## The Omegas ### Or: Standard Foundations ### Or: Can Mathematics be Formalized (Friedman, 1997) ### Or: Does mathematics need new axioms (Feferman, 2000) ### Or: Less painful than hell (Hirst, 2000, slightly misquoted) ### Or: These Hell Torments ### Or: The official doctrine of mathematics ### Or: Very few people seem to have a problem with that ### Or: The ~~Current~~ Old Foundations ### Or: Numbers are sets ### Or: Booleans are sets ### Or: Functions are sets ### Or: ZFC > _These axioms are the official doctrine - Remarkably, this is not just the official doctrine for set theory, it turns out that this is the official doctrine for mathematics! - Very few people seem to have a problem with that which I find quite remarkable._ > -NJ Wildberger > _For some random reason, set theory won._ > -Kevin Buzzard 0: Well ZFC has never really been _used_ as foundations in any real sense. 1: What do you mean "used as foundations." 0: I mean nobody uses it. 1: What?! 0: Or almost nobody. They usually just talk about it or assert that ZFC proves such-and-such. It's extremely rare to see mathematicians actually working _within_ ZFC as a formal system. 1: That doesn't seem so bad. I mean we programmers don't really write in machine code, like ever. But it's sort of the "foundations" of any language that compiles to it, in a sense. 0: No no, I mean even books about ZFC don't work within ZFC. 1: WHAT?! 1: Why would books on ZFC not work within ZFC? 0: Mathematicians don't like to. 1: Even the ones who choose to write books about it? 0: Especially those ones. Here look. This is from a book published in the year 2000 by an extremely accomplished logician. Brilliant guy named Jeff Hirst. Someone who chose to go into foundations and did some damn good work there. So you might expect he'd be working inside the formal system. Here's what actually happens. Now this is one of the best presentations of ZFC in my opinion, but it's also an example of how ZFC isn't exactly "used as foundations." Come read it with me. ![[zfc-hirst-01.jpg]] 1: "These hell torments"? 0: Yeah that's axiomatic set theory. 1: What's this book about? 0: Axiomatic set theory. 1: What?! 0: That's the point. No one within mathematics really "likes" ZFC. Not foundational people like Hirst who chose foundations as their favorite area. Not even set theorists like Thomas Jech who chose set theory specifically as their favorite area. Now to be fair, the Hirst book here isn't entirely about ZFC. But a third of it is devoted to set theory and "standard foundations." Hirst isn't acting unusual here. Like I said, this book is one of the best presentations of this stuff at an undergrad level in my opinon. This is just sort of how ZFC is viewed and how it's used. The "paradise" in that quote is pretty clearly "the paradise Cantor created," the one David Hilbert is always quoted as talking about: informal set theory, the kind that uses standard mathematical reasoning and eventually leads to paradoxes. In contrast to that paradise are "these Hell torments," which is naturally, well, axiomatic set theory. 1: And this is a book _about axiomatic set theory?_ 0: Yep. At least this part is. This book is extremely good at choosing quotes for section headings. A lot of the quotes in this section are about hell. Or L, which is a thing Gödel invented in the course of studying, well, axiomatic set theory. ![[zfc-hirst-07.jpg]] 1: Damn, seems like mathematicians really don't like axiomatic set theory. 0: Yeah, there are a bunch of funny dog whistles throughout this part of the book about how mathematics probably needs some new foundations. Can't disagree. ![[zfc-hirst-10.jpg]] 1: Why not just come up with new foundations? 0: People have. This guy Harvey Friedman, the one Hirst is quoting in the image below, he kicked off a revolution that completely changed our understanding of what mathematics is at the most fundamental level. ![[zfc-hirst-13.jpg]] 1: What kind of revolution? 0: It turns out most of mathematics is equivalent to one of five sentences. 1: Most of mathematics is WHAT?! 0: Equivalent to one of five sentences. 1: HOLY F--- goto: [[The Sigmas]]