Created 2012-10-14 / Edited 2012-10-14

"In other words, there are true statements of mathematics which cannot be proved." depends on what you mean by "true": you are outside the system of mathematics and I wouldn't dare to say that something that cannot be proved is "true". Such truth can only be thought of a philosophical problem outside mathematics.

Moreover, there have been some statements unprovable in axiomatic systems (for example the continuum hypothesis in ZFC set theory). Mathematicians, instead of trying to philosophically argue whether such statements are true or false, try to prove independence of some unprovable results from others. For example "every Parovicenko space is homeomorphic to $\beta\omega$" is equivalent to the continuum hypothesis. Set theoretic topology is full of such results that prove consistency instead of logical implication.

-- Rodrigo Hernandez Gutierrez 2012-10-15 00:14 UTC