What is an Axiom? Definition & Examples

[Silviu]

Hello! I was wondering how does a mathematical statement come to be an axiom? I understand that an axiom can't be proven using other mathematical statements. But how does one know that a statement can or can not be proven? For example, why isn't Riemann Hypothesis considered an axiom? I also read some stuff about Godel's incompleteness theorem and if I got it right, he proved that using a finite number of axioms there will always be stuff we can't prove, so does't this mean that at a point we can come across a statement that we can't prove using the axioms we have by now? So how can we say that a conjecture can or can't be proved and how can we decide if it can be considered an axiom?

 

[danieldf]

I don't know about the requirements of a statement to be considered an axiom, but as I see it, seems axioms are to be more fundamental truths (which we consider to be). RH is such a complex proposition it seems to be pushing too much to consider it a axiom. Though there are a number of things mathematicians know to be true depending on RH to be correct, they won't simply take that for granted just because RH seems to be true.

I believe Continuum Hypothesis is one example of a problem that was proved to be undecidable. But I heard of that too long ago..

 

[jbriggs444]

Hello! I was wondering how does a mathematical statement come to be an axiom?

By accepting it as an axiom and using it within a formal system as such. There is no requirement of any deeper truth.

Axiom: Given a line and a point not on that line there is exactly one line containing that point and not intersecting the given line.
Axiom: Given a line and a point not on that line there are no lines containing that point and not intersecting the given line.
Axiom: Given a line and a point not on that line there are infinitely many lines containing that point and not intersecting the given line.

All three are valid axioms. Each is "true" in some contexts.

Before tackling the meaning of Godel's work, you should probably start with a better understanding of formal systems. With that under your belt, you should next learn about models and how "truth" is judged relative to a model. (Google "tarski")

 

[Silviu]

By accepting it as an axiom and using it within a formal system as such. There is no requirement of any deeper truth.

Axiom: Given a line and a point not on that line there is exactly one line containing that point and not intersecting the given line.
Axiom: Given a line and a point not on that line there are no lines containing that point and not intersecting the given line.
Axiom: Given a line and a point not on that line there are infinitley many lines containing that point and not intersecting the given line.

All three are valid axioms. Each is "true" in some contexts.

Before tackling the meaning of Godel's work, you should probably start with a better understanding of formal systems. With that under your belt, you should next learn about models and how "truth" is judged relative to a model. (Google "tarski")

Thank you for reply and references. However, this doesn't really answer my question (I will look over the sources you suggested meanwhile). What I was wondering is how can we know that the axioms you listed can't be proved using other mathematical statements (in which case they wouldn't be axioms, but theorems). I understand that we accept them and build a certain type of geometry on them, but why can't we do the same with Riemann hypothesis and build the number theory on it, without attempting to prove it and instead accepting it as an axiom (unless, of course, somehow it was proved that the Riemann hypothesis can be proved, so we know that it has a solution, but I am not aware of such a proof, yet).

 

[ericjericj]

Axioms aren't merely unprovable statements, they play a foundational and generative role for a branch of mathematics. The choice of axioms postulated at the beginning of geometry, for example, dictates the character and potential richness of the geometry that you can develop from them.

As an analogy, you could maybe think of them as the fundamental forces that govern the particles in a possible universe. In this analogy, the particles are primitive terms of a mathematical model. The more sophisticated molecules and complex interactions between them are like the theorems that can be deduced in that particular branch of mathematics. Not a perfect analogy, but it may illustrate why something like the Riemann Hypothesis wouldn't fit into the category of axiom.

 

[jbriggs444]

why can't we do the same with Riemann hypothesis and build the number theory on it, without attempting to prove it and instead accepting it as an axiom (unless, of course, somehow it was proved that the Riemann hypothesis can be proved, so we know that it has a solution, but I am not aware of such a proof, yet).

For much the same reasons that we do not have an axiom that the 10⁹⁹th decimal digit of pi is 9: Because it might be disprovable and because there is not a lot of rich theory that you could deduce from that starting point.

 

[TeethWhitener]

why can't we do the same with Riemann hypothesis and build the number theory on it, without attempting to prove it and instead accepting it as an axiom (unless, of course, somehow it was proved that the Riemann hypothesis can be proved, so we know that it has a solution, but I am not aware of such a proof, yet).

We can: https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences
But the problem then becomes: all of these consequences follow from the Riemann hypothesis and all the rest of the axioms. If it turns out that we can prove that the Riemann hypothesis is false using the rest of the axioms, then including Riemann as an axiom with the rest of them gives an inconsistent system.

To state the issue slightly differently, if you have a statement that you know is false given a set of axioms, it can be instructive to figure out how many axioms you'd have to abandon for the original statement to be consistent (not even true, but merely consistent) with the system.

 

[Silviu]

For much the same reasons that we do not have an axiom that the 10⁹⁹th decimal digit of pi is 9: Because it might be disprovable and because there is not a lot of rich theory that you could deduce from that starting point.

I am not sure I get the example with $\pi$ (finding any decimal of $\pi$ is possible if you calculate enough terms of a series that converge to something related to $\pi$ and it was done up to high numbers using supercomputers, so in this case there is no point in making that an axiom, as it is something that can be calculated, and thus proved). About the richness, in the case of Riemann hypothesis, I think there is more than enough rich number theory content that you can obtain from it (and that was already obtained, assuming that it is correct).

 

[Silviu]

We can: https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences
But the problem then becomes: all of these consequences follow from the Riemann hypothesis and all the rest of the axioms. If it turns out that we can prove that the Riemann hypothesis is false using the rest of the axioms, then including Riemann as an axiom with the rest of them gives an inconsistent system.

To state the issue slightly differently, if you have a statement that you know is false given a set of axioms, it can be instructive to figure out how many axioms you'd have to abandon for the original statement to be consistent (not even true, but merely consistent) with the system.

Thank you for your reply. I understand what you mean. But isn't this the risk with any axiom? When a set of axioms is chosen, how can you know they are all consistent, except by developing the theory and seeing if you get a contradiction at a point? Of course if you can prove or disprove Riemann hypothesis, there is no discussion to be done, but until then, why can't we take it as a theorem, and see how stuff are going?

 

[Stephen Tashi]

When a set of axioms is chosen, how can you know they are all consistent, except by developing the theory and seeing if you get a contradiction at a point?

That is indeed a problem and mathematics often proceeds in the way that you describe - namely by developing the theory and hoping that no contradiction is reached.

It is sometimes possible to find examples of physical systems that satisfy a set of axioms. If we trust that Nature does not contain utter contradictions, the existence of a physical example that satisfies a set of axioms shows that the axioms (by themselves) are not contradictory. Likewise if we have confidence that a certain specific example from mathematics is not contradictory, then a set of axioms that the example satisfies isn't contradictory.

 

[TeethWhitener]

When a set of axioms is chosen, how can you know they are all consistent, except by developing the theory and seeing if you get a contradiction at a point?

As @Stephen Tashi said, this is a problem. There's an important point to be made here, which is that , e.g., something as sophisticated as ZF set theory can't prove itself consistent (Godel first showed this). The standard response here is that you can prove ZF consistency by assuming the existence of certain large cardinals. Of course, the problem here is that this new system (ZF + large cardinal axiom) can't prove itself consistent either, and up and up we go.

why can't we take it as a theorem, and see how stuff are going?

That was the point of the link. People have proven a lot of things, given the truth of the Riemann hypothesis. However, if it turns out Riemann is false, then you'd have to prove those corollaries a different way (assuming they aren't inconsistent).