Welcome to MathOverflow

Question

3

2

The real numbers can be axiomatically defined (up to isomorpism) as a Dedekind-complete ordered field.

What is a similar standard axiomatic definition of the integer numbers?

A commutative ordered ring with positive induction?

flag	asked May 1 2010 at 19:30 Victor Makarov 271●1●9

A free commutative ring with 1 of rank 0? An initial object in the category of rings with 1? – Arturo Magidin May 1 2010 at 19:55

I don't think that this question belongs to MO. your description is right and characterizes Z. also Z is characterized by a univeral property: it is initial in the category of commutative rings. – Martin Brandenburg May 1 2010 at 19:55

1

I cannot answer this question, but I believe that “being the initial object in the category of rings” characterises the integers up to isomorphism; I was wondering if this property can be used somehow in order to construct a (second order?) categorical theory of integers. – Antonio E. Porreca May 1 2010 at 20:05

4

Isn't it considerable overkill to talk about categories? – Gerald Edgar May 1 2010 at 20:12

3

Here is another characterization: $\mathbb{Z}$ is a ordered ring (commutative, nontrivial, with identity) with "double induction". Every subset $S$ such that $0 \in S$ and $x \in S \Rightarrow x \pm 1 \in S$ is already $\mathbb{Z}$. – Martin Brandenburg May 1 2010 at 20:34

show 3 more comments

François G. Dorais · Accepted Answer · 2010-05-01 23:39:06Z UTC

10

It's the unique commutative ordered ring whose positive elements are well-ordered.

Edit: Oh, François basically already said this, didn't notice. Should I delete this?

edited May 1 2010 at 23:39

François G. Dorais♦

answered May 1 2010 at 23:36

Harry Altman
980●1●10●16

1

It's an elegant formulation. Don't delete. – François G. Dorais♦ May 1 2010 at 23:39

I would be very grateful for any references on proofs of the uniqueness theorem. – Victor Makarov May 2 2010 at 1:25

4

you should prove it by yourself. – Martin Brandenburg May 2 2010 at 9:27

François G. Dorais · Answer 2 · 2010-05-01 23:32:15Z UTC

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.

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.

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 this question for related information.

Axiomatic definition of integers

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.