I was reading a bit about Gödel's incompleteness theorems. I haven't took the time to really study it, but I'm very curious about statements like these:
In other words, if our axioms are consistent then in every model of the axioms there is a statement which is true but not provable. source
Given any system of axioms that produces no paradoxes, there exist statements about numbers which are true, but which cannot be proved using the given axioms.
What I don't understand is this. How can you show that such a statement is true, without proving it ? This seems like a contradiction in itself to me.
Can someone give me an example of such a statement (about numbers) that we know is true, but which cannot be proven to be true ? And how then do you conclude that such a statement is true ? Because of the relations that numbers have with the real world ?