Is it possible that there is an axiom we have not discovered yet?

Question

For example, the completeness of real numbers was discovered around 1800 and now $\mathbb{R}$ is considered to be a Dedekind-complete ordered field. Is it possible that a new axiom about $\mathbb{R}$ will be discovered?

Answer 1

While the OP specifically asks about $\mathbb{R}$, I'm going to say a bit about the more general problem of additional axioms for mathematics in general. The case of $\mathbb{R}$ plays a special role here of course, but I think it helps to view it in the broader context.

Because this answer is rather long, I'm going to mention right here the various papers/slides around Harvard's EFI program; I think these are quite approachable - mostly - and will be of interst to the OP, and there's certainly lots there.

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There is indeed extensive debate about whether mathematics in general needs additional axioms. The topic of the continuum problem is of special interest, since it's arguably the most natural problem in set theory following Cantor's discovery of different infinite cardinalities but known to be independent of the usual set-theoretic foundation of mathematics:

Is there a set of real numbers which is uncountable but not in bijection with all of $\mathbb{R}$?

For example, results in descriptive set theory indicate that any such set would have to be extremely complicated; does that constitute evidence for a negative answer?

But there are plenty of other points of interest, and some common axiom candidates include V=L, large cardinals (and their "generic" versions), forcing axioms, and the inner model hypothesis. And this debate - both the specific one about CH (or the set theory of the reals more generally) and the broader one about general mathematics is tied to the question of justifying the ZFC axioms themselves. The axiom of choice is the obvious target (with the axiom of determinacy being the main competitor) but not the only one, and there are dramatically contrasting alternative foundational systems which have been seriously proposed. Although ZFC is currently dominant that could change (especially since category- or type-theory-based systems seem to play better with computer proof systems, which is a phenomenon which couldn't have been predicted when ZFC was introduced and adopted), so this side of the issue shouldn't be thought of as over-and-done-with.

There has been far too much ink (physical and digital) spilled around all this to summarize here; the best I can do is mention some sources of interest.

• The paper Does mathematics need new axioms? is an important starting point, and in my opinion Maddy's "Believing the axioms" $1$ and $2$ should be read along with these: it's not constructive to talk about criteria for adding new axioms without recalling the criteria we used (or now retroactively use) for justifying the existing axioms.

• As mentioned above, CH plays a central role in these debates; the debate at this Mathoverflow question is quite technical, but a wonderful wealth of sources on the topic.

• Also as mentioned above, Harvard's EFI event brought together a number of people to discuss questions of new axioms for mathematics and meaningfulness of mathematical statements; the many sources there will be of interest, even if they focus more on questions of meaningfulness.

• Many of the above-mentioned sources also discuss large cardinal axioms, but those are central enough in the debate that it's worth mentioning them specially here. Large cardinal principles have the interesting feature of provably not deciding CH one way or the other (at least, the vast majority of them), so the study of large cardinals as potential axioms is orthogonal to the CH problem. There are a couple stackexchange questions discussing the cases both for and against large cardinals; see e.g. here.

• There are examples of arguably-natural concrete combinatorial problems which cannot be solved in ZFC and indeed are intimately tied to large cardinals; this has been studied extensively by Harvey Friedman, and many sources on the topic can be found on his website. Unfortunately the results are quite technical, but the principles he studies can be understood without advanced background or too much effort (whether they're "natural," though, is another question ...).