soft question - The Importance of ZF

There is a very active ongoing debate within set theory about whether mathematics needs new axioms, and philosophers of mathematics are weighing in on all sides. Relevant considerations include many very deep topics in set theory, including independence, forcing and the large cardinal hierarchy. Some of these topics are at once highly technical and philosophical at the same time. It is fair to say that there is an emerging field called the philosophy of set theory that is grappling precisely with these issues.

Let me try to mention just a few of the considerations. First, the historical fact remains that the ZFC axioms are sufficiently powerful to carry out almost all of the construction methods that arise in mathematics outside set theory. Indeed, the ZFC axioms are provably far more powerful than necessary for the vast majority of ordinary mathematics. This is proved by the stunning results of the field of Reverse Mathematics (see Steve Simpson, Harvey Friedman etc.), which calculates for a huge collection of classical mathematical theorems exactly which axioms are needed to prove them. Reverse Mathematics proceeds by proving the axioms from the theorem as well as the theorem from the axioms (over a very weak base theory), thereby showing the necessity of those axioms, and it turns out that most all of the classical theorems of mathematics can be proved in relatively weak theories.

Nevertheless, within set theory, set theorists have discovered the ubiquitous independence phenomenon, by which an enormous number of set-theoretical assertions turn out to be independent of the ZFC axioms. This means that they are neither provable nor refutable in ZFC. We now have thousands of instances of fundamental set theoretic propositions that are known to be independent of ZFC. This includes almost any nontrivial statement of infinite cardinal arithmetic (such as the Continuum Hypothesis), as well as an enormous number of statements in infinite combinatorics, and so on. This phenomenon supports the view that ZFC is a weak theory, unable to decide these questions.

But of course, by the Incompleteness Theorem we know that any theory we can write down will exhibit this independence phenomenon. It is impossible in principle to avoid it.

Large cardinals are strong axioms of infinity, some of which go back to the time of Cantor (so they are not new), which are not provable in ZFC and which transcend ZFC in consistency strength, forming a vast hierarchy of consistency strength above it. Thus, they tend to make up for the weakness of ZFC (although there remains extensive independence even with large cardinals). Some set theorists make the case that the existence of large cardinals have numerous attractive regularity consequences, even for low down for sets of reals, that they seem to point the way towards the finally true set theory, which must remain elusively hidden from us because of the Incompleteness theorem. Making sense (or nonsense) of this view is a central concern of the emerging Philosophy of Set Theory.