Gödel's incompleteness theorem
Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.
What this effectively means is that there will always be a true but unprovable statement within any system of mathematics.
What is so amazing about the statement is that someone was able to prove it!
No comments:
Post a Comment