In 1931, Kurt Gödel published two theorems known as his ‘Incompleteness Theorems’. They are theorems of mathematical logic that establish the limitations of the set of axioms that establish the foundations of mathematics and allow us to do arithmetic. Thus, they are important in both mathematical logic and in the philosophy of mathematics. The results of these theorems are widely interpreted as giving a negative answer to Hilbert’s second problem.
First Incompleteness Theorem
The first theorem states that:
“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.”
The meaning of this theorem is that it shows that there are true statements expressible in our current mathematical language that cannot be proved within our axiomatic system. This theorem highlighted that even if an axiom is ever added that makes the system complete, it does so by making it inconsistent.
Therefore, no formal system can characterize the natural numbers, as there will be true number-theoretical statements that the system cannot prove. This had devastating consequences on the program proposed by Gottlob Frege and Bertrand Russell, who aimed to define the natural numbers in terms of logic.
Second Incompleteness Theorem
The second of Gödel’s theorems says that:
“For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.”
This strengthens the first incompleteness theorem, as the first theorem does not necessarily express the consistency of the theory.
To read about the details of the proofs click here.
Gödel’s conclusions are only proven for the formal theories that satisfy the necessary hypotheses. However, not all axiomatic systems satisfy them.
Furthermore, his second theorem only shows that the consistency of some theories cannot be proven from the axioms of those theories themselves. It doesn’t show that consistency cannot be proved from other axioms. For example, the consistency of Peano arithmetic can be proved in Zermelo-Fraenkel Set Theory.
Hope you enjoyed this post, sorry it was a bit short! M x