These notes on Simple Sets are a grabbag about the simple sets of recursion theory. If you don’t know what those are, you probably are not interested, but the Wikipedia article is nice and short and gives the basics.
The last result uses the “shiny black box” (see below), which seems like cheating, but isn’t!
I wrote up the notes sometime in the 1980s based on two papers, and on a lecture by Michael Stob at MIT (reporting on joint work with Maas and Shore). They discuss effectively simple sets and promptly simple sets.
Why do we (mostly) say atomic bomb instead of nuclear bomb, which is technically more correct? This was asked on the History of Science and Math stackexchange. Here’s my answer.
Filed under History, Physics
MW: Time to talk about the Paris-Harrington theorem. Originally I thought I’d give a “broad strokes” proof, but then I remembered what you once wrote: keep it fun, not a textbook. Anyway, Katz and Reimann do a nice job for someone who wants to dive into the details, without signing up for a full-bore grad course in model theory. So I’ll say a bit about the “cast of characters” (i.e., central ideas), and why I think they merit our attention.
JB: So, you were going to tell me a bit how questions about the universe of sets cast their shadows down on the world of Peano arithmetic.
MW: Yup. There are few ways to approach this. Mainly I want to get to the Paris-Harrington theorem, which Enayat name-checks.
First though I should do some table setting of my own. There’s a really succinct way to compare ZF with PA: PA = ZF − infinity!
MW: Our goal for the next few posts is to understand Enayat’s paper
• Ali Enayat, Standard models of arithmetic.
MW: I’m going to take a leisurely approach, with “day trips” to nearby attractions (or Sehenswürdigkeiten, in the delightful German phrase), but still trying not to miss our return flight.
Also, I know you know a lot of this stuff. But unless we’re the only two reading this (in which case, why not just email?), I won’t worry about what you know. I’ll just pretend I’m explaining it to a younger version of myself—the one who often murmured, “Future MW, just what does this mean?”
Recursively enumerable (r.e.) sets are “semi-decidable”: if x belongs to an r.e. set W, then there’s a terminating computation proving that fact. But there may not be any way to verify that x does not belong to W. The founding theorem of recursion theory—the unsolvability of the Halting Problem—furnishes an r.e., non-recursive set. For this set, we have a program listing what’s in W, but no program listing what’s out.
JB: It’s interesting to see how you deploy various philosophies of mathematics: Platonism, intuitionism, formalism, etc. For a long time I’ve been disgusted by how people set up battles between these, like Punch-and-Judy shows where little puppets whack each other, instead of trying to clarify what any of these philosophies might actually mean.
For example, some like to whack Platonism for its claim that numbers “really exist”, without investigating what it might mean for an abstraction—a Platonic form—to “really exist”. If you define “really exist” in such a way that abstractions don’t do this, that’s fine—but it doesn’t mean you’ve defeated Platonism, it merely means you’re committed to a different way of thinking and talking.