What is the difference between an axiom and a definition?

Question

Now I know that this question has been asked before here, but the reason I'm asking this again is because the example given in the question there, namely one of Peano's Axioms is very clearly an axiom to me, given that Peano's Axioms propose the existence of a certain set with certain properties, while I find I still do not see the difference in other examples.

Sometimes I feel the word axiom is used where the word definition should be used instead. The most glaring example coming to me right now are the field axioms. The field axioms do not give us a statement that we assume to be true. They do not propose the existence of anything or determine something to be universally true. They just give us a definition for a certain type of set, and say that if a set fulfills these properties, then we can call it a field. Isn't this exactly what a definition is?

I shall compare this to an example from linear algebra. If $A$ is a square matrix and $A^TA=I$, then $A$ is an orthogonal matrix. We do not say that this is the orthogonal matrix axiom, but rather call it a definition for the orthogonal matrix. In the same way, why do we not call the field axioms field definitions instead?

Answer 1

In my opinion, it is a matter of perspective.

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If we work with fields as objects within mathematics (say, within the theory of ZFC), then the field axioms are basically definitions that decide which objects in our universe are fields, and which ones aren't.

On the other hand, if we view fields as models for a certain theory, then the field axioms describe the properties such a model has to satisfy. In this case I would argue that calling those properties axioms is more appropriate.

If I construct some object in a more powerful theory, and show that it satisfies the axioms, I could argue that the object is something that satisfies some definitions. On the other hand, from the viewpoint of the object itself, it is some structure that satisfies the axioms, it does not "know" that it is constructed as part of a larger, more powerful theory.

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Similarly, the Peano Axioms could be taken as axioms that tell us which properties arithmetics has, but on the other side, we could see it as a collection of properties that together define a class of objects that behave arithmetical. For example, we could show that the natural numbers are such an object, and thus satisfy the definition of a "Peano object"

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It is even quite natural within set theory to switch between these viewpoints. For example, you commonly encounter countable models of ZFC, which are just sets inside the universe and thus an object defined by the axioms of ZFC (although we could not prove its existence within ZFC, but that's another story).

On the other hand, such sets are models of ZFC, so from the perspective of the model itself, it looks as if you have a complete universe. Then the axioms of ZFC are actual axioms, telling you which statements are true, constructions are allowed to be made, etc.

Answer 2

My two cents: The "field axioms" are the axioms upon which we build field theory. They say what a field is, the same way the Peano axioms say what the natural numbers are, or the ZF axioms say what a set is. So it's not wrong to call them axioms.

But I can agree that it's a blurry line.

Answer 3

The linked question (with its answers) deals more with the formal aspects of definitions.

I'll try with a more "easy" approach.

Consider again set theory [ref. to Herbert Enderton, Elements of set theory (Academic Press, 1977)].

We will start from an intuitive explanation of what the topic of the theory is :

A set is a collection of things (called its members or elements), the collection being regarded as a single object. We write "$t \in A$" to say that $t$ is a member of $A$ [page 1].

This is not a definition : we cannot define everything and we have to start somewhere. This "elucidation" gives us the basic stuff of the piece of "Mathematical world" the theory will speak of : (some) objects and a (binary) relation between them.

Then a first "principle" is stated [page 2] : the Priciple of Extensionality. It states the fundamental property of sets : they are identified only through their members, i.e. only the membership relation is relevant for the world of sets.

This principle is used to show that the empty set (the set with no members at all) is unique, i.e. tehre are no two different sets that are both empty.

This little proof assumes that in the world of sets there is a set that is empty.

The next move is use the Axiomatic Method [page 10-on] to develop in a rigorous way the theory of sets. The method is well-known in mathematics :

we are going to state the axioms of set theory, and we are going to show that our theorems are consequences of those axioms.

The first axiom of the theory is the (previously stated) Axiom of Extensionality [page 17].

Next we have two existence axioms: the Empty Set Axiom, followed by the Pairing Axiom.

Nothing in principle has changed from the previous intuitive approach : we presuppose a "universe of discourse" for our theory and we call sets its objects. The objects are (in some cases) conected by the relation of membership, and this relation is "extensional".

In the universe of sets there is a "distinguished" object called the empty set.

Finally, for every two sets : $a$ and $b$, the universe of sets has also a new set (called its pair) whose elements are exactly $a$ and $b$.

The set existence axioms can now be used to justify the definition of symbols [previously] used informally. First of all, we want to define the symbol "$\emptyset$" [page 18].

