What makes maths beautiful?
Use of the phrase &quot;not-necessarily-all-that-well-founded&quot; is unfortunate since standard set theory studies the properties of extensional, well-founded objects in general. Though clearly you didn&#x27;t have the technical understanding of well-foundedness in mind. Just might confuse ppl because there is a branch of set theory called &quot;non-well-founded set theory&quot; that is perfectly respectable -- I.e., and &quot;well-founded&quot; in the sense you intend -- but not about the same objects studied in the standard context.<p>I believe what you have in mind regarding set theory as &quot;not ideal for &#x27;foundations&#x27;&quot; (why is the word foundations in quote, btw?) is that there is a recent push to reexamine the prospects more intensional foundations based on type theory. The most promising and currently fashionable version of this push is Homotopy Type Theory (HoTT), which has been given press and prestige from the IAS at Princeton and interest from Fields Medalist Vladimir Vovodsky (sp?). There are plausible reasons given, though often polemical, for preferring some sort of type theoretical or category theoretical foundations over set theoretical ones. Perhaps most importantly, for readers of HN, the comparatively easy formalization into computer-checkable proof and computer-assisted proof (look up, for instance, Coq).<p>That said, there is a long way to go before HoTT (or something like it) displaces traditional foundations and set theory. For one thing, there is no obvious axiomatization or way to confirm that HoTT is a suitable foundation for extant non-foundational mathematics. For one thing, standard type theory (MLTT) plus Vovodsky&#x27;s univalence axiom is proof-theoretically equivalent to a subsystem of analysis. I would guess that, for instance, you can&#x27;t prove that all Borel sets are measurable using MLTT plus univalence (but this is totally speculative and I would appreciate being corrected).<p>Last -- I suspect the only mathematicians who find set theory proofs beautiful are set theorists. Most other mathematicians don&#x27;t really give af about set theory.