User none - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:49:29Z http://mathoverflow.net/feeds/user/24853 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101160/axiom-to-exclude-nonstandard-natural-numbers/101203#101203 Answer by none for Axiom to exclude nonstandard natural numbers none 2012-07-03T03:28:21Z 2012-07-03T03:53:34Z <p>By Gödel's incompleteness theorem, there can't be any such axiom in a first-order, recursively enumerable theory.</p> <p>You can axiomatize $\mathbb N$ by adding an infinitary rule of inference, the Hilbert $\omega$-rule:</p> <p>$$P(0)\wedge P(1) \wedge P(2) \wedge \cdots \over \forall n P(n)$$</p> <p>to PA for each arithmetic predicate P. This says, if predicate P holds for each of the natural numbers (0,1,2...) then you can deduce the formula $\forall n P(n)$. The resulting system is called <a href="http://en.wikipedia.org/wiki/%CE%A9-consistent_theory#.CF.89-logic" rel="nofollow">ω-logic</a>. </p> <p>Obviously this is something of a "cheat" since you no longer have an effective theory. As one example (maybe there are better ones) of how it can be used, Michael Rathjen's article "The Art of Ordinal Analysis" describes using the $\omega$-rule to analyze stronger and stronger arithmetic theories, and is pretty interesting.</p>