Is everything in math either an axiom or a theorem?

In summary: I'm sorry, I misunderstood your previous statement. I thought you were asking for an example of something that is assumed to be true but cannot be proven. As for your question about the difference between definitions and axioms, definitions are used to clarify and define specific mathematical objects or concepts, while axioms are more general assumptions that serve as the foundation for a particular mathematical system.
  • #71
Feynstein100 said:
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.
 
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  • #72
gmax137 said:
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.
 
  • #73
Rfael said:
you have 'axioms' 'theorems' 'lemmas' and 'corollaries' :D
Just when I thought I was done, they pull me back in 😁
 

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