Author Archives: Michael Weiss

Non-standard Models of Arithmetic 8

JB: So, you were going to tell me a bit how questions about the universe of sets cast their shadows down on the world of Peano arithmetic.

MW: Yup. There are few ways to approach this. Mainly I want to get to the Paris-Harrington theorem, which Enayat name-checks.

First though I should do some table setting of my own. There’s a really succinct way to compare ZF with PA: PA = ZF − infinity!

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Non-standard Models of Arithmetic 7

MW: Our goal for the next few posts is to understand Enayat’s paper

• Ali Enayat, Standard models of arithmetic.

JB: Yee-hah!

MW: I’m going to take a leisurely approach, with “day trips” to nearby attractions (or Sehenswürdigkeiten, in the delightful German phrase), but still trying not to miss our return flight.

Also, I know you know a lot of this stuff. But unless we’re the only two reading this (in which case, why not just email?), I won’t worry about what you know. I’ll just pretend I’m explaining it to a younger version of myself—the one who often murmured, “Future MW, just what does this mean?”

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Friedberg’s Enumeration without Duplication

Recursively enumerable (r.e.) sets are “semi-decidable”: if x belongs to an r.e. set W, then there’s a terminating computation proving that fact. But there may not be any way to verify that x does not belong to W. The founding theorem of recursion theory—the unsolvability of the Halting Problem—furnishes an r.e., non-recursive set. For this set, we have a program listing what’s in W, but no program listing what’s out.

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Non-standard Models of Arithmetic 5

MW: John, you wrote:

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

and you gave a good elucidation in post 2 and post 4. But I’d like to defend my right to “true arithmetic” and “the standard model \mathbb{N}“.

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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”.

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Non-standard Models of Arithmetic 3

[Reminder: JB=John Baez, MW=Michael Weiss.]

MW: Besides Kaye and Kossak & Schmerl., I should mention the book by Hájek and Pudlák, but I don’t have a copy of that. Thanks muchly for the Enayat paper, which looks fascinating.

What you and Enayat are calling the “standard” model of arithmetic is what I used to call “an omega”, i.e., the omega of a model of ZF. Is that the new standard terminology for it? I don’t like it, for philosophical reasons I won’t get into. (Reminds me of the whole “interpretations of QM” that books have to skirt around, when they just want to shut up and calculate.)

Leaving ZF out of it, a friend in grad school used to go around arguing that 7 is non-standard. Try and give a proof that 7 is standard using fewer than seven symbols. And of course for any element of a non-standard model, there is a “proof” of non-standard length that the element is standard. I think he did this just to be provocative. Amusingly, he parlayed this line of thought into some real results and ultimately a thesis.

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Non-standard Models of Arithmetic 2

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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