What is the canonical way to extend Peano's axioms to the set of all integers?

Question

My first idea on how to do this would be:

1. $0\in\mathbf Z$
2. $\forall x\in\mathbf Z,Sx\in\mathbf Z\land Px\in\mathbf Z$
3. $P$ and $S$ are injective
4. $\forall x\in\mathbf Z,PSx=SPx=x$
5. some induction axiom that holds for decreasing sequences as well (intuitively, if $\phi$ holds for $x_0\in\mathbf Z$ and $\forall x\in\mathbf Z,\phi(x)\to\phi(Px)$ then $\phi$ holds for all numbers less than $x_0$)
6. addition (with the two standard axioms $\forall x\in\mathbf Z,x+0=x$ and $\forall x,y\in\mathbf Z,Sx+y=S(x+y)$ and an extra axiom $\forall x,y\in\mathbf Z,Px+y=P(x+y)$)
7. multiplication

defining $-1=P0$, $-2=PP0$, etc.

Is this the canonical way to do it (if so, what is it called) or is there a problem I haven't noticed (if so, what would be the "right way")?

Answer 1

The canonical way to extend Peano arithmetic to the integers is not by changing the language or axioms, but instead by treating integers as equivalence classes of pairs of natural numbers.

We think of an integer $x$ as any pair $(a,b)$ of natural numbers with $a-b=x$. More formally, we translate arithmetic statements about integers by first replacing the integers with pairs of natural numbers, and then transforming the results by \begin{align} 0_Z& \to (0,0)\\ S(a,b)& \to (Sa,b)\\ (a,b)+(c,d)& \to (a+c,b+d)\\ (a,b)(c,d)& \to (ac+bd,ad+bc)\\ (a,b)=(c,d)& \to a+d=b+c \end{align}

This process turns every arithmetic statement about integers into an arithmetic statement about natural numbers instead. Eg $$(\exists x,y,z\in\mathbb{Z})x^3+y^3+z^3=30$$ can be translated idiomatically (i.e. after performing the mechanical translation and then some of the usual simplifications) as \begin{align}(\exists a,b,c,d,e,f)& a^3+3ab^2+c^3+3cd^2+e^3+3ef^2=\\ &b^3+3ba^2+d^3+3dc^2+f^3+3fe^2+30\end{align}

Answer 2

Here is a somewhat different way to think about it, although the result is equivalent to the theories in the other answers.

Begin with the observation that the structures $\langle\mathbb{N},+,\cdot,0,1,<\rangle$ and $\langle\mathbb{Z},+,\cdot,0,1,<\rangle$ are bi-interpretable as first-order structures. What this means, to say it simply, is that each can define a copy of the other in such a way that when one composes the interpretations, each model can see how it is copied into the interpretation-of-the-interpretation.

To define a copy of the integers in the natural numbers, one can think of every integer as the difference of two natural numbers (as in Matt's answer), and then use pairs of numbers under the same-difference relation $$(a,b)\sim(c,d)\quad\text{ if and only if }\quad a+d=b+c.$$ The integers are the resulting quotient structure (but in fact one can avoid the quotient by definably picking least representatives). The converse interpretation is simply to take the non-negative integers.

The importance of the bi-interpretation perspective is that the constructions mentioned in the other answers can in fact be undertaken inside the models, rather than in some larger metatheoretic context. Being bi-interpretable, these two structures are thus revealed to have exactly the same semantic content, merely presented in a different form.

Now, the main point is that this bi-interpretation works in any model of PA, not just the standard model, and PA proves that it is a bi-interpretation. Every model of PA is bi-interpretable with the ring arising via the interpretation with the same-difference quotient.

Because of this, it is natural to take the desired theory of the integer ring to be exactly the theory that PA proves about $\mathbb{Z}$ through this interpretation. The resulting theory will be bi-interpretable with PA, and the bi-interpretation of the structures $\mathbb{N}$ and $\mathbb{Z}$ are an instance of that bi-interpretation.

In fact, this theory will be equivalent to the theories offered in the other answers.

Answer 3

I don't think there is one canonical system. Some version of what you wrote would do (but you need to do something about ordering). In model theory of arithmetic, where it is often convenient to work with variants of first-order arithmetic theories that include negative numbers, people commonly opt for a base theory with a clear algebraic meaning rather than a minimalistic set of axioms, hence the following version of Peano arithmetic is sort of canonical:

• $(\mathbf Z,0,1,+,\cdot,\le)$ is a discretely ordered commutative ring.

• Usual induction: $\phi(0)\land\forall x\,(\phi(x)\to\phi(x+1))\to\forall x\ge0\,\phi(x)$.

Answer 4

The free unital ring over the unital semiring $(\mathbb{N}, 0, 1, {+}, {\times})$, or in layperson's terms “just add subtraction”.

Answer 5

This is a rambling collection of thoughts which are tangential to your question--I am posting in the hope that they might interest you. Let me begin by recalling that the reals can be charaterised as a totally ordered, Dedekind complete field. This is often used as the basis of an analysis course. There are various possibilities for its presentation--one can simply postulate the existence of such a structure and then bash on, or one can prove its existence based on a postulated model of Peano´s axiom. This involves extending the positive integers to the integers, then to the rationals, finally to the reals. Each of the extensions are similar in method (the first one has been expounded in the above thread), the last one being, as one would expect, rather more sophisticated (equivalence classes of Cauchy sequences of rationals).

However, there is an alternative approach which only uses the order structure and which has certain advantages. We work backwards from the reals with the following claims:

I. The real line is a totally ordered, Dedekind complete space with a countable order dense subset and without a largest or smallest element. One can also incorporate open intervals (same axioms), half open intervals, closed intervals or the positive and negative half lines by assuming the existence of least or greatest elements, in this system.

II. The rationals form a countable totally ordered space without gaps and no smallest or largest element. (A gap is a pair $x$ and $y$ with $x<y$ but with no elements between them).

III. The positive integers form a countable, Dedekind complete, totally ordered space with a smallest element but no largest one and the property that if $x<y$, there are at most a finite number of elements between them (this is not a circular argument--the notion of a finite set is internal to set theory). In order to get the integers, one simply drops the corresponding condition and assumes that there is no smallest element.

If one is not prepared simply to postulate the existence (and uniqueness) of such structures, one is then faced with two tasks--to demonstrate the consistency and categoricity of these axioms,i.e., that they have models which are are unique in the natural sense. This can be done with about the same effort as for the standard method.

I would claim that there are several advantages in these characterisations. Mathematically, the simplicity of using only one structure, the order, rather than three (order, addition and multiplication with a list of axioms). Also, these characterisations explain the magic appearance of the real numbers to measure physical quantities, much discussed in the philosophy of physics. Many (most?) natural quantities have a natural order structure but no sensible algebraic one. The classical examples are entropy and temperature (nobody would want to multiply temperatures). The question of going from the ordering "being warmer than" to a numerical value (empirical temperature, then absolute temperature) was of considerable interest to the pioneers of thermodynamics and is discussed in some detail in Maxwell´s "Theory of Heat". The corresponding difference between an ordering (preference) and a utility function is also fundamental in economics (fundamental enough to result in Debreu in being made a laureate).

Of course, once you have the reals, it is routine to then show that you can deduce the algebraic structure from the axioms. This involves some arbitrary choices and so is not natural or unique, but this a desirable property for reasons too intricate to expound here.

I will add a caveat. I am dragging these results from memory. They are certainly known but I have been unable to pinpoint explicit occurrences in the literature. Hence I have called them claims rather than results, since I haven´t sat down to write out the proofs.