Non-standard Models of Arithmetic 4

MW: I wrote: “I don’t like calling the omega of a model of ZF a standard model, for philosophical reasons I won’t get into.”

JB: I like it, because I don’t like the idea of “the” standard model of arithmetic, so I’m happy to see that “the” turned into an “a”.

MW: Well, on second look, I see Enayat uses “ZF-standard” in the body of the paper. I’m fine with that.

Anyway, back when I was in grad school, I wondered whether there were models of true arithmetic that are not omegas. Answer: yes. I wrote up a short note, just for myself. I’m sure the result is well-known, maybe even in one of the three books we’ve mentioned. (Also, at this remove I no longer remember if I came up with the argument on my own, or if my advisor gets the credit.)

JB: What’s “true arithmetic”?

MW: Just the theory of the standard model, i.e., all closed formulas satisfied by . Now, if you don’t believe the term “the standard model” means anything, then I’d have to write a much longer and more confusing definition.

JB: What’s ? (Most of the time I act like I know, but when I’m doing logic I admit that there are lots of different things one could mean by it.)

I don’t think “the standard model” means anything unless it’s defined. So please give me your definition!

If I pick a set theory, say ZFC or whatever, I can prove in there that’s there’s an initial model of PA (one with a unique embedding in any other), and I’m happy to call that the “standard model of PA in ZFC”. Then I can talk about the theory consisting of all closed sentences in this model. All that’s fine with me.

MW: Hmmm… It looks like we’re not going to be able to avoid all philosophy. Fair warning: my philosophy of math is a mish-mash of intuitionism, formalism, and platonism. And I’m not using those terms in the usual way! But I think this is a topic for the next post. After that, I promise we’ll get back to Real Math.