The approach is quite clear : we have some assumptions (the universe of sets and the membership relation) that are so basic that we cannot state/define in the theory itself and with them we formulate axioms : some of them express basic properties of sets (Extensionalty) while other state the existence of specific sets.

When we have assumed the existence of a specific set (and showed its uniqueness) we may introduce a new symbol to denote it (a "name" for it).

Here the difference between axiom and definition is subtle : we have an axiom asserting the existence of a set with no memebers, and we define a name for it : empty set.

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Conclusion : we have not defined what "set" and "membership" are. In the context of the specific theory we are developing, our knowledge of them is through the axioms.

We have stated axioms expressing properties of sets and membership and asserting the existence of sepecific sets.

We have introduced new names for those specific sets.

Answer 4

My POV: A theory has a certain set of primitives: symbols or terms that are not definable because at the start there is nothing to define them by. "Point", "Line" are classical primitive terms of Euclidean geometry.

• Axioms are statements about the primitives. They determine the content of the theory. (An axiom may be expressed in terms of later-defined concepts, but is reducible to primitives, and expresses some new relationship between them.) In a sense, the axioms can be thought of as defining the primitives by determining how they relate to each other. Axioms do not introduce new symbols or terms to the theory.
• Definitions introduce new symbols or terms to the theory by expressing them in terms of pre-existing symbols or terms. Definitions are intended to make the theory easier to understand and use, but do not introduce anything that was not already expressible.

For example, one can define a plane as the span of 3 non-colinear points (the span of a collection of points being the smallest set containing the collection and also containing the line through any two points in it). Euclidean geometry is much easier to develop with the concept of a "plane". But you could swap out "plane" with "span of 3 non-colinear points" everywhere, and still get the same theory.

Answer 5

This is how I understand the word "axiom":

When we talk about axioms of a certain object, we are listing some properties the object shall have. Peano axioms are several properties that the object, the set of natural numbers, shall have. Field axioms are several properties that "fields" shall have. Yes, they are axioms, but also collectively defines a field. So field axioms are both axioms and definition of a field. The two things are not exclusive.

Axioms are used in a way that they themselves imply the object satisfying the axioms has many interesting extra properties not listed among the axioms. Whenever you have a object $X$ and you want to show that it has all those interesting properties, you don't have to prove one by one, but instead you just need to prove that $X$ satisfies the axioms, and hence would have the interesting extra properties.

By the way, I don't think Peano axioms propose the existence of natural numbers, but only say what fundamental properties natural number shall have. You actually have to prove that there is something that satisfies Peano axioms. Though in practice it is not a concern for most people.

Answer 6

I think the distinction is loose, but may be more definite given the text or the discipline.

In Euclid's The Elements, he defines a Point as "That which has no part". A Line is "A breadthless length", with no definition given to either breadth or length. These are fundamental entities largely deriving their meaning from basic elements of experience.

Since terms are defined using other terms, there is an infinite regress or one relies on undefined terms. So Line and Point derive their meaning less from statements about them than pictorial representations.

Once the basic elements are defined in terms of very basic representations considered self-evident, relationships between these elements are described in axioms. Consider the first axiom: "A Line may be drawn between any two Points." This justifies one use of a Straightedge and establishes a relationship between sets and points. The second axiom asserts that any line can be extended indefinitely in any direction [along a straightedge].

Here the major difference between a definition and an axiom is whether the statement introduces an entity or establishes relationships between previously introduced entities.

A theorem is deduced from axioms, definitions, and previously established theorems, even if the statements themselves are fundamental. Consider the Compass Equivalence Theorem. It asserts that any Line Segment can be moved anywhere in the plane and oriented in a new direction while preserving the length. Why is this not an axiom? Euclid doesn't merely assert when its possible to prove. We can suspect the possibility of a theorem given the complexity of the relationship asserted. This rule doesn't apply to the Fifth Postulate which looks more complicated than some theorems.

In the case of Field Axioms, we have already been given definitions for elements, sets, set membership, binary operation, etc, as baseline, undefined elements and the axioms are relationships between them. But, the Field itself is a name given to the entity described by the relationships of the fundamental elements. We have a definition of a field in terms of axioms. Definitions occur at different levels of abstraction of phenomenon under consideration.

So less a key difference than a useful rule of thumb, a definition is a new concept introduced in terms of undefined terms, an axiom usually describes without proof relationships in terms of previously defined terms.