Peano Arithmetic and the Field of Rationals - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:45:47Z http://mathoverflow.net/feeds/question/65664 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65664/peano-arithmetic-and-the-field-of-rationals Peano Arithmetic and the Field of Rationals Ali Enayat 2011-05-21T18:01:13Z 2011-05-22T21:00:35Z <p>In 1949 Julia Robinson showed the undecidability of the first order theory of the field of rationals by demonstrating that the set of natural numbers $\Bbb{N}$ is <em>first order definable</em> in $(\Bbb{Q}, +, \cdot$).</p> <p>It is not hard to see that Robinson's result can be reformulated in the following symmetric form.</p> <p><strong>Theorem A.</strong> <em>The structures</em> ($\Bbb{N}, +, \cdot$) <em>and</em> $(\Bbb{Q}, +, \cdot$) <em>are bi-interpretable.</em></p> <p>The following generalization of Theorem A is considered folkore (I am not aware of a published reference).</p> <p><strong>Theorem B.</strong> <em>If $(M, +, \cdot)$ is a model of $PA$ (Peano arithmetic), then the field of rationals $\Bbb{Q}^M$ of $(M, +, \cdot)$ is bi-interpretable with $(M, +, \cdot )$.</em></p> <p>Let $EFA$ denote the <em>exponential function arithmetic</em> fragment of $PA$, a fragment also known as $I\Delta_{0}+exp$. </p> <p>Based on <em>a posteriori</em> evidence <em>classical</em> theorems of Number Theory do not require the full power of $PA$ since they can be already verified in $EFA$ (indeed Harvey Friedman has conjectured that even FLT can be verified in $EFA$, with a proof that would be very different from Wiles'). </p> <p>This suggests that in Theorem B one should be able to weaken $PA$ to $EFA$, hence my question:</p> <p><strong>Question</strong>. <em>Is there a published reference for the strengthening of Theorem B, where $PA$ is weakened to $EFA$?</em></p> <p>P.S. The following paper provides an excellent expository account of Robinson's theorem (and related results).</p> <p>D. Flath and S. Wagon, <em>How to Pick Out the Integers in the Rationals: An Application of Logic to Number Theory</em>, American Mathematical Monthly, Nov. 1991.</p>