Axioms & Theorems: What's the Difference?

[Red_CCF]

Hi

Can anyone help me define the axioms and theorems and what the differences are? I know axioms are suppose to be statements that are considered true based on logic (ex. x+y=y+x) but cannot be proven. Can someone explain why it can't be proven? Thanks

 

[slider142]

An axiom does not have a proof. Theorems are proven based on axioms and some set of logical connectives. Most definitions are axioms. Ie., in set theory the existence of an empty set is an axiom. They are like statements one assumes to be true for the sake of seeing what the logical consequences are.
It is possible for some axioms to be considered theorems and vice versa depending on how the mathematician wants to approach a problem. Ie., the existence of a least upper bound for every set or real numbers that is bounded below is axiomatic in some texts whereas the Archimedean property is a theorem based on it, while in other texts, the Archimedean property is taken to be an axiom and the least upper bound property is a theorem based on it.

 

[Red_CCF]

An axiom does not have a proof. Theorems are proven based on axioms and some set of logical connectives. Most definitions are axioms. Ie., in set theory the existence of an empty set is an axiom.
It is possible for some axioms to be considered theorems and vice versa depending on how the mathematician wants to approach a problem. Ie., the existence of a least upper bound for every set or real numbers that is bounded below is axiomatic in some texts whereas the Archimedean property is a theorem based on it, while in other texts, the Archimedean property is taken to be an axiom and the least upper bound property is a theorem based on it.

But why can't axioms be proven? For example with something like xy=yx, why can't we plug in numbers and find out?

 

[slider142]

But why can't axioms be proven? For example with something like xy=yx, why can't we plug in numbers and find out?

You need to first define the symbol '=' (an axiom), and the number that results from the product xy and yx (another axiom). You are assuming quite a lot of axioms! (In fact, I believe you are assuming x and y are elements of a field, as those are the axioms taught in your first math lessons; that is they satisfy the field axioms). There are many useful number systems where xy is not equivalent to yx. Ie., the non-abelian groups.

 

[Elucidus]

But why can't axioms be proven? For example with something like xy=yx, why can't we plug in numbers and find out?

Unfortunatley, no amount of showing examples proves that a relation is true for all products. So how would you try to show that xy = yx for all possible combinations?

--Elucidus

 

[Preno]

Trivially, axioms can be proven: their formal proof consists merely in stating the axiom.

Also note that the division of the theorems of, say, arithmetic between axioms and non-axioms is conventional. One could easily pick a different (but equivalent) set of axioms.

I know axioms are suppose to be statements that are considered true based on logic (ex. x+y=y+x) but cannot be proven.

No, that's not true. If axioms were "true based on logic", they could be proven and there would be no reason to include them as axioms of our theories. x+y=y+x is not a theorem of logic.

 

[HallsofIvy]

Trivially, axioms can be proven: their formal proof consists merely in stating the axiom.

I suspect I agree with you but I wouldn't put it that way! Axioms are given as true. That is, the axioms define which mathematical system we are talking about. The axiom "through any point not on a line there exist exactly one line through the given point parallel to the given line" is an axiom of Euclidean Geometry simply because if it isn't an axiom then whatever you are talking about is not Euclidean geometry! In hyperbolic geometry it is accepted that "through any point not on a line there exist no line through the given point parallel to the given line" while in elliptic geometry it is "through any point not on a line there exist more than one line through the given point parallel to the given line". And, of course, you can have other kinds of geometry by asserting that one of those is true at one point, and another one is true at another point.

Also note that the division of the theorems of, say, arithmetic between axioms and non-axioms is conventional. One could easily pick a different (but equivalent) set of axioms.

[originally posted by Red_CCF] I know axioms are suppose to be statements that are considered true based on logic (ex. x+y=y+x) but cannot be proven.

No, that's not true. If axioms were "true based on logic", they could be proven and there would be no reason to include them as axioms of our theories. x+y=y+x is not a theorem of logic.

I would cut him a bit of slack on that one. It's an old (very old!) point of view but Euclid himself took axioms as statements from logic and arithmetic while postulates were specifically about geometry. The Euclidean "axiom" I stated above, Euclid called a "postulate". That may have been the distinction Red_CCF was making. Modern mathematics does not recognize any difference between "axioms" and "postulates", calling them all "axioms".

