# Reference Request: Non-Standard Models of PA

I am attempting to write an expository paper on non-standard models of PA that is accesible to students taking an introductory graduate course in mathematical logic (covering Godel's incompleteness theorems, the diagonalization lemma, models, etc.). In this paper I want to give an explanation of some results such as Tennanbaum's Theorem (there does not exist a countable recursive model of PA that is not isomorphic to the standard model). By, "give an explanation of", I mean to actually work through an explanation of the proof, some of the techniques involved, and the general overlap between techniques used in the proofs of Tennanbaum's theorem, some theorems proven by Rosser on extensions of PA, Robinson's overspill lemma, etc. (Note: I want to avoid digressing into an explanation of forcing if possible).

My question is, what books or online resources do you know of that would be useful for me? That is, do you happen to know of surveys of these topics that are around on arXiv or JSTOR? I have been digging through the mathematical logic section of arXiv for papers and I found a few that are useful, but I thought that some mathematicians/logicians on MO might know of some papers that give a just survey of the introductory results regarding non-standard models of PA.

Thank you!

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Richard Kaye's book Models of Peano Arithmetic is good and accessible. – Ed Dean Mar 2 '12 at 0:41
@EdDean: It seems that it is not accessible to purchase as it is not available on amazon or apparently anywhere else on the internet... That book is exactly what I am looking for though, care to make your suggestion an answer so I can accept it? – Samuel Reid Mar 2 '12 at 6:41
Regarding the nonexistence of computable nonstandard models, see mathoverflow.net/questions/12426/…. Essentially the same argument as for ZFC works for PA. Tennenbaum's theorem is stronger than that, asserting that neither $+$ nor $\codt$ separately is computable. – Joel David Hamkins Mar 3 '12 at 0:59
Hi, this response may be a bit late, but I've found Andrey Bovykin's expository papers and his online video lectures very useful in getting a down-to-Earth understanding of non-standard models. He doesn't specifically talk about Tennanbaum's theorem to my recollection, but I've found him very understandable and intuitive. – Everett Piper Mar 11 '12 at 1:43

## 4 Answers

Richard Kaye's book Models of Peano Arithmetic is good and accessible. And I know that what Frank said in his comment, about its availability as a pdf online, is indeed true; though like Frank, I shan't give a link here.

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It’s also available as a djvu. – Emil Jeřábek Mar 2 '12 at 13:15

I've found Kaye's web pages here pretty enlightening.

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Peter Smith has a pretty good handout on Tennenbaum's theorem that I found useful when learning that material. As others have mentioned, Richard Kaye's Models of Peano Arithmetic is the go-to reference work here. Kossak and Schmerl's The Structure of Models of Peano Arithmetic gives the state of the art, but you probably won't need this one.

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The Kossak/Schmerl book is excellent. – Joel David Hamkins Mar 2 '12 at 23:33

Boolos and Jeffrey's book Computability and Logic has a nice account of Tennenbaum's theorem, at least in the third edition.

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