John Baez and I have been having a conversation about nonstandard models of Peano arithmetic (PA). It started with “John’s dream”, as I’ll call it, which begat a goal: understand a paper by Enayat. For more on the dream and the goal, check out the second post.
Many aspects of nonstandard arithmetic (NSA) don’t lie along our itinerary. This series of posts will be a home for them (or at least the ones that strike my fancy). Other posts may serve as footnotes or appendices to the conversation—stuff that doesn’t lend itself to the dialog format. Neither series is a prerequisite for the other. For convenience, I’ll refer to the other series as “the Conversation” here, and the two series will share a TOC.
One more bit of throat-clearing (or table-setting, pick your favorite metaphor) before I get down to business in the next post. I will mostly write in the Platonic style, where we pretend that there really is a great big universe of sets out there. The “real”, “actual”, “true” universe. It means I can say, for example, “ℕ denotes the system of the actual natural numbers”, or “true arithmetic is the set of formulas that hold in the standard model of Peano arithmetic”. Locutions like this litter the standard literature. For that matter, Platonic style licenses “The actual value of 2ℵ0”, though I doubt I’ll be writing anything like that.
This looks like philosophy, but it’s really just style. People tend to ask questions like, “Where is this reasoning taking place? In what system is there supposed to be this complete theory/model of ZFC/notion of truth…”. It’s nice to answer, “Why, in the actual universe of sets!” Now, people (like me) who talk this way don’t need an actual universe that really exists and is the one and only true universe. We just need an outermost universe we can talk about.
What is that outermost universe? We don’t really care, so long as it obeys a reasonable bunch of axioms. (Usually ZFC. Which is why you won’t hear me talking about “the actual value of 2ℵ0”.) This suggests “Hilbert’s gambit”, aka the formalist turn: all that matters is what we can formally prove from the axioms. I’m sure you’ve heard the motto, “Math is a game with symbols.” (Apparently Hilbert never said this, and it misstates his position. I thought I should note that.) All this blather about the actual universe of sets is “merely corroborative detail, intended to give artistic verisimilitude to an otherwise bald and unconvincing narrative.” (W.S.Gilbert)
Philosophically, I’m ok (or at least equally ok) with both Platonism and formalism. (Not that those are the only two choices!) I don’t know that Cantor and Gödel and Cohen were wrong—maybe there really is “one true universe” out there! But I don’t know that the formalists are wrong, either, and maybe it is just a symbol game! Or maybe one of my old professors was right, and some kind of “mathematical multiverse” best describes the state of affairs.
When discussing this stuff, it’s nearly always way more pleasant to talk as if the ZFC universe really exists, than to be formal. “Naive set theory” gets the job done. That’s what I mean by the Platonist style. I’ve been doing this long enough so I know how to stay safely inside the ZFC boundary—and I’m not enough of a masochist to try actually formalizing stuff (at least fully).
A warning: this is a blog, not a textbook; the order of topics will be driven by whim, not careful organization. If I use a term before defining it, Wikipedia and search engines lie at your fingertips. Sometimes I’ll provide a link, and sometimes I’ll be lazy.
Diagonal Argument 