The Further Adventures of Hero and Hera.

Posted on September 27, 2017 by

Those familiar with the neo-logicism literature, may also be familiar with the characters Hero and Hera. Hero was introduced by Crispin Wright in the late `90s, and the story Hero and his sister Hera was fleshed out by Philip Ebert and Marcus Rossberg in 2007*.

In that paper, we learn that Hero and Hera both studied at Ohio State University before parting ways for their PhDs. Hero went to St. Andrews to work with Crispin Wright, and Hera went to Stanford to work with Ed Zalta. Their story ends with both of them having to defend their PhDs by proving/ arguing that 2+2=4.

What follows is a continuation of their stories that I wrote to illustrate a point in my dissertation. I’m not sure if it will make it into the final draft, so here it is for your bemusement. But be warned, there are quite a few bad math jokes.

After graduating with his doctorate from St. Andrews, Hero landed a couple of short postdocs and teaching appointments, but the Tory government was making it more and more difficult for him to stay in the UK, so he had a tough decision to make. Hero eventually chose stability and landed a job at a prestigious management consultancy firm in New York City that valued his PhD and experience with logic and the philosophy of mathematics. But, despite his career change Hero never stopped caring about neo-logicism, and his regular conversions with his beloved sister—now a tenured assistant professor at the University of Alberta (she was willing to put up with the weather for a TT job at a research institution)—kept him reasonably up to date with the world of  academic philosophy.

After being cooped up for a couple of days due to a freak snowstorm (the second one that year), Hero decided to put his mind to work considering the notion of set. He had lost his copy of Jech’s Lectures in Set Theory, but that was okay, he wanted to work from first principles anyway. So hero started with a couple axioms he knew he would need: extensionality and infinity (because, still thinking like a neo-logicists, he wanted to create a mathematically fruitful theory). Hero now has the basis of an identity criterion for sets, and has guaranteed the existence of enough sets to at least model arithmetic. At this point our Hero isn’t sure how to proceed, but intuitively thinks that the mathematical universe is pretty big (but not superhuge), and as a platonist, thinks that there are sets outside Gödel’s constructible universe. Recall that Hero’s facility with second-order logic was second only to that of his undergrad advisor at Ohio State University, so he was able to reverse engineer a set-theoretic axiom that guarantees that $V \neq L$. That task took him longer than it took the MTA to get the subways running again, so Hero had little time to devote to set theory after that.

In the mean time, Hera had been invited to speak at a workshop on set theoretic foundations in the City, and of course jumped at the chance to get out of Edmonton in the middle of winter and see her brother. So a couple of weeks later Hero and Hera were drinking Scotch in Hero’s tiny Bushwick apartment and got talking about his set theory project. Hera quickly worked out, despite the Scotch, that Hero had somehow posited the existence of a Ramsey cardinal mostly accidentally (they are, after all, accomplished mathematical logicians).

“Extensionality and infinity make sense (though have you read Aczel?), but why on earth would you take the existence of any large large cardinal as axiomatic? And besides, what does that get you other than lots of sets?”

“Well, I’m not entirely sure, or, rather, I haven’t the foggiest..” replied Hero, “but the universe must be extra-constructible, right?”

“Well, I think so, but. . . ” she paused “you can’t just decide how big the universe is! Why not supercompact, or hell, superhuge?”

“Now hold on!” protested Hero.

“At least posit a measurable Woodin for comedy’s sake.”  (The Scotch was starting to have an effect.)

After a a worryingly long tangent involving lots of poor and slightly distasteful jokes about the names of large cardinals, Hera brought them back to Hero’s set theory project. “But seriously, what makes you think you can just declare that the universe is a certain size, and what good is that going to do anyway?” Hero gestured as if he were about to speak. He sipped his drink instead. Hera continued, “It’s not even clear that without other axioms, what positing any large large cardinal would get you in terms of mathematical strength. Even if your theory does turn out to be fairly strong, surely the size of the universe–”

“Don’t call me ‘Shirley.”

“Oy! the size of the universe should be determined more than your vague intuitions that it’s ‘pretty big’. We generally think that there are accessibles because that assumption implies the consistency of ZFC which we assume anyway. ZFC, by the way, tells us a lot more about sets than ‘there’s a Ramsey’. Anyway, think about that while I top us up.”

“Don’t you have to give a talk tomorrow?” inquired Hero. A couple of minutes passed and Hera looked to be ready to get back to their discussion. Hero dove back in.

“First of all, it’s name is ‘Frank’. More importantly, this is an unfinished project. I intended to add more axioms. I suppose the task is to fill in my theory in a way that justifies Frank’s existence.”

“I’ll grant that that it might be possible. Still though..”

The conversation devolved from there, but rest assured that Hera gave a
brilliant talk, and was only slightly late to the workshop due delays on the L.

*Ebert and Rossberg (2007), “What is the Purpose of Neo-Logicism?”, Travaux de Logique, 18, pp. 33-61