Harvey Friedman bringing incompleteness and infinity out of quarantine
There is a lot of ordinary mathematics that is outside of ZFC, which Friedman is well aware of but the writer of this article may not be. Grothendieck was not interested in abstract set theory when he introduced what are now called &quot;Grothendieck Universes&quot;. He merely wanted to do algebraic geometry at a high level of abstraction. Similarly, Conway was apparently a bit dismayed when the formalization of his simple idea of &quot;surreal numbers&quot; required deeply unnatural encodings in ZFC...<p>My opinion is still that ZFC itself is unnatural as a foundation for mathematics, precisely because we have to do so much encoding to get anything useful out of it. And whenever you iterate &quot;large&quot; encodings you leave the universe of &quot;ordinary ZFC&quot;. Conway suggested - and this is realized in modern type theory - that we should instead allow arbitrary &quot;free&quot; constructions, such as his surreal numbers, to extend the basic universe of mathematics. To some extend this can be encoded in set theory, but only with ridiculously large (Mahlo) cardinals.<p>This is not a direction that Friedman considers worthwhile, because he thinks that first-order logic and ZFC are inevitable. It&#x27;s a shame that so many people on the FOM mailing list share the same view. There are good reasons why you don&#x27;t find a lot of category theorists on that list anymore...