Gödel Incompleteness Theorem

Context:
- It can shows that within any formal mathematical system that is capable of expressing basic arithmetic, there are statements that cannot be proven to be true or false using the rules and axioms of that system.
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Example(s):
- a Godel First incompleteness theorem, that no consistent system of axioms whose theorems can be listed by an effective procedure is capable of proving all truths about the arithmetic of natural numbers / or that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms.
- a Godel Second incompleteness theorem, that the system cannot demonstrate its own consistency.
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See: Halting Problem, Theorem, Mathematical Logic, Kurt Gödel, Philosophy of Mathematics, Hilbert's Program, Number Theory, Effective Procedure, Algorithm, Natural Number.

References

(Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems Retrieved:2023-6-19.
- Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
  The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., 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. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
  Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.