Paris Harrington Theorem
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematical logic, the Paris Harrington theorem states that a certain combinatorial principle in Ramsey theory is true, but not provable in Peano arithmetic. This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic; it was already known that such statements existed by Gödel''s first incompleteness theorem. The strengthened finite Ramsey theorem is a statement...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematical logic, the Paris Harrington theorem states that a certain combinatorial principle in Ramsey theory is true, but not provable in Peano arithmetic. This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic; it was already known that such statements existed by Gödel''s first incompleteness theorem. The strengthened finite Ramsey theorem is a statement that is not provable in Peano arithmetic. (It should not be confused with the Paris Harrington theorem, which states that the strengthened finite Ramsey theorem is not provable in Peano arithmetic.)
