What is the definition of axiom (mathematically speaking)

Question

I'm wondering about the exactly definition of axiom (mathematically speaking). The definition of this term seems a little blur in my mind. For example, the definition of point in the Euclidean Geometry is considered by the common sense as an axiom. Thus it seems to me that every definition can be regarded as an axiom.

I'm a little confused. The definition of point in Euclid's book is indeed an axiom? Is there another examples of axioms outside Euclidean Geometry? What is the definition of axiom (mathematically speaking)?

Thanks

Answer 1

Consider the following image:

The axioms are, as Stephen Douglas Allen said: The first in a line of logic, just look at the picture. It means that they are assumptions made beforehand, and theorems are consequences of these assumptions (See Thm, 1.1) and consequences of other theorems (See Thm, 3.2) or consequences of theorems and axioms (See Thm, 2.1). That is, these consequences can be shown (proved) to be true under the assumption of those axioms. Mathematics works under an epistemological system which does not allow infinite regresses. That means that you can take a theorem that is far away from the axioms, and then ask why it is true but it will eventually reach a proposition that answer all the chain that you've been asking, namely the axiom and from this point (grossly) no further questions can be asked.

You're right to think that any definiton can be an axiom (it's important to know what you mean with definition. Some definitions could work as axioms, some of them are just describing the existence of some objects. Axioms are usually interactions between some of the objects but sometimes these interactions are simply statements about their existence). Let's think a little about how axiomatization is done: Why would you use an arbitrary definition as an axiom? Perhaps, there are deeper ideas that are more primitive than these definitions and the object of these definitions can be built via these more primitive ideas, take for example set theory. It's believed that all mathematics can be built with it. Set theory has it's very primitive ideas in which all other mathematical ideas can be done.

But what I wrote is from a global standpoint. I'm talking about all mathematical ideas and another idea that can be used to write all these mathematical ideas. Usually there is also a local standpoint: If you read some analysis books, you'll see that, for example, commutativity and associativity are axioms in some of these books, but they actually can be proved with more primitive ideas. If you construct the real numbers from the very beginning, each of those axioms are actually theorems (if you look from the global standpoint). Then, for example, it's possible that what I'm calling as theorems $1,1; 1,2; 2,1$ are seen by another person as axioms. The reason of why they do that seems to be pragmatical: If you assume that some theorems are axioms, then you don't need to bother proving them. I took a long course on the construction of numbers, from $\Bbb{N}$ to $\Bbb{Z}$ to $\Bbb{Q}$ to $\Bbb{R}$ and then, for the matter of a course in analysis, it's usually irrelevant to prove these most basic ideas.

In the work of axiomatization, there is always this idea of local vs. global, perhaps the mathematical structures you're seeing now are built via some axioms but It's also possible that these axioms are actually consequences of more primitive ideas.

Answer 2

As axiom is an assumption we make that we consider to be true. That is, we decide that it is true. Because of this, an axiom is unprovable. It is true because we say it is true. All other laws, theorems, etc must be proven from the base set of axioms.

An axiom can simply be a definition or it can be a theorem. A definition only identifies something, gives it a name, and does contain any real information about it (as in the definition of a point). A theorem, on the other hand, says what something can/can't do (eg parallel lines can never cross).

For example: Einstein's postulates of relativity.

If you accept certain facts about the speed of light as true (axioms), then the rest of the theory of relativity follows from these facts. These facts can be observed in the real world, but they can never be proven from logic alone.

The bottom line is that nothing is absolutely true (or false). Everything relies of a previous assumption. An axiom is simply the first assumption and begins the chain of further logic.

Answer 3

I quote myself from here:

In my opinion, we should try to refrain from speaking of "axioms", since the term is basically meaningless. Admittedly, I tend to violate this recommendation all the time. Anyway, the way I see it, there are sentences. A collection of sentences can entail another sentence. If we have a collection $S$ of sentences, we can write $\mathrm{cl}(S)$ for the collection of all sentences entailed by the sentences of S. In words, $\mathrm{cl}(S)$ is the theory generated by $S$. That's all. Calling the elements of $S$ "axioms" gains us nothing. For some reason, however, it seems very difficult to refrain from calling things axioms.