StarkEffects.com

Concise Articles, Tutorials & Primers on Science, Math & Technology

Articles by Subject Category

Enter your email address to subscribe to the StarkEffects NewsLetter.


Troy Stark's Science & Society Opinion Blog


Troy Stark's Linked In Profile -


Physics & Electro-Optics Consulting Services: Advance your business or product development with these Experienced, professional physicists, engineers & entrepreneurs.

Now you can put a face with the name. This is the guy that runs this website. All the errors are his fault.




Buy the books online and pay less than $10. Save Money and Shelf Space! For Amazon's Kindle, click here!



affiliate_link



Welcome to the StarkEffects.com Article on the incompleteness theorem of Kurt Gödel!

A short description of a very upsetting limitation.

Gödel's Incompleteness Theorem

The Incompleteness Theorem states a limitation on what we can know.

Gödel's incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating the inherent limitations of all but the most trivial formal systems for arithmetic.

During Gödel's lifetime the German mathemetician Hilbert was pursuing a complete and consistent set of axioms for all of mathematics. Gödel's work states that such a set is impossible.

Gödel's first theorem:
For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true but not provable in the theory can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

According to Gödel, any set of axioms and rules must contain true statements that cannot be proven within that set. It may be possible to augment the set with new axioms and/or rules, but then you obtain a new set which is larger but still incomplete having statements that are true but unprovable within the system. This means that any system of axioms and rules will contain statements that cannot be proven or disproven based solely on the rules and axioms of that system. This, by extension, means that you cannot be certain that the set of rules and axioms are consistent, meaning they will never lead to contradictory statements according to the rules of the system.

Gödel's second theorem:
For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Comments, Suggestions, Criticisms and Complaints Welcome: Webmaster at StarkEffects
Related Info & Products

StarkEffects, Excited by Science!
Google
 
Web www.StarkEffects.com
© Copyright 2006  StarkEffects, All Rights Reserved
The SiteMap   Privacy Policy   Contact Us