Gödel proved that, within any axiomatic framework for mathematics there are mathematically true statements that we will never be able to prove are true within that framework.“
Source : Marcus du Sautoy, What We Cannot Know: Explorations at the Edge of Knowledge
Gödel developed two theorems dealing with the subject :
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.
The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
Further info at Wikipedia
Note: This item is one of a special case Known Unknowables i.e. in an area where it can be proved that we will never be able to resolve an answer.
Ideas for new topics, and suggested additions / corrections for old ones, are always welcome.
If you have skills or interests in a particular field, and have suggestions for Wikenigma, get in touch !
Or, if you'd like to become a regular contributor . . . request a login password. Registered users can edit the entire content of the site, and also create new pages.
( The 'Notes for contributors' section in the main menu has further information and guidelines etc.)
You are currently viewing an auto-translated version of Wikenigma
Please be aware that no automatic translation engines are 100% accurate, and so the auto-translated content will very probably feature errors and omissions.
Nevertheless, Wikenigma hopes that the translated content will help to attract a wider global audience.