Open AccessThe modified Ramsey theorem is not a G_del sentence
Author(s) -
Я. Хинтикка
Publication year - 2013
Publication title -
logičeskie issledovaniâ
Language(s) - English
Resource type - Journals
eISSN - 2413-2713
pISSN - 2074-1472
DOI - 10.21146/2074-1472-2013-19-0-33-38
Subject(s) - undecidable problem , sentence , arithmetic function , peano axioms , atomic sentence , independence (probability theory) , mathematics , order (exchange) , arithmetic , computer science , discrete mathematics , decidability , artificial intelligence , statistics , finance , economics
G_delian 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.