All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Paradoxes
Provably unprovable? (Posted on 2004-08-22) Difficulty: 4 of 5
Gödel proved that there are true sentences that cannot be proved.

Suppose I told you that the Goldbach conjecture is one of those. (The Goldbach conjecture supposes that every even integer number can be expressed as the sum of two odd primes.)

Is that logically possible? (And, no, I haven't proved it!)

See The Solution Submitted by Federico Kereki    
Rating: 3.4167 (12 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: My thought is.... | Comment 18 of 36 |
(In reply to My thought is.... by Penny)

Let's be clear about what Godel's theorem asserts.  It says that in any formal system of axioms and numbers rich enough to include all the integers, multiplication, and addition, and a few other familiar properties, that there exist statements in that system that are true in that system, but cannot be formally derived from the axioms of that system. 

In fact, Godel demonstrated how to construct such a statement--lets call it G.  Godel showed that G is true by creating a correspondence between statements about numbers and the numbers themselves.  This, essentially, "steps outside" the formal system to demonstrate the truth of G.  Statement G depends on what set of axioms you choose as the basis of your system.  G is a very long statement, but its existence only demonstrates that statements exist which are true but have no formal proof within their own system.

Is the Goldbach conjecture such a statement?  Until someone proves it or finds a counter-example, we'll never know.


  Posted by Ken Haley on 2004-10-03 02:34:25
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (14)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information