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#flashcards/stem/math
Two theorems of logic that are concerned with the limits of provability in formal axiomatic theories.
Shows that Hilbert’s program to find a complete and consistent set of axioms for all mathematics is impossible.
First theorem
No consistent system of axioms whose theorems can be listed by an algorithm is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true but that are unprovable within the system.
Second theorem
The system cannot demonstrate its own consistency.
Questions
What does the theorems prove?::Finding a complete and consistent set of axioms for all mathematics is impossible