# Are there any new axioms being developed?

Are any mathematicians working on finding new axioms, either for ZFC or another foundational theory of math?

I know that because of Godel's incompleteness theorem, it's impossible to construct a consistent set of axioms, complex enough to permit multiplication, for which every statement is decidable. By extension, ZFC must be incomplete. Adding axioms to the ones we already have is, of course, a dangerous process: if we add an axiom that turns out to be false (because it can be used to prove a contradiction), then all work that relies on that axiom is on shaky ground, and has to be re-evaluated.

At the same time, however, results like the Riemann Hypothesis are given almost the same status as axioms. They're widely accepted to be true; they're used to prove other results in mathematics; but no one has been able to prove them. The problem with declaring the Riemann Hypothesis an axiom is that the Riemann Hypothesis is a very high-level, nontrivial statement, which is why we want a proof of it.

The Axiom of Determinacy is one example of a possible new axiom.

• The Riemann Hypothesis is certainly not widely accepted to be true. It is widely suspected to be true but that is very different. Jun 5 '18 at 1:11
• I dk of anyone using the Riemann Hypothesis as an axiom, although there is much that has been published about the consequences of it. There is continuing research on "large cardinals" and elementary embeddings , using assumptions that are not consequences of ZFC (unless ZFC is inconsistent) and in fact could be (consistently) assumed to be inconsistent with ZFC. Jun 5 '18 at 1:12
• I didn't mean to imply that anyone thought of the RH as an axiom. I meant that a lot of people have published results contingent upon the RH being true, and then other people have published results on top of those results. Jun 5 '18 at 1:18

Very briefly:

Yes, there are several programs being developed that can be understood as pursuing new axioms for set theory. For the question itself of whether pursuing new axioms is a reasonably line of inquiry, see the following (in particular, the paper by John Steel):

MR1814122 (2002a:03007). Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, John R. Does mathematics need new axioms? Bull. Symbolic Logic 6 (2000), no. 4, 401–446.

I

A significant part of the effort here is in the study of large cardinal axioms and their consequences. Large cardinal axioms state the existence of cardinals with strong properties that cannot be established on the basis of $\mathsf{ZFC}$ alone. Years ago, people referred to them as "axioms of strong infinity". The cumulative hierarchy is defined recursively by setting $$V_\alpha=\bigcup_{\beta<\alpha}\mathcal P(V_\beta)$$ for all ordinals $\alpha$. We have that $$V_0\subsetneq V_1\subsetneq\dots\subsetneq V_\alpha\subsetneq\dots$$ and the universe of all sets is $V=\bigcup_{\alpha}V_\alpha$, where the union runs over all ordinals. At a minimum, a large cardinal $\kappa$ is uncountable, regular (meaning that if $\kappa$ is split into fewer than $\kappa$-many pieces, at least one of the pieces has size $\kappa$), and strong limit. Cardinals $\kappa$ with these three properties are called strongly inaccessible, a property which gives us, among others, that $V_\kappa$ is a model of $\mathsf{ZFC}$ (this already takes us beyond what we can prove from the $\mathsf{ZFC}$ axioms, by the second incompleteness theorem).

In $\mathsf{ZFC}$ we can prove a powerful reflection theorem. This result gives us that any (first-order) property of the universe $V$ is true of arbitrarily large stages $V_\alpha$ (we say that the truth of the property is reflected from $V$ to $V_\alpha$). Most significant large cardinals studied today satisfy strong reflection properties (specifically, certain properties of $V$ hold of $V_\kappa$, or properties of $V_\kappa$ hold of smaller $V_\rho$, among others, and usually we can ensure this sort of behavior even for properties not expressible in first-order), a consequence of the fact that these cardinals $\kappa$ are measurable, meaning that there is an elementary embedding $j:V\to M$ from the universe of sets into a transitive class $M$ with the property that $j(\kappa)>\kappa$ and $\kappa$ is the first ordinal moved by $j$. By requiring that $M$ be close to $V$ (for instance, by being closed under $\rho$-sequences or by containing $V_\rho$ for certain $\rho$), or by requiring that $j(\kappa)$ be "high above" $\kappa$ one can obtain even stronger reflection properties, and this provides us with a template for studying strengthenings of measurability.

For an introduction to the beautiful theory of these cardinals, and their history, I recommend

MR2731169. Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Second edition. Paperback reprint of the 2003 edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2009. xxii+536 pp. ISBN: 978-3-540-88866-6.

In a famous paper on the continuum hypothesis, Gödel proposed the study of large cardinal axioms as a way to understand properties of the universe of sets beyond those granted by $\mathsf{ZFC}$ (precisely as in this question).

