axiom vs. conjecture vs. postulate

[cyclohexane]

How exactly is each one defined? (ie. what makes certain statements in mathematics conjectures and others postulates?) Is this completely arbitrary?

 

[Cogman]

axiom = truth
conjecture = not proven false
postulate = assumed true (might not be), IE using conjecture in a proof to make another conjecture. Not totally worthless really, so long as the conjecture is decently hard to disprove.

Conjectures have nothing that proves them, but also nothing that disproves them. For example, the twin prime conjecture (infinite number of twin primes), nobody has disproved this, however, nobody has a solid way of proving either.

The infinitude of primes is an example of an axiom, it has been proven.

I don't have a good example of a postulate.

 

[iCyborg]

The infinitude of primes is an example of an axiom, it has been proven.

Erm, no. That's a theorem. Axioms are never proved (well, sometimes a set of axioms can be redundant), they're supposed to be obvious and self-evident truths that form a basis from which you can prove other statements, as you gotta have something to start from in order to prove anything. Otherwise, nothing would ever be proved. Any mathematical truth is really only a truth within some axiomatic system, like ZFC.

Postulate to me is the same as an axiom, though perhaps it can be more tentative, or even known to be wrong but used to define a discipline, like Euclid's 5th postulate - with it we have Euclidean geometry, without it, we have Non-Euclidean. Though an axiom can be used for the same purpose...

In some sense, both axioms an postulates are conjectures, in the sense that they're assumptions, but conjectures are usually used for something that's potentially provable, as opposed to axioms/postulates which are true by definition.

 

[Cogman]

Erm, no. That's a theorem. Axioms are never proved (well, sometimes a set of axioms can be redundant), they're supposed to be obvious and self-evident truths that form a basis from which you can prove other statements, as you gotta have something to start from in order to prove anything. Otherwise, nothing would ever be proved. Any mathematical truth is really only a truth within some axiomatic system, like ZFC.

Postulate to me is the same as an axiom, though perhaps it can be more tentative, or even known to be wrong but used to define a discipline, like Euclid's 5th postulate - with it we have Euclidean geometry, without it, we have Non-Euclidean. Though an axiom can be used for the same purpose...

In some sense, both axioms an postulates are conjectures, in the sense that they're assumptions, but conjectures are usually used for something that's potentially provable, as opposed to axioms/postulates which are true by definition.

woops, my mistake. Though, I did always think of axioms and postulates as a bit out of the same league of truthiness (whatever you want to call it.).

Conjectures, though, are more like unproven theorems.

 

[cyclohexane]

Thanks, that was helpful.