The modified Ramsey theorem is not a Gödel sentence

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J. Hintikka


Gödelian sentences are self-referential first-order sentences in the language of arithmetics. Perhaps the most celebrated one is the sentence which asserts its own unprovability. It is well known that this sentence is neither provable nor refutable in PA (Peano Arithmetics). Some logicians and philosophers have complained that such a sentence is difficult to grasp given its ‘meta-theoretical’ content and they started to look for undecidable arithmetical statements which have a combinatorial content. One such sentence is a variant of Ramsey’s sentence: the Paris-Harrington theorem asserts its undecidability. In the present paper I shall argue that such a sentence is not first-order expressible and thereby it does not provide the desired example of a combinatorial, undecidable arithmetical sentence. Instead I shall argue that it is expressible in Independence-friendly (IF) logic.


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J. Hintikka. The modified Ramsey theorem is not a Gödel sentence // Логические исследования / Logical Investigations. 2018. Т. 19. № 1.