TITLE: Why does one have to check if axioms are true?
QUESTION [28 upvotes]: In Tao's book Analysis 1, he writes:

Thus, from the point of view of logic, we can define equality on a [remark by myself: I think he forgot the word "type of object" here] however we please, so long as it obeys the reflexive, symmetry, and transitive axioms, and it is consistent with all other operations on the class of objects under discussion in the sense that the substitution axiom was true for all of those operations.

Does he mean that, if one wants to define define equality on a specific type of object (like functions, ordered pairs, for example), one has to check that these axioms of equality (he refers to these four axioms of equality as "symmetry", "reflexivity", "transitivity", and "substitution") hold in the sense that one has to prove them? It seems so, because of these two passages:

[In section 3.3 Functions] We observe that functions obey the axiom of substitution: if $x=x'$, then $f(x) = f(x')$ (why?).

(My answer would be "because that's an axiom", but Tao apparently wouldn't accept that.)
And after defining equality of sets ($A=B:\iff \forall x(x\in A\iff x\in B)$), Tao writes (on page 39):

One can easily verify that this notion of equality is reflexive, symmetric, and transitive (Exercise 3.1.1). Observe that if $x\in A$ and $A = B$, then $x\in B$, by Definition 3.1.4. Thus the "is an element of" relation $\in $ obeys the axiom of substitution

So he gives the exercise to prove the axioms of equality for sets. Why does one has to prove axioms? Or, put differently: if one can prove these things, why does he state them as axioms?

REPLY [2 votes]: Austin Mohr's answer is excellent; however, as it led to an extensive argument in the comments I wish to put forth a slight simplification and restatement of it which may add something.
(I posted this as a comment originally, but wanted to expand on it.)


So he gives the exercise to prove the axioms of equality for sets. Why does one has to prove axioms? Or, put differently: if one can prove these things, why does he state them as axioms?

The basic point to realize here is that English words have meaning outside of math.
You're free to invent any relation you wish, with whatever properties you wish.  You can label it with whatever made-up term you want without any necessity to prove anything about it.
If you do this, you are just defining what you are talking about, and you can then proceed to say something using your stated definitions.
However, if you use an English word to name or describe your relation, then you should take into account the English meaning of the word.  That is, in choosing an English word as a name, you should choose one which aligns with the properties your relation has.
And the corollary: If you want to use a particular English word as a name for your relation, you should be sure your relation has the properties that would be implied by that name.
This applies whether we are naming relations, operators, or anything else.

If I define a unary operator and call it the "inverse," but my operator has the property that repeated application of that operator will never produce the original input, I have misnamed it.
If I define a relation and call it the "equality relation," but it is not transitive nor symmetric, I have again chosen the wrong name.
(Can you misname your relations and operators?  Of course you can.  The only thing it will break down is your communication with other people, which is of course the only reason to have names for things in the first place.)
A relation which is not reflexive, symmetric, and transitive, will violate the English-language meaning of the word "equal," so if your relation may not have those properties then use another name for it instead.

In this case, the "equality relation" between sets has been defined in your textbook, and it is now your task to show the appropriateness of the label "equality relation" for the defined relation, by showing that it has the properties one would expect out of the English-language meaning of the word "equality."