Unprovable allegations

Unprovable statements in any theory are statements that can neither be proved nor disproved within the framework of this theory. Gödel’s incompleteness theorem says that in every fairly complex consistent theory that includes formal arithmetic, there is an unprovable [and undeniable in it] statement. However, finding sufficiently simple statements of this kind and proving their unprovability is a difficult task.

The most famous and important results here are:

• The 5th postulate of Euclid is unprovable using the remaining axioms of classical geometry .
• The axiom of choice and the continuum hypothesis are unprovable in set theory with the Zermelo – Frankel axiomatics (ZF).
• The Paris – Harrington theorem is unprovable in Peano arithmetic .