Axiomatic definition of integers - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:06:02Z http://mathoverflow.net/feeds/question/23193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23193/axiomatic-definition-of-integers Axiomatic definition of integers Victor Makarov 2010-05-01T19:30:13Z 2010-05-02T02:19:53Z <p>The real numbers can be axiomatically defined (up to isomorpism) as a Dedekind-complete ordered field.</p> <p>What is a similar standard axiomatic definition of the integer numbers?</p> <p>A commutative ordered ring with positive induction?</p> http://mathoverflow.net/questions/23193/axiomatic-definition-of-integers/23200#23200 Answer by François G. Dorais for Axiomatic definition of integers François G. Dorais 2010-05-01T20:50:56Z 2010-05-01T23:32:15Z <p>The ring $\mathbb{Z}$ is the unique ordered ring which satisfies full second-order induction: $$\forall X(0 \in X \land (\forall n \geq 0)(n \in X \to n+1 \in X) \to (\forall n \geq 0)(n \in X)),$$ where $X$ varies over all subsets of $\mathbb{Z}$ (or even all sets). In the comments, Martin Brandenburg has given yet another characterization of $\mathbb{Z}$ which does not assume the ordering. </p> <p>A dual characterization is that every nonempty subset of $\mathbb{Z}$ which is bounded below has a minimal element. This is closer to the characterization of $\mathbb{R}$. Note that all of these characterizations only make sense in standard second-order logic, but the proposed characterization of $\mathbb{R}$ has the same problem. </p> <p>The ring of integers also has categorical characterizations. For example, as proposed in the comments, $\mathbb{Z}$ is initial object in the category of (ordered) rings. See <a href="http://mathoverflow.net/questions/17483/defining-free-monoid-without-nat" rel="nofollow">this question</a> for related information.</p> http://mathoverflow.net/questions/23193/axiomatic-definition-of-integers/23212#23212 Answer by Harry Altman for Axiomatic definition of integers Harry Altman 2010-05-01T23:36:20Z 2010-05-01T23:39:06Z <p>It's the unique commutative ordered ring whose positive elements are well-ordered.</p> <p>Edit: Oh, François basically already said this, didn't notice. Should I delete this?</p>