Is everything in math either an axiom or a theorem?

[FactChecker]

I've been meaning to ask about that. It seems to me that what Gödel actually discovered is that self-reference is different kind of thing that isn't compatible with normal everyday logic i.e. the pattern of true/false doesn't necessarily apply to self-referential statements. I have yet to see an example of it in a non-self-referential context i.e. a statement that isn't self-referential and not true/false. It seems that as long as you stay away from self-reference, you should be fine.

I think it is important to distinguish between the number of counterexamples that are easily proven to be counterexamples versus the number of counterexamples that exist. There are $\aleph_1$ transcendental numbers, but far fewer proven ones.

 

[Henryk]

I'm certainly not a mathematician, can you give an example of something known to be true but unproven? In my mind, axioms are "given" -- assumed to be true. But they may not "really" be true. Things like flat space.

There is no such thing as not "really" true in mathematics. Maybe the general theory states that the physical space is not actually flat, but that doesn't mean we can't construct a true flat space in mathematics.

 

[Feynstein100]

you have 'axioms' 'theorems' 'lemmas' and 'corollaries' :D

Just when I thought I was done, they pull me back in