Defining \$\mathbb{Z}\$ in \$\mathbb{Q}\$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:48:07Z http://mathoverflow.net/feeds/question/86861 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86861/defining-mathbbz-in-mathbbq Defining \$\mathbb{Z}\$ in \$\mathbb{Q}\$ Math-player 2012-01-27T21:17:05Z 2012-01-28T08:24:04Z <p>It was proved by Poonen that \$\mathbb{Z}\$ is definable in \$\mathbb{Q}\$ using \$\forall \exists\$ formula. Koenigsmann has shown that \$\mathbb{Z}\$ is in fact definable by universal formula. What is the simplest geometric interpretation of these results?</p> <p>EDIT: It is important to note, as Joel says, that the first result in this direction was that of Julia Robinson in 1948. The references for the latest results are: <a href="http://arxiv.org/abs/1011.3424" rel="nofollow">http://arxiv.org/abs/1011.3424</a> (Koenigsmann's paper), and <a href="http://www-math.mit.edu/~poonen/papers/ae.pdf" rel="nofollow">http://www-math.mit.edu/~poonen/papers/ae.pdf</a> (Poonen's paper).</p> <p>Thank you</p>