What actually constitutes a *definition* for a function?

Question

I'm reading Enderton's text on logic trying to justify ( to myself ) the use of induction on recursively defined sets. This is of course used repeatedly in trying to prove results about well-formed formulas in propositional calculus. Now in the authors digressive explanation in formulating the general concept of induction he first defines the "inductive functions" on a universal set. For an example he defines the formula building operations on the set of all expressions (any strings with letters from the Language of Propositions). And then proceeds to say the smallest set which contains the propositional symbols and is closed under the said functions is the set of all formulas. This to me makes perfect sense. But then we come to the examples. And the first one is the recursive definition of the natural numbers. The author first defines the successor function for all real numbers. That is, he defines $S : \Bbb R \to \Bbb R$ to be $S(x) = x + 1$. Then says the smallest set which contains $1$ and is closed under the successor function is $\Bbb N$. Which again is perfectly fine. But that got me thinking about the successor function used in other constructions such as in Landau's Foundations of Analysis. It seemed clear to me then but now when I look again I realise no clear domain was defined for the successor function.

All this time I was under the impression that to define a function one must specify a domain and then assign a value to each element in the domain. But in the case of the successor function in $\Bbb N$ it does not seem to be true. But Enderton in his book makes the point to define the function on a super set of what we wish to use it on. So,

What constitutes the definition of a function? Must not the domain always be specified clearly, unambiguously, distinctly and separately before making use of the said function for something else?

Answer 1

Regarding Edmund Landau, Foundations of analysis : The arithmetic of whole rational, irrational, and complex numbers (ed or : 1930), its approach is the "standard" axiomatic one.

See pages 1-2 :

We assume the following to be given : A set (i.e. totality) of objects called natural numbers, possessing the properties — called axioms — to be listed below. [...]

Axiom 1 : $1$ is a natural number. That is, our set is not empty; it contains an object called $1$ (read "one").

Axiom 2 : For each $x$ there exists exactly one natural number, called the successor of $x$, which will be denoted by $x'$.

The second axiom implicitly define the function $S$ ("successor") such that :

$S(n)=n'$ for all $n$.

The domain of the function is clearly the totality of natural numbers itself.

The relation of the above definition with the inductive definition (cited into Enderton's book as an example, used only - I think - in order to show how inductive definitions "work") can be better understood supplementing it with Herbert Enderton, Elements of Set Theory (1977), page 66-on :

There are, in general, two ways of introducing new objects for mathematical study: the axiomatic approach and the constructive approach. The axiomatic approach is the one we have used for sets. The concept of set is one of our primitive notions, and we have adopted a list of axioms dealing with the primitive notions.

Now consider the matter of introducing the natural numbers :

$0, 1, 2, \ldots$

for further study [see : Landau].

An axiomatic approach would consider "natural number" as a primitive notion and would adopt a list of axioms. Instead we will use the constructive approach for natural numbers. We will define natural numbers in terms of other available objects (sets, of course). In place of axioms for numbers we will be able to prove the necessary properties of numbers from known properties of sets.

First of all, Enderton defines $0$ as the $\emptyset$ [Landau "starts from" $1$, but the choice is immaterial] and then defines a successor operation [page 68] :

Definition. For any set $a$, its successor $a'$ is defined by

$a' = a \cup \{ a \}$.

A set $A$ is said to be inductive iff $\emptyset \in A$ and it is "closed under successor", i.e. :

$(\forall a \in A) a' \in A$.

In terms of the successor fucntion, the first few natural numbers can be characterized as :

$0=\emptyset, 1=\emptyset', 2=\emptyset'',\ldots$

In this definition, the successor function is defined by way of the union operation : $\cup$, which is already defined in set theory : its domain is the "universe" of sets.

Answer 2

What constitutes the definition of a function? Must not the domain always be specified clearly, unambiguously, distinctly and separately before making use of the said function for something else?

Yes.

f is said to be a function (of one variable) mapping all elements of set A to elements of set B iff

1. $f\subset A\times B $

2. $\forall x\in A: \exists y \in B:(x,y)\in f$

3. $\forall x,y_1,y_2: [(x,y_1)\in f \land (x,y_2)\in f \implies y_1=y_2]$

The author first defines the successor function for all real numbers. That is, he defines $S : \Bbb R \to \Bbb R$ to be $S(x) = x + 1$. Then says the smallest set which contains $1$ and is closed under the successor function is $\Bbb N$.

It can be shown that the function $S$ restricted to this "smallest set" $\mathbb{N}$, call it the function $S'$, is indeed a function mapping $\mathbb{N}$ to itself. Furthermore, $S'$ can be shown to satisfy all the requirements for a successor function given by Peano's axioms for the natural numbers.

Another approach, of course, is to start with Peano's axioms and construct the real numbers -- a much larger project.