In Terence Tao's book "Analysis 1", in definition 3.3.1 (function definition), he says

Let $X, Y$ be sets, and let $P(x, y)$ be a property pertaining to an object $x \in X$ and an object $y \in Y$, such that for every $x \in X$, there is exactly one $y \in Y$ for which $P(x, y)$ is true. Then we define the function $f: X \rightarrow Y$ defined by $P$ on the domain $X$ and range $Y$ to be the object which, given any input $x \in X$, assigns an output $f(x) \in Y$, defined to be the unique object $f(x)$ for which $P(x, f(x))$ is true.

This definition is on a chapter on set theory foundations, which starts by postulating the existence of objects and sets (he does impure set theory, so not all objects are sets).

Here, under the conditions of the definition, he says that there exists an object $f$, which has the property that for any $x\in X$, $f(x)$ denotes the unique $y\in Y$ such that $P(x,y)$ is true. Shouldn't this be an axiom that postulates the existence of an object?

In another question Equality of functions: axiom or definition? it is said that "Tao does mention that strictly speaking, this definition is an axiom that posits the existence of a function given sets $X,Y$ and property $P$." However, I did not find such a statement in the book.

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  • $\begingroup$ It is a definition; we have a binary relation $P(x,y)$ that satisfies an additional property (of being "functional"). In this case we stipulate to call it function and we introduce to special symbol $f(x)$ to denote the unique object such that $P(x,f(x))$ holds. $\endgroup$ Nov 25 at 7:01
  • $\begingroup$ I agree with this. However, he says that this function is an object. So, isn't this an axiom? The fact that the function is an object is important since he later deals with sets of functions, so a function must be an object. $\endgroup$
    – Plemath
    Nov 25 at 17:00
  • $\begingroup$ Agreed; see Tao's Intro (page 19: "In this text we will be studying many objects" (I assume: mathematical objects). The issue is that, IMO, Ax.3.1 is quite useless. $\endgroup$ 2 days ago
  • $\begingroup$ Anyway, in the def quoted above Tao writes: "Then we define the function $f : X→Y$ ... to be the object which..." Thus, it is an object and so we can use it to the left of the $\in$ sign. $\endgroup$ 2 days ago

Based on Tao’s system, this is a combination of a definition and an axiom.

A more formal treatment would work like this:

Given sets $X$ and $Y$, there is a kind of object called a “function from $X$ to $Y$”. The statement “$f$ is a function from $X$ to $Y$” is denoted $f : X \to Y$.

If $f : X \to Y$ and $x \in X$, then $f(x) \in Y$.

Axiom: Let $P(x, y)$ be a property on $x \in X, y \in Y$. If for all $x \in X$, there is a unique $y \in Y$ such that $P(x, y)$, then there is a function $f : X \to Y$ such that for all $x \in X$, $P(x, f(x))$.

Note that Tao then defines two functions to be equal if their outputs agree.

  • $\begingroup$ Before you say what the axiom is, you postulate the existence of some object (given sets $X$, $Y$. So, isn't this also an axiom? (I am sorry, I am very new to this treatment of sets...) $\endgroup$
    – Plemath
    Nov 25 at 16:54
  • $\begingroup$ @Plemath No, I postulated the existence of a kind of object, together with its properties. If there are no objects of this kind, then I haven’t really postulated anything. $\endgroup$ Nov 25 at 17:01
  • $\begingroup$ Is the "kind of object" a formal thing? What about this phrasing: Axiom: Let $P(x,y)$ be a property on $x\in X, y\in Y$ st for all $x\in X$, there is a unique $y\in Y$ st $P(x,y)$ is true. Then, there exists an object $f$, st $f(x)$ denotes the unique $y\in Y$ st $P(x,y)$ is true. We write $f:X\rightarrow Y$, and we call $f$ a "function". $\endgroup$
    – Plemath
    Nov 25 at 17:18

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