In 1931, Kurt Godel
released a paper titled "On Formally Undecidable Propositions of Principia Mathematica and Related Systems
" (well the title was originally in German
). The paper contains a proof of what is commonly referred to as Godel's Theorem
. This theorem is in two parts.
In a nutshell, the first part states that any consistent axiomatic mathematical system which is capable of talking about number theory will be incomplete. That is to say there will be statements in the system which we cannot prove true and cannot prove false (within the system). The second part states that for any such system the statement "This system is consistent" is unprovable (within the system). That is we cannot prove that it is impossible to derive a contradiction in the system.
One tool used by Godel in proving his theorem is Godel numbering. He came up with a procedure to assign a postive whole number (called the Godel number) to each symbol, formula and proof in the system. No two symbols etc. could have the same Godel number, so there was a direct correspondence between statements in the system and a subset of the positive whole numbers. The reason this is important is : statements referring to statements within the system can now be thought of as statements referring to positive whole numbers. More accurately (though more technically) : meta-mathematical statements became statements in number theory.