Is it possible to prove that you can't prove some theorem? For example: Prove that you can't prove the Riemann hypothesis. I have a feeling it's not possible but I want to know for sure. Thanks in advance
[Math] Can we prove that a statement cannot be proved
incompletenesslogic
Best Answer
From the time of Euclid on, there was interest in showing that Euclid's Fifth Postulate follows from the rest of Euclid's axioms. (Roughly speaking, the Fifth Postulate says that through a given point there is a unique line parallel to a given line.)
Finally, in the $1830$'s, Bolyai and Lobachevsky independently showed that the Fifth Postulate does not follow from the rest, by discovering hyperbolic geometry. To use modern language, they found a model of the remaining axioms in which the Parallel Postulate fails.
The work of Bolyai and Lobachevsky is, arguably, the first independence result. It may have helped to change the notion of what one means by an axiomatic system.