Noob needs help understanding what is an axiom.

[General_Sax]

Wikipedia says this: "In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths."

I don't understand this part: "... or subject to necessary decision".

So, one can "create" an axiom that isn't self-evident? How does that work?

 

[zhermes]

I think they just mean, you decide its true--even if just for the sake of an argument. We're getting into just the semantics of it, the important part is that an axiom is any truth taken as a priori.

An axiom doesn't have to actually be true. For instance, if we're debating the usefulness of unicorns, we might take as an axiom the existence of unicorns--just so we can discuss what we're really interested in: whether or not they're useful.

 

[SW VandeCarr]

I don't understand this part: "... or subject to necessary decision".

So, one can "create" an axiom that isn't self-evident? How does that work?

"Self-evident" is old fashioned, boring and uncool. "Consistency" is where it's at. You never write down just one axiom. You write several, and together with a set of rules of inference, you go ahead and prove theorems. The theorems may or may not mean anything (that is, they may or may not have an interpretation) but they can be proved with the small caveat that you can't also prove the theorem's negation. If you can, that's called a contradiction, and your whole system falls apart. Of course, sometimes that's what you want because your aim might be to show your axioms are not consistent. You can do this but can't prove your axioms are consistent unless you embed your logic into a higher logic where your axioms effectively become theorems in that higher logic (and then your "axioms" can be shown to be consistent if and only if the higher logic's axioms are consistent).

Clear?