Gödel’s first incompleteness theorem was the first to suggest that Hilbert’s programme to create a complete and consistent system of axioms of all mathematical truths is impossible. It says, in particular, that any consistent system of “strong enough” axioms about arithmetic on the natural numbers that can be listen by any effective procedure is incomplete. A theory (P) is complete if either (P) can be proven or (\neg P) can be proven. (P) is consistent if not both (P) and (\neg P) can be proven. The theorem also says that if such a consistent set of axioms is infinite and can be enumerated by a computer program, it is still incomplete. As a result, for any axiom added to the system which maintains its consistency, there will always be other truths that cannot be proven with the system’s own language. If an axiom is added to the system such that it makes it complete, it does so at the cost of making it inconsistent. If a system is inconsistent (i.e., contains (P) and (\neg P)), then one may be able to prove that anything is true.
These results are rather intriguing. It means that there are mathematical truths of arithmetic that we cannot prove with our current system, which may implicitly promote the Platonist account for mathematics. Perhaps this is because we are unable to capture certain mathematical facts, perhaps due to their independence. The implications of the theorem was also devastating for Hilbert’s programme.
Hilbert’s programme was inspired by the fundamental crisis of mathematics which encompassed many paradoxes including Russell’s paradox. He wanted to ground all existing mathematical theories to a finite, consistent, and complete set of axioms. He proposed that this could be reduced to just doing so for arithmetic as it appeared easier and more reasonable.
Gödel’s first incompleteness theorem showed, however, that even for arithmetic, it is impossible to completely axiomatize all truths of arithmetic. In fact, Smorynski (1977) argued that the first theorem alone is sufficient for a fatal blow to Hilbert’s programme because Hilbert required that for any real formula (P), it must be able to be finitistically proven, which was shown to be impossible.
It is quite convincing that if there is no consistent and complete axiomatization of arithmetic, then there is none for mathematics in general. However, I do believe there are weaker statements of Hilbert’s programme that doesn’t require completeness so we could axiomatize only some existing results. However, this is far from his original goal and such a small axiomatization would be meaningless.
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