Topics Mathematics

Question Typical statements (first order) of the Peano Axioms puzzle me. Neither a mathematician nor logician, I find myself thinking the following: One would hope that arithmetic is consistent with the world as it is. So the axioms of arithmetic should be true in a domain containing the items that populate reality, e.g., a domain containing this keyboard upon which I now type. But this keyboard is neither identical to zero nor is it the successor (or predecessor) of any whole non-negative number. So what's with, e.g., (Ax)((x = 0 v (Ey)(x = Sy))? On what would think its intended interpretation, the axiom (theorem in some versions) seems false "of reality." And some other typical items of (first order) expositions seem either false or at least meaningless, e.g., (Ax)(Ay)(x + Sy = S(x + y)). What could be meant by "the sum of this keyboard and the successor of 6 is equal to the successor of the sum of this keyboard and the positive integer 6? Unless one has already limited the domain to exclude typical non-arithmetic items, then stating the (first order) Peano Axioms with leading universal quantifiers seems to produce false and false or meaningless statements. So how would one try to change/complicate the (first order) axioms to avoid this? I recall reading somewhere that in some of his work Tarski would use a predicate for non-negative integers to limit the scope, something like "for all x, if x is a member of the non-negative numbers, then...." But how else might I think about this? Thanks for helping un-confuse me. Or don't we care if the Peano Axioms are not true of the world we live in? Wayne W.

Accepted:
November 26, 2011

Comments

You've pretty much answered your own question.

There are two ways of thinking about this. On the first, the "domain" of the theory being axiomatized is taken to consist only of the natural numbers (i.e., the non-negative integers). So it is, in a way, like when the coach says to the driver, "Everyone is on the bus". She doesn't really mean that everyone is on the bus, only that everyone on the team, or whatever, is on the bus. We speak this way all the time. It's not exactly the same phenomenon, but it's close enough to get the idea.

The second way, which you mention in connection with Tarski, is to introduce an explicit restriction to the natural numbers into the axioms. So let "Nx" be a predicate letter meaning: x is a natural number. Then the idea is to "relativize" the axioms to Nx: We replace each universal quantifier (∀x) by: (∀x)(Nx → ...); each existential quantifier (∃x) by: (∃x)(Nx & ...) So the addition axioms will take the form:

(∀x)(Nx → x + 0 = x)

(∀x)(∀y)(Nx & Ny → x + Sy = S(x + y))

We'll also need axioms about N itself:

N0

(∀x)(Nx → NSx)

(∀x)(∀y)(Nx & Ny → N(x + y))

(∀x)(∀y)(Nx & Ny → N(x × y))

These assure us that the objects the original theory speaks about all fall under Nx. Induction will be reformulated as:

A(0) & (∀x)(Nx & A(x) → A(Sx)) → (∀x)(Nx → A(x))

The translation here is thus pretty straightforward. (Note that the last two axioms about N will be provable given induction, but in general we need them, as one can see by considering Robinson arithmetic, which doesn't have induction.)

Indeed, the idea here is obviously quite general, and one can give a general treatment of this kind of translation. E.g., one can show that if some formula is a theorem of the original theory, then its "relativization"—the result of replacing each quantifier by its relativization to Nx—will be a theorem of the new theory, and conversely.

So the usual way of formulating the Peano axioms is just a sort of simplification, allowing us to suppress all the Ns that would otherwise clutter the formulae. But there are cases where one wants the Ns, say, if we wanted to mix set theoretic ideas with arithmetical ones in a single theory.