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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

user75086
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    A thorough reading of this,- https://en.wikipedia.org/wiki/Axiom - should be sufficient. See some references in the same link to learn about the philosophy of math regarding axioms. – Panglossian Oporopolist Jul 01 '15 at 02:34

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Consider the following image:

enter image description here

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.

Red Banana
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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.

Stephen Douglas Allen
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  • So in this sense can I say for example the definition of a limit of a function is an axiom? – user75086 Jul 01 '15 at 03:17
  • Not really, An axiom is the first in a line of logic. The limit of a function doesn't introduce anything new to the field of mathematics. It is just applying algebra to a certain type of problem. That is: it is made things that are already well defined. To find an axiom you have to go back to the first definition (the one with no prior definitions). – Stephen Douglas Allen Jul 01 '15 at 03:31
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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.

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goblin GONE
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  • Why do we gain nothing by calling the elements of $S$ axioms? It is linguistically useful to define what our assumption are even if there is no mathematical difference between $x\in S$ and $x\in cl(S)$ – Stephen Douglas Allen Jul 01 '15 at 04:18
  • @StephenDouglasAllen, well, they're just elements of $S$. I don't understand your second sentence. – goblin GONE Jul 01 '15 at 05:25
  • $x\in S$ means $x$ is an element of $S$. Mathematically speaking it makes no difference if $x$ came from $S$ or if it came from $cl(S)$. However, I'm saying that linguistically speaking, it does. We need to be clear about what our assumptions are. – Stephen Douglas Allen Jul 01 '15 at 05:42
  • @StephenDouglasAllen, the statement $\varphi \in S$ is, in general, a much stronger statement than $\varphi \in \mathrm{cl}(S)$. For instance, if $S$ consists of the axioms of $\mathrm{ZFC}$, then $\varphi \in S$ means that $\varphi$ is an axiom of $\mathrm{ZFC}$, while $\varphi \in \mathrm{cl}(S)$ means that $\varphi$ is a theorem of $\mathrm{ZFC}$. – goblin GONE Jul 01 '15 at 15:28
  • then I really don't understand your statement "gains us nothing". You just said that $\varphi\in S$ is stronger than $\varphi\in cl(S)$, thus we gain something by knowing if an element is an axiom or not. Please explain your position further. – Stephen Douglas Allen Jul 02 '15 at 00:00
  • @StephenDouglasAllen, saying that $\varphi$ is an axiom of $S$ gains us nothing, because its the same as saying that $\varphi$ is an element of $S$. We don't need the word "axiom" at all. – goblin GONE Jul 02 '15 at 00:03
  • If I say to you "$x$ is an axiom", then you immediately know what I mean. On the other hand, if I say to you "$x\in S$ then you don't know anything until I explain "$S$ is a set of axioms". Hence I would say that "we gain nothing by saying $x$ is an element of $S$". – Stephen Douglas Allen Jul 02 '15 at 00:07
  • @StephenDouglasAllen, arguably, if I say "$x$ is an axiom" I am not being clear at all. An axiom of what? Oh, this thing called $S$. Okay, then its an element of $S$. – goblin GONE Jul 02 '15 at 00:08
  • Let us continue this discussion in chat. – Stephen Douglas Allen Jul 02 '15 at 00:15
  • @StephenDouglasAllen, I can't access chat from here, so lets just continue the discussion here. You write: "aren't you proposing an alternative to calling them axioms?" Kind of. What I'm saying is, rather than calling them "axioms" -- a term that cannot be usefully defined -- lets just call them sentences, since this is a technical term with a precise technical meaning. But note that "sentence" carries no connotations regarding the truthfulness or falseness of the statement under question. – goblin GONE Jul 02 '15 at 17:19
  • Let me add another perspective. Let $G$ denote a group. Then some subsets of $G$ are generating sets for $G$, and some are not. Suppose $A \subseteq G$ is a generating set for $G$. Would you prefer I call the elements of $A$ "elements of $A$" or "generators of $G$"? Clearly, the former is more precise and much clearer. Similarly, suppose $T$ is a first-order theory, and suppose that $S \subseteq T$ is a collection of sentences that generate $T$. Okay, would you prefer I call the elements of $S$ "axioms of $T$" or would you prefer I call them "elements of $S$"? – goblin GONE Jul 02 '15 at 17:24
  • Thanks for clarifying your position better. Let me see if I can summarise:

    1. I'm saying that at some point you have to specify what is a "building-block" and what is built from those blocks. Using a word to describe them is useful (you say "generator" I say "axiom", let's call the whole thing off).

    2. You say there is no need to call them anything as everything can be generated from this vast sea of sentences we call logic. As you say "some sets are generating, some are not", hence there is no single basis that can be called "the grand set axioms".

    How is that so far?

     – Stephen Douglas Allen Jul 06 '15 at 00:58
  • @StephenDouglasAllen, yes, that seems to be a reasonable description of my position. Being a generating set for a group $G$ is a well-defined property of subsets of $G$, but being a generator of $G$? The term is basically meaningless, unless by "$g$ is a generator of $G$" we mean "${g}$ is a generating set for $G$." But this is definitely not what we mean by "axiom"! If you say: "$\varphi$ is an axiom of ZFC" you're probably not trying to say that from $\varphi$ alone, every theorem of ZFC can be deduced! – goblin GONE Jul 06 '15 at 20:29
  • OK I think I understand you, your point is valid. We're arguing different cases. I agree that there is no point arguing if "$x\in G$ is an axiom of $G$" because in most cases a basis for $G$ can be found with (or without) $x$. Axioms are mostly arbitrary (as long as they're independent). However (and this is my point) there are times when you must choose your generators and then proceed from there. They must be clearly designated at the outset. Many will say "My chosen axioms are...". At some point, our language must specify our assumptions and this is why we have words like "axiom". – Stephen Douglas Allen Jul 08 '15 at 00:30
  • @StephenDouglasAllen, a related point is that axioms don't need to be independent. For instance, my preferred axioms for group theory are: ${1x=x,x1=x,(xy)z = x(yz), x^{-1}x = 1, xx^{-1} = 1}$. Those last two can each be proved from the other, in the presence of the remaining axioms. But I prefer to include both of them in my axioms for group theory. – goblin GONE Jul 08 '15 at 23:50
  • @StephenDouglasAllen, also, note that logicians use of the word "independent" is quite different from how ring theorists use the word. In ring theory, a subset $S$ of a module is said to be independent iff $0$ cannot be expressed as a non-zero linear combination of elements of $S$. This is a stronger condition than being a minimal generating set for the submodule generated by $S$. I refer to this weaker condition as "pseudo-independent." If you think about it, when logicians say "independent", what they really mean is "pseudo-independent." – goblin GONE Jul 08 '15 at 23:55