• Non-standard Models of Arithmetic 19: We marvel a bit over Enayat’s Prop. 6, and especially Cor. 8. The triple-decker sandwich, aka three-layer cake: ω* ^{U}*⊂

What’s the quest?

I’ll say this very vaguely, since you’re wisely asking for a quest instead of a theorem. (To understand what a mathematician is doing you need to know their quest, not their theorems.)

I think the concept of “standard” model of PA is circular in a subtle way.

Very *very* crudely, the standard natural numbers are 0, 1, 1+1, 1+1+1, … But what does the “…” mean? It means that we keep on going. But *how long* do we keep on going? What sort of expressions 1+…+1 count as natural numbers? Only those where there’s a *standard natural number* of plus signs!

But this is circular.

There’s a lot of evidence that every model of PA “feels standard to itself”, but I want to make this clearer.

To do this, I want a way to exhibit situations where I think I’m dealing only with standard natural numbers, while *you* think some of my numbers are *nonstandard*. That is: I want a framework where in some context some model of PA counts as “standard”, but in some other context that model counts as “nonstandard”.