To Settle Infinity Question, a New Law of Mathematics (2013)
The article is fine as long as it goes but thinking of set theory as the &quot;foundational laws&quot; of mathematics is sort of a mid-20th century understanding of math. We have a better understanding of what axioms are for and how foundations should be structured. I am confident the author understands the more subtle framing I&#x27;d like to discuss, but I think it&#x27;s worth describing here for readers who were confused.<p>The main difference: this is not a new &quot;law of mathematics&quot; so much as a new law of set theory. No one but set theorists will be affected, and no other mathematicians care much.<p>Foundations are not nearly as set as undergrad math teaches you. I don&#x27;t mean the Axiom of Choice (which pretty much no one has any problems with anymore). However, various large cardinal axioms are pretty standard in advanced Algebraic Geometry (for example, the existence of Grothendiek Universes). Set theorists work with all sorts of large cardinal axioms all the time, and for a while it was fashionable to assume various generalized continuum hypotheses. Graph theorists, for some applications, assume &quot;V=L&quot;, because that makes things tidy. It&#x27;s fine, and no one really has a problem with it, nor do these disagreements really cause problems, even though some of these axioms are contradictory. (And even when various large cardinal axioms are assumed, often people work with an intuitive ill-founded set theory with the assumption that it can be made well-founded by ramifying using the large cardinals instead.)<p>The 21-century understanding is something like: there&#x27;s no need for there to be &quot;ultimate&quot; foundations. As mentioned, there is little impact on &quot;ordinary&quot; mathematics from different axioms of set theory, so maybe the set theorists should choose axioms of set theory, but that needn&#x27;t affect other mathematicians. Obviously someone should study set theory, it&#x27;s cool and there are interesting theorems and so on; and if they find that &quot;V = ultimate L&quot; is a useful axiom for their field of study, nothing wrong with that, much like commutative algebra is a fine field of study though of course non-commutative algebra is interesting too. But there&#x27;s no realistic sense in which group theory is &quot;defined in terms of&quot; set theory—you can tell, because group theorists tend not to care about set theoretic axioms, and have no dog in choosing between &quot;V=ultimate L&quot; and forcing.<p>The argument between &quot;V = ultimate L&quot; and the forcing axioms are a debate over what &quot;set theory&quot; should study, and each mathematician involved thinks that it should study the sort of things that they study—whether that means forcing or large cardinal axioms. Large cardinal axiom people have a well-developed kind of work they want to do: defining various forms of large cardinals, proving implications between them, and then discovering that all large cardinal axioms are totally ordered by proof strength. It&#x27;s good, interesting work, and &quot;V = ultimate-L&quot; will really complete the research program they&#x27;ve been pursuing; or, proving independence results for various theorems, and a forcing axiom would really put the machinery they work with deep into the foundations of set theory, making that work much simpler and at the same time deeper. That, too, is a valuable field of research, in fact it has led to the only Fields medal in logic.<p>I assure you that <i>your personal</i> understanding of sets will not be affected much by which axiom which mathematicians choose to adopt.