Non-standard Models of Arithmetic 1

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 a 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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Filed under Conversations, Peano arithmetic

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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“Under Construction”

In the ancient days of the internet, nearly all webpages had an “under construction” graphic on them. It became something of a joke.

Originally, I was using this site mainly for “non-blog” features (the tabs other than the Home tab). Since I had no blog posts, the Home was reading “Not Found”. I felt that was even worse than “Under Construction”. Hence this stub.

Now that I’ve started to do some actual posts, the title of this post no long fits. But I’m too busy/lazy to figure out what my first post should really say. So I guess it really still is “under construction”.

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