we live), Peano's axioms (intended to describe the natural numbers), and Zermelo's axioms (intended to describe the universe of sets).

For a long time, all axiom systems were of this sort, but toward the end of the 19th century people began to use axioms for a different purpose. Axioms of this second sort are intended to describe important properties of a whole class of structures. The motivation for this was the observation that sometimes structures that are clearly very different (for example the integers versus the polynomials in one variable over the complex numbers) nevertheless share some properties and, further, that proofs of these properties look very similar in these different contexts (e.g., the proof that principal ideal domains have unique prime factorization). So it became useful to write down axioms summarizing some basic properties common to many systems (e.g., axioms defining what a principal ideal domain is) as the starting point for deductions (e.g., of unique factorization) that would apply to many contexts.

For axioms of the second sort, it makes no sense to ask about their truth or falsity. One can talk about truth or falsity in a particular structure (e.g., the principal ideal domain axioms are true in the ring of integers but false in the ring of polynomials in $2$ or more variables over the complex numbers). Since these axioms are intended to summarize important properties of some structures, they will be true in those structures and false in others. Deducing other properties from the axioms is an efficient way to prove those properties for all structures that satisfy the axioms. In this way, axioms are particularly useful if many structures satisfy them (and therefore satisfy everything deducible from these axioms), and then it is to be expected that some additional statements are satisfied by some but not all of the many structures that satisfy the axioms. Such additional properties will be undecidable from the axioms.

The situation is quite different for axiom systems of the first sort, systems intended to describe one particular structure. Their purpose is not aided by being satisfied in other structures; indeed, the purpose may be hindered, because one might then want to add more axioms to exclude those extraneous structures and thus describe the desired structure more precisely.

For axiom systems of this first sort, one is usually interested primarily in truth or falsity in the one structure that the axioms were designed to describe. When people talk about "truth" in the context of such axiom systems, they mean truth in that one particular structure. Deductions from the axioms, which were the whole purpose of the second sort of axioms, are also a very useful tool for establishing that some statement is true in this sense (i.e., true in the original structure that the axioms were designed to describe), and for convincing other people of that, but they are not the only tool. For example, someone who knows that the ZFC axioms are true of the universe of sets also knows that the statement "ZFC is consistent" is true, even though it cannot be deduced from ZFC (by Gödel's second incompleteness theorem).

In connection with axioms of the first sort, people (very probably including you) are much more likely to say that "$2+2=4$" is true than to say that it's deducible from the Peano axioms. The reason "true" is meaningful in this context is that we have a common understanding of the structure that the Peano axioms are intended to describe (the natural numbers), and we've known that $2+2=4$ since we were little children, long before we heard of Peano or deductions.

I view the situation with ZFC and sets as analogous to the situation with the Peano axioms and natural numbers. I have an understanding of what ZFC is intended to describe, the universe of sets, and "true" means truth in that universe. Notice also that non-trivial theorems about sets were proved before Zermelo formulated the axiom system that would grow to become ZFC. For example, it was known that no set can be mapped onto its power set well before Zermelo found it necessary to state axioms about sets.

Of course, I don't claim to understand the universe of sets as well as I understand the natural number system, but this looks to me like a quantitative difference: How much do I understand? Qualitatively, the two are the same sort of understanding. So I'm willing to talk about truth in the set-theoretic context just as I'm willing to talk about truth on the number-theoretic context. And I would be very happy if I could improve my understanding of the set-theoretic universe enough to know whether the cardinality of the continuum is $\aleph_1$ or not.

Finally, let me echo Joe Shoenfield's statement in the preface of his wonderful book "Mathematical Logic": "I will offer no apology, however, if I have occasionally stated a philosophical opinion without observing that the contrary opinion is also widely held."