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Kurt Gödel

The development of mathematics towards greater exactness has, as is well-known, lead to formalization of large areas of it such that you can carry out proofs by following a few mechanical rules. The most comprehensive current formal systems are the system of Principia Mathematica (PM) on the one hand, the Zermelo-Fraenkelian axiom-system of set theory on the other hand. These two systems are so far developed that you can formalize in them all proof methods that are currently in use in mathematics, i.e. you can reduce these proof methods to a few axioms and deduction rules. Therefore, the conclusion seems plausible that these deduction rules are sufficient to decide all mathematical questions expressible in those systems. We will show that this is not true, but that there are even relatively easy problems in the theory of ordinary whole numbers that can not be decided from the axioms. This is not due to the nature of these systems, but it is true for a very wide class of formal systems, which in particular includes all those that you get by adding a finite number of axioms to the above mentioned systems, provided the additional axioms don't make false theorems provable.
Kurt Gödel, in the opening paragraph of "On formally undecidable propositions of Principia Mathematica and related systems - I." (For those who are not mathematicians, this paper by Gödel, published in 1931, in Monatshefte für Mathematik, proved by entirely elementary means that any formal system of mathematics which was sufficient to contain the natural numbers — integers beginning with zero — must also include statements which are true but cannot be proven to be true within the system. It has come to be known as the paper that introduced his 1st and 2nd Incompleteness Theorems. To summarize, his theorems proved that the study of mathematics not only can never be complete, it can never be founded upon a single axiomatic system that, even in principle, covers all true theorems. This is arguably, humankind's first formal demonstration that the first part of Einstein's famous dictum, "God is subtle, but not malicious." is true.)
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