The Nuances of Truth in Axioms and Premises

[starstruck_]

Hey everyone,
I’m taking my first discrete math course this term and am kind of struggling with determining the difference between different terminology. As the title says, it’s specifically with premises and axioms. My professor’s notes begin with an introduction to premises as one of the two major components of a deductive argument. They state that the premise is a statement from a previous body of knowledge and is assumed to be true, and that the conclusion in a deductive argument is based on the assumption that the premise is true. As for axioms, axioms are briefly defined when discussing formalization. The notes state that an axiom is a statement that holds true. I’m trying to understand the difference between the two.
I’m not sure how correct my understanding is but it appears to me that an axiom and premise are similar?
The difference being that an axiom is simply a statement that holds true and is assumed to be true irrespective of proof. However for a premise, a premise may be a statement that comes from another proof (hence the previous body of knowledge), but an axiom may also be a premise if used within a deductive argument?

is this understanding correct, any clarification would be incredibly helpful!

 

[PeroK]

I thought an axiom was a fundamental statement that you assumed to be true as the basis for the mathematics. A theorem is something that you have previously proved, using the axioms. A premise is a provisional assumption that you assume as part of a proof. For example:

In standard number theory we have the Peano axioms as the basis of number theory. A theorem might be that addition is commutative, or that there are infinitely many primes. A premise might be that some number ##p## is a prime.

PS I've no way to know whether your professor is using this terminology.

 

[Dale]

Any axiom is a premise, but there are some premises that are not axioms. For example, sometimes you will use a premise that you know to be false and reason from that false premise to some contradiction, thus disproving the premise. Such a premise would not be used as an axiom.

 

[Hill]

Axioms are definitions of a specific system of knowledge. Premises are conditions of a specific argument. IMHO.

 

[jbriggs444]

I thought an axiom was a fundamental statement that you assumed to be true as the basis for the mathematics.

As @PeroK seems quite aware, the relevant notion of "truth" is more nuanced than one might suppose.

Consider Euclid's parallel postulate for example: "For any line and point not on that line there is exactly one line parallel to the given line that contains the given point". Is that axiom true or false?

It turns out that there are spherical geometries where that axiom is false. Given a point not on a given line, there are no parallel lines containing that point. It also turns out that there are hyperbolic geometries where the axiom is also false. Given a point not on a given line, there are infinitely many parallel lines containing that point. Meanwhile the flat two-space contemplated by Euclid is a [as far as we know] consistent geometry where the parallel postulate holds true.

Instead of asking whether an axiom is true we can ask instead about "truth in a model". The parallel postulate is false in a spherical model, false in a hyperbolic model and true in a flat model.

Then you have the problem of determining whether a model exists that satisfies the axioms.

Edit: Checked and found that Euclidean geometry is provably consistent.