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 Godel's first incompleteness theorem.The strengthened finite Ramsey theorem is a statement that is not provable in Peano arithmetic.

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