| There are a couple of statements that have been proven to be unprovable in the standard axiom set (ZF). The provability of the unprovability of unprovable statements is not guaranteed, but happens to be true for many of the more useful axioms. The most common is the Axiom of Choice, which if true is incredibly convenient. For example, the Axiom of Choice implies that all Vector Spaces have a basis, which makes linear algebra tractable. Mathematicians generally accept it as true because it's useful and intuitively true, but it does have some consequences that are intuitively false, like the Banach-Tarski paradox. We even have a saying: "The Axiom of Choice is obviously true, the Well-Ordering Principle is obviously false, and Zorn's Lemma no one even knows." The implicit punchline being, these three things are logically equivalent because each implies the other. Other axioms that are independent of ZF include the Continuum Hypothesis and the Axiom of Foundation. The Continuum Hypothesis says that there is no type of infinity that is larger than the cardinality of the Rationals and also smaller than the cardinality of the Reals. It doesn't make a damn lick of sense for such a type of infinity to exist, but it's not technically possible to disprove it. The Axiom of Foundation, in very simple terms, disallows objects like "the set that contains itself." Such sets are not constructable in ZF, and are logically impossible, but they are not technically disallowed. Using "stronger" axiom sets to investigate the properties and consequences of these unprovable statements is common. Additionally, many automated theorem-provers include several of the more convenient axioms that aren't included in "regular" mathematics. More reading: http://en.wikipedia.org/wiki/List_of_statements_undecidable_... (Note: ZFC is ZF plus the Axiom of Choice; Choice is so damn convenient that the mathematics establishment usually accepts it by default, although proving that you don't need Choice to prove something is often seen as a worthwhile accomplishment.) |