MR0023780 (9,403h). Gödel, Kurt F. What is Cantor’s continuum problem?, Amer. Math. Monthly 54, (1947), 515–525. Errata, Amer. Math. Monthly 55, (1948), 151. Revised and expanded version in Philosophy of Mathematics: Selected Readings, Paul Benacerraf, and Hilary Putnam, eds., Prentice-Hall, Inc., Englewood Cliffs, N.J, 1964 vii+536 pp.; see MR0166055 (29 #3333); second edition, Cambridge University Press, Cambridge, 1983. viii+600 pp. ISBN: 0-521-22796-8; 0-521-29648-X, pp. 470–485.; see MR0742474 (85h:00015).

II.

Complementary to the study of large cardinal axioms is the pursuit of the inner model program, which aims at identifying canonical minimal models for these axioms. A nice not-too-technical introduction to this area can be found in

MR1369169 (96j:03071). Jensen, Ronald B. Inner models and large cardinals. Bull. Symbolic Logic 1 (1995), no. 4, 393–407.

Gödel's paper mentioned above is also seen as one if the original motivations for the inner model program. The program has been quite successful. For instance, it has identified several generic absoluteness results (verifying that the truth of statements of certain logical complexity cannot be changed by the technique of forcing), in the presence of large cardinals. Also, we now know that determinacy, although incompatible with the axiom of choice, is true (in the presence of large cardinals) in natural inner models. Together with this, there is an understanding that progress in the inner model program is intimately tied up with an understanding of determinacy for large classes of sets. As a result, there is active work studying strong extensions of determinacy axioms (a kind of consistency strength ladder on the determinacy rather than the large cardinal side of things).

An excellent introduction to this area can be found in

MR3087400. Sargsyan, Grigor. Descriptive inner model theory. Bull. Symbolic Logic 19 (2013), no. 1, 1–55.

III.

Together with these advances, there is also a good understanding of the limitations on what can be settled just from large cardinals. In particular, large cardinals (as commonly understood) cannot decide the continuum hypothesis. Instead, most work into axioms that in particular settle the size of the continuum has centered around understanding forcing axioms and, more generally, strong reflection principles. There are several good surveys on this topic nowadays, and a kind of informed consensus that these principles give us that the continuum should be $\aleph_2$. See

MR3184691. Todorcevic, Stevo. Notes on forcing axioms. Edited and with a foreword by Chitat Chong, Qi Feng, Yue Yang, Theodore A. Slaman and W. Hugh Woodin. Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore, 26. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. xiv+219 pp. ISBN: 978-981-4571-57-9

and

MR3656315. Moore, Justin Tatch. What makes the continuum $\aleph_2$. In Foundations of mathematics, A. E. Caicedo, J. Cummings, P. Koellner and P. B. Larson, eds., 259–287, Contemp. Math., 690, Amer. Math. Soc., Providence, RI, 2017.

IV.

A separate line of research into new axioms, mainly championed by Matt Foreman, concentrates on generic embeddings, see

MR1648052 (2000a:03086), Foreman, Matthew. Generic large cardinals: new axioms for mathematics? Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 11–21.

A variant has been introduced quite recently, already showing good potential, the study of so-called virtual large cardinals, see

Gitman, Victoria and Schindler, Ralf. Virtual large cardinals, to appear in the Logic Colloquium 2015 proceedings volume (Åsa Hirvonen, Jouko Väänänen, Dag Westerståhl, eds.).

I think these are the main 4 lines of research in set theory that one can describe as at least partially motivated by the pursuit of new axioms. Naturally, there is much more. Recently, diverse formalizations of a multiverse of sets have been considered. And there is also work understanding other aspects of infinitary combinatorics that typically lead to the isolation of new principles (axioms), but the 4 lines I mentioned seem to be the best established, at least currently.

It should also be mentioned that Woodin has for at least 10 years now been researching what he calls Ultimate-$L$. Besides fitting naturally within the inner model program, this research has suggested that attention should also be paid to the study of very strong large cardinal axioms that are incompatible with the axiom of choice. Woodin describes this as a result of the so-called $\mathrm{HOD}$-dichotomy, indicating that either the inner model $\mathrm{HOD}$ of hereditarily ordinal definable sets is close to $V$, or else there is a limit to the inner model program, and a genuine hierarchy of large cardinals incompatible with choice.

For Ultimate-$L$, see

MR3632568. Woodin, W. Hugh. In search of Ultimate-$L$: the 19th Midrasha Mathematicae Lectures. Bull. Symb. Log. 23 (2017), no. 1, 1–109.

In the first paragraph of the question it is suggested that the search for new axioms may perhaps be in the context of alternative (non-set-theoretic) foundations. In this regard, by far the most prominent alternative foundational program at the moment is the one studying homotopy type theory, a program with strong ties to computer science and some versions of constructive mathematics. See

MR3204653. The Univalent Foundations Program. Homotopy type theory—univalent foundations of mathematics. The Univalent Foundations Program, Princeton, NJ; Institute for Advanced Study (IAS), Princeton, NJ, 2013. xiv+589 pp.

• What are large cardinal axioms? Jun 5 '18 at 19:44
• Jorge, I edited the answer the add references and to incorporate the comments I left on large cardinals. Jun 6 '18 at 13:46