 

[Preno]

I suspect I agree with you but I wouldn't put it that way! Axioms are given as true. That is, the axioms define which mathematical system we are talking about. The axiom "through any point not on a line there exist exactly one line through the given point parallel to the given line" is an axiom of Euclidean Geometry simply because if it isn't an axiom then whatever you are talking about is not Euclidean geometry! In hyperbolic geometry it is accepted that "through any point not on a line there exist no line through the given point parallel to the given line" while in elliptic geometry it is "through any point not on a line there exist more than one line through the given point parallel to the given line". And, of course, you can have other kinds of geometry by asserting that one of those is true at one point, and another one is true at another point.

Yeah, that's right, but my main point was that the division between axioms and (non-axiomatic) theorems is a matter of convention. There are lots of equivalent ways of defining the same system (i.e. axiomatizing the same set of theorems), so (outside formal proof theory) it really is not important which ones we decide to call axioms. One person's axiom is another person's theorem and vice versa.

I would cut him a bit of slack on that one. It's an old (very old!) point of view but Euclid himself took axioms as statements from logic and arithmetic while postulates were specifically about geometry. The Euclidean "axiom" I stated above, Euclid called a "postulate". That may have been the distinction Red_CCF was making. Modern mathematics does not recognize any difference between "axioms" and "postulates", calling them all "axioms".

I suppose you have a point there, but even according to Euclid's conception, axioms include non-logical truths. Axioms in the Euclidean sense may be necessarily true, but (if I understand it correctly) they are not necessarily logical truths, at least not what we would call logical truths today.

 

[Elucidus]

Caveat: I believe the following comment is true, but I am unable at the moment to verify it.

For formal systems, it is possible to create several equivalent but non-identical systems of axioms that form the foundation for that system. But I think it the case that for some formal systems these sets of equivalent sets of axioms contain one or more "essential" axiom that occur in all formalizations of that system.

These would be, in my opinion, truly axiomatic of that system.

I cannot think of such off the top of my head, but it is highly possible that these are truly axiomatic:

(1) There exists a set. (From Set Theory)

(2) There exists an inductive set. (From Number Theory)

--Elucidus

 

[Hurkyl]

But I think it the case that for some formal systems these sets of equivalent sets of axioms contain one or more "essential" axiom that occur in all formalizations of that system.

As stated, this is trivially false. Let P is any axiom in some set of axioms, here are a few ways to produce other sets of axioms that generate the same theory:

1. Replace "P" with "not not P"
2. Choose any statement "Q". Replace "P" with the two axioms "if Q, then P" and "if not Q, then P"
3. Choose any other "Q" in your set of axioms. Replace "P" with "if Q, then P"
4. Choose any tautology "Q". Replace "P" with "Q implies P"
5. Choose any contradiction "Q". Replace "P" with "P or Q"


A particular example is that I have very rarely seen "there exists a set" listed as an axiom of set thery. The axiom of the empty set seems to be a more common existential axiom to use. (Or an awkwardly phrased axiom of infinity)

 

[Elucidus]

As stated, this is trivially false. Let P is any axiom in some set of axioms, here are a few ways to produce other sets of axioms that generate the same theory:

1. Replace "P" with "not not P"
2. Choose any statement "Q". Replace "P" with the two axioms "if Q, then P" and "if not Q, then P"
3. Choose any other "Q" in your set of axioms. Replace "P" with "if Q, then P"
4. Choose any tautology "Q". Replace "P" with "Q implies P"
5. Choose any contradiction "Q". Replace "P" with "P or Q"


A particular example is that I have very rarely seen "there exists a set" listed as an axiom of set thery. The axiom of the empty set seems to be a more common existential axiom to use. (Or an awkwardly phrased axiom of infinity)

I agree that any axiom P can be replaced with these and would make my assertion trivially false. Given more thought, I'd have worded my comment more carefully. Typically when someone is devising a system of axioms, the intent is to create the simplest list (i.e. fewest axioms, axioms that are mutually independent, and are logically simple). Admitedly this isn't always easy (e.g. Parallel postulate).

My comment, which could be incorrect, was that some systems have simple axiom lists that have identical (or at least very similar) axioms. Obviously one could devise an axiom list that contained completely dissimilar axioms but they may not be as simple as the others.

--Elucidus