Reference Request: Non-Standard Models of PA - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T20:58:31Z http://mathoverflow.net/feeds/question/90002 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90002/reference-request-non-standard-models-of-pa Reference Request: Non-Standard Models of PA Samuel Reid 2012-03-02T00:32:10Z 2012-03-03T00:05:00Z <p>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).</p> <p>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.</p> <p>Thank you!</p> http://mathoverflow.net/questions/90002/reference-request-non-standard-models-of-pa/90008#90008 Answer by Steven Landsburg for Reference Request: Non-Standard Models of PA Steven Landsburg 2012-03-02T01:36:28Z 2012-03-02T01:36:28Z <p>I've found Kaye's web pages <a href="http://web.mat.bham.ac.uk/R.W.Kaye/papers/tennenbaum/tennenbaum.xhtml" rel="nofollow">here</a> pretty enlightening.</p> http://mathoverflow.net/questions/90002/reference-request-non-standard-models-of-pa/90034#90034 Answer by Ed Dean for Reference Request: Non-Standard Models of PA Ed Dean 2012-03-02T13:04:46Z 2012-03-02T13:04:46Z <p>Richard Kaye's book <em>Models of Peano Arithmetic</em> 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.</p> http://mathoverflow.net/questions/90002/reference-request-non-standard-models-of-pa/90081#90081 Answer by Timothy Chow for Reference Request: Non-Standard Models of PA Timothy Chow 2012-03-02T22:48:43Z 2012-03-02T22:48:43Z <p>Boolos and Jeffrey's book <i>Computability and Logic</i> has a nice account of Tennenbaum's theorem, at least in the third edition.</p> http://mathoverflow.net/questions/90002/reference-request-non-standard-models-of-pa/90083#90083 Answer by Benedict Eastaugh for Reference Request: Non-Standard Models of PA Benedict Eastaugh 2012-03-02T23:02:52Z 2012-03-03T00:05:00Z <p>Peter Smith has a pretty good <a href="http://www.logicmatters.net/resources/pdfs/TennenbaumTheorem.pdf" rel="nofollow">handout on Tennenbaum's theorem</a> that I found useful when learning that material. As others have mentioned, Richard Kaye's <em>Models of Peano Arithmetic</em> is the go-to reference work here. Kossak and Schmerl's <em>The Structure of Models of Peano Arithmetic</em> gives the state of the art, but you probably won't need this one.</p>