To Settle Infinity Question, a New Law of Mathematics (2013)
This sort of thing just strikes me as so pointless.  How could one possibly decide?  Obviously, if we&#x27;re talking about new axioms that aren&#x27;t provable from the old ones, you can&#x27;t use those old ones to settle the matter.  The axioms under discussion don&#x27;t really affect normal math so you can&#x27;t appeal to their usefulness there (unlike with, say, the axiom of choice, which famously was controversial for a while).  And I don&#x27;t think people can really have much of a claim to have any good intuition for such things, or else we&#x27;d have had such an axiom a long time ago.  My bet is neither of these will become widely accepted.<p>That said, if I had to bet on one of the two becoming accepted, I&#x27;d go with &quot;V = ultimate L&quot;.  Why?  Because mathematics already has an axiom roughly analogous, that doesn&#x27;t actually affect normal mathematics but prunes away sets we don&#x27;t need so people don&#x27;t waste their time with them.  I&#x27;m talking, of course, about the axiom of foundation.  It&#x27;s like, why is that even in ZFC?  Ordinary math doesn&#x27;t actually care about the material content of sets; it really affects nothing.  The only thing it does, is that, well, without it, non-well-founded sets are something of a problem, in that they could exist, or they could not, and you can&#x27;t really prove much about them either way.  There are non-well-founded set theories that replace the axiom of foundation with an axiom of antifoundation, allowing you to actually prove things about non-well-founded sets -- Peter Aczel wrote a good book on the subject -- but if you just take it out and don&#x27;t put anything in to replace it, then, well, you just can&#x27;t really say anything about the matter.  The axiom of foundation prunes these irrelevant sets away and tells you to focus on actual math rather than this mystery that&#x27;s not really much of a mystery at all.  So I&#x27;d bet on another axiom along those lines becoming accepted, rather than one that does the opposite.  After all, if we want to take the opposite perspective that anything that can exist ought to... well, then why not non-well-founded sets, too?<p>(Note: Not a logician, my understanding of the two axioms being discussed is based pretty much entirely on this article, so the analogy I claim may not actually work as well as I think it does.)