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There are plenty of mathematicians - mostly set theorists - who are actively working on finding new axioms of mathematics to resolve questions which can't be resolved by ZFC. Projective Determinacy is probably the most important example of a new axiom of mathematics which goes far beyond what can be proved in ZFC, but which has become widely accepted by the experts. (See [1] for some discussion about the arguments in favor of projective determinacy, and [2] for a taste of Steel's position on the subject.) I suggest reading Kanamori's book [3] if you want to learn more about this direction. (There are plenty of recent developments in the field which occured after the publication of that book - for an example of cutting edge research into new axioms, see the paper [4] mentioned in one of the answers to [5].) If you are only interested in arithmetic consequences of the new axioms, and if you feel that consistency statements are not too interesting (even though they can be directly interpreted as statements about whether or not certain Turing machines halt), you should check out some of the research into Laver tables [6], [7], [8], [9]. Harvey Friedman has also put a lot of work into finding concrete connections between advanced set-theoretic axioms and more concrete arithmetic statements, for instance see [10]. [1] https://mathoverflow.net/questions/479079/why-believe-in-the...
[2] https://cs.nyu.edu/pipermail/fom/2000-January/003643.html
[3] "The Higher Infinite: Large Cardinals in Set Theory from their Beginnings" by Akihiro Kanamori
[4] "Large cardinals, structural reflection, and the HOD Conjecture" by Juan P. Aguilera, Joan Bagaria, Philipp Lücke, https://arxiv.org/abs/2411.11568
[5] https://mathoverflow.net/questions/449825/what-is-the-eviden...
[6] https://en.wikipedia.org/wiki/Laver_table
[7] "On the algebra of elementary embeddings of a rank into itself" by Richard Laver, https://doi.org/10.1006%2Faima.1995.1014
[8] "Critical points in an algebra of elementary embeddings" by Randall Dougherty, https://arxiv.org/abs/math/9205202
[9] "Braids and Self-Distributivity" by Patrick Dehornoy
[10] "Issues in the Foundations of Mathematics" by Harvey M. Friedman, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3137697 |