Axiomatic definition of integers

Question

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

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

A commutative ordered ring with positive induction?

Answer 1

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

Answer 2

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.

Answer 3

Part 1: Abelian group introduction

(See Part 2 for the final answer).

I feel that for a fully elegant definition of integers one needs to readdress the definition of an Abelian group. For the sake of communication I will even introduce a synonym minusop for abelian groups to stress the independence of their definition from the general groups. The general groups have an elegant definition (perhaps more than one). Then adding the commutativity axiom one gets Abelian groups. This seems to me unnecessarily complex. Let me define Abelian groups directly.

Symmetric operations

Symmetric operations should not be confused with much more special commutative operations.

By definition, an operation   $\#:X^2\rightarrow X$   is called symmetric   $\Leftarrow:\Rightarrow$

$$ \forall_{x\ y\ z\in X}\qquad x\ \#\ (y\ \#\ z)\ \ =\ \ z\ \#\ (y\ \#\ x) $$

This property will appear in the definition of minusop as an axiom. Both addition and subtraction are symmetric operations in any Abelian group. Every commutative operation is symmetric.

Minusop

By definition, a minusop is an ordered pair   $(X\ -)$,   where   $X$   is an arbitrary set, and   $-$   is a binary operation in   $X$,   such that the following three axioms hold:

1. $x-(y-z)\ =\ z-(y-x)$
2. $x-x\ =\ y-y$
3. $x-(x-x)\ =\ x$

for arbitrary   $x\ y\ z\in X$.

Every Abelian group admits a standard interpretation as a minusop; and every non-empty minusop admits its standard interpretation as an Abelian group, so that Abelian groups and non-empty minusops are essentially the same objects.

(I am not writing the last obvious statement in any detail to keep this post sensibly short).

REMARK

A class of operations even more general than symmetric is still useful--I call operation   $\#:X^2\rightarrow X$   insider trading (in mathematics it's legal)   $\Leftarrow:\Rightarrow$

$$\forall_{u\ w\ x\ y\ \in\ X}\qquad (u\ \#\ w)\ \#\ (x\ \#\ y)\ \ =\ \ (u\ \#\ x)\ \#\ (w\ \#\ y)$$

Every symmetric operation is an insider trading.

A simplification by Emil Jeřábek

The last two of the three minusop axioms above (in the previous section before REMARK) can be replaced by one, as pointed out Emil Jeřábek, as now presented below. A minusop can be axiomatized by the following two conditions:

1. Symmetry:     $x-(y-z)\ =\ z-(y-x)$
2. Zero:     $x-(y-y)\ =\ x$

for arbitrary   $x\ y\ z\ \in\ X$.

It can be seen instantly that the earlier three axioms imply the two above (the new first axiom is a repetition of the old first axiom).

In the other direction, a substitution of $y$ by $x$ in axiom zero gives the old axiom 3. Furthermore, assuming the above two axioms we obtain:

$$ x-x\ =\ (x-x)-((y-y)-(y-y))\ =\ (y-y)-((y-y)-(x-x)) $$ $$ (y-y) - (y-y)\ =\ y-y $$

which proves the old axiom 2. The new 2-axiom system is equivalent to the old 3-axiom system.

Answer 4

Part 2: Cyclands and integers

(This text is a continuation of Part 1 from the same thread).

First I'll axiomatize here the cyclic groups--I'll call them cyclands (to avoid any confusion during the definition stage; later one can go back to the standard naming; the same goes for Abelian groups and minusops). Then there will come time for integers.

Cyclands

By definition, a cycland is an ordered triple   $(X\ -\ 1)$   such that   $X$   is an arbitrary set,   symbol   $-$   stands for a binary operation in   $X$,   $1\in X$, and   the following (algebraic induction-like) axiom holds:

$$ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$$

Integers

DEFINITION   Integers (i.e. the system of integers) is a cycland   $(\mathbb Z\ -\ 1)$   such that the following (additional) two axioms hold:

1. $1-x\ne x$
2. $1-x=x-1\quad\Rightarrow\quad x=1$

for arbitrary   $x\in\mathbb Z$.

Integers from scratch

To have an easy overview of the definition of integers let me list all the relevant axioms directly (without mentioning minusops and cyclands).

The system of integers is an ordered triple   $(\mathbb Z\ -\ 1)$   such that the following six axioms hold:

1. $ x-(y-z)\ =\ z-(y-x)$
2. $ x-x\ =\ y-y$
3. $ x-(x-x) = x$
4. $ 1-x\ne x$
5. $ \left(1-x=x-1\right)\ \Rightarrow\ \left(x=1\right)$
6. $ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$

for arbitrary   $x\ y\ z\ \in\ \mathbb Z$.

Other standard constructions or notions

We define in   $\mathbb Z$   the usual, like:

• $0\ :=\ 1-1$
• $\mathit{Neg}(x)\ :=\ 0-x$
• $x+y\ :=\ x -\mathit{Neg}(y)$

Furthermore, in every cycland, in particular in   $\mathbb Z$,   there is exactly one binary operation   $\cdot$   which has the following two properties:

• $x\cdot 1\ =\ x$
• $x\cdot(y-z)\ =\ x\cdot y - x\cdot z$

for arbitrary   $x\ y\ z\ \in\ \mathbb Z$.

Answer 5

This has been beaten to death, but here goes...

The positive numbers in a ring are the sums of ones, i.e. the set of positive numbers is the smallest inductive subset of the ring. The negative numbers are the negatives of the positive numbers. A ring is minimal if every number is positive, negative, or zero.

$\bf Z$ is the unique minimal ring in which zero is neither positive nor negative.

${\bf Z}/n\bf Z$ is the unique minimal ring in which zero is either positive or negative.

Similarly, the positive numbers in a field are the sums of ones or their ratios. A field is minimal if every number is positive, negative, or zero.

$\bf Q$ is the unique minimal field in which zero is neither positive nor negative.

${\bf Z}/p\bf Z$ is the unique minimal field in which zero is either positive or negative.

Answer 6

The integers can also be characterized as a well-ordered abelian group Theorem 20.14 in Warner:

Warner, Seth, Modern algebra. Vols. I, II, Prentice-Hall Mathematics Series. Englewood Cliffs, N.J.: Prentice-Hall, Inc. x, 806 p. (1965). ZBL0134.24903.

Answer 7

The Grothendieck completion (https://en.wikipedia.org/wiki/Grothendieck_group) of the smallest inductive set.

Answer 8

Here is another definition that I believe should work: $\mathbb{Z}$ is the smallest ring into which $\mathbb{N}$ can be embedded as a semiring. This can be made rigorous by the following universal property: whenever $R$ is a ring and $f:\mathbb{N} \to R$ is a semiring embedding (an injective semiring homomorphism), there is a unique ring embedding $g: \mathbb{Z} \to R$ such that the diagram $\require{AMScd}$ \begin{CD} \mathbb{N} @>c>> \mathbb{Z}\\ @. {_{\rlap{f}}\style{display: inline-block; transform: rotate(30deg)}{{\xrightarrow[\rule{4em}{0em}]{}}}} @VVgV\\ @. R \end{CD} commutes (here $c$ is the unique map from $\mathbb{N}$ into $\mathbb{Z}$, since $\mathbb{N}$ is the initial semiring). This defines $\mathbb{Z}$ up to unique isomorphism, and captures the intuitive notion that $\mathbb{Z}$ is obtained from $\mathbb{N}$ by simply "adding inverses for every element" to make the resulting structure a ring.