I read online a statement to the effect that (I'm paraphrasing): Goedel's incompleteness theorem shows that we cannot even have a complete and consistent theory for the natural numbers. I am ...
Freeman Dyson has claimed that $\not \exists m,n Reversed(2^n) = m ^ 5 $ (where Reversed(l) just is the reverse of the digits of l in base 10), is probably an example of an unproveable truth ...
There is a recursive definition of the set of squares which uses only addition: $CS(x,y) := IS(x) \wedge IS(y) \wedge x \lt y \wedge \forall z: (x \lt z) \wedge (z \lt y)⇒\neg IS(z)$ $IS(x)⇔ x=0 ...
Maybe this is a dumb question, but I have to admit that it is not really clear to me what the theory is, in which incompleteness of PA and stronger theories is proved. The texts I did study so far are ...