Monthly Archives: May 2019

Non-standard Models of Arithmetic 2

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JB: The only books I know on models of Peano arithmetic are Kaye’s Models of Peano Arithmetic and Kossack and Schmerl’s more demanding The Structure of Models of Peano Arithmetic, and I’m trying to read both. But I have a certain dream which is being aided and abetted by this paper:

• Ali Enayat, Standard Models of Arithmetic.

Roughly, my dream is to show that “the” standard model is a much more nebulous notion than many seem to believe.

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Non-standard Models of Arithmetic 1

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This is the first in a series of posts, recording an e-conversation between John Baez (JB) and me (MW).

JB: I’ve lately been trying to learn about nonstandard models of Peano arithmetic. Do you know what a “recursively saturated” model is? They’re supposed to be important but I don’t get the idea yet.

MW: What books and/or papers are you reading? I used to know this stuff, indeed my thesis (1980) was on existentially complete models of arithmetic. When I looked at it a couple of years ago, I was amazed at how much I’d forgotten. Talk about depressing.

Anyway, I’ll toss out a few vague ideas, to see if they help. Maybe this will be the push I need to get back to Kaye’s book, or even Kossak & Schmerl. I picked them up a few months ago, hoping to revisit my youth, but I didn’t make it past the prefaces.

As Hodges puts it, model theory is “algebraic geometry minus fields”. If you have an algebraic number r in an extension field K/F, it’s natural to look at all the polynomials in F[x] which have r as a root. It turns out that this is a principal ideal, generated by the minimal polynomial.

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The Lambda-Calculus and the Plotkin/Scott Model

I won’t define the (untyped) λ-calculus; you have the rest of the internet for that. But the basic formalism is remarkably simple. Instead of writing x\mapsto 2\cdot x, for example, we write λx.(2·x). The λ-term ( λx.(2·x))7 stands for the application of the doubling function to 7, and we say that ( λx.(2·x))7 reduces to 2·7=14. (This is called β-reduction or β-conversion.)

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