August 30, 2002

Mathematics Incomplete?

Mainstream mathematicians tend to like to believe that everything in this world can be explained, and proved, by use of logic or mathematics if you want. I guess I did too, atleast up until a professor gave me the name of Kurt Gödel...

Kurt Gödel (1906-1978) was the name of an Austrian mathematician that focused mainly on metamathematics. His one big 'hit' of the era was 'On Formally Undecidable Propositions'.

He proved it impossible to establish the internal logical consistency of a very large class of deductive systems - elementary arithmetic, for example - unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves ... Second main conclusion is ... Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set... Even if the axioms of arithmetic are augmented by an indefinite number of other true ones, there will always be further mathematical truths that are not formally derivable from the augmented set.
- Nagel and Newman, Gödel's Proof

The result is actually seen quite early on in Mathematics. Given two integers a and b, it is not necessarily possible to solve a/b within the integers Z. Naturally one could resort to extend the system from Z to e.g. Q. In Q all numbers are defined by a/b, for all a and b in Z.

So far so good. But what about the square root of a (a still in Z)? Is it in Z, or is it in Q? Most likely it is in R. Atleast if a is in Z+. And one can go on.

Slightly frustrating to know that mathematics, the art of mathematics, is incomplete, that not all problems can be solved even in an infinitely complex calculus. My world is falling apart...

- Gödel's Incompleteness Theorem
- The Kurt Gödel Society

Posted by ludvig at August 30, 2002 12:09 AM | TrackBack
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