@article{KANAMORI198723,
    Abstract = {G{\"o}del's paper on formally undecidable propositions [3] raised the possibility that finite combinatorial theorems could be discovered which are independent of powerful axiomatic systems such as first-order Peano Arithmetic. An important advance was made by J. Paris in the late 1970's; building on joint work with L. Kirby, he used model-theoretic techniques to investigate arithmetic incompleteness and proved theorems of finite combinatorics which were unprovable in Peano Arithmetic [11]. The Paris-Harrington paper [13] gives a self-contained presentation of the proof that a straightforward variant of the familiar finite Ramsey Theorem is independent of Peano Arithmetic. In this paper, we consider a simple finite corollary of a theorem of infinite combinatorics of Erd{\"o}s and Rado [1] and show it to be independent of Peano Arithmetic. This formulation avoids the Paris-Harrington notion of relatively large finite set and deals with a generalized notion of partition. This shift of focus also provides for simplifications in the proofs and directly yields a level-by-level analysis for subsystems of Peano Arithmetic analogous to that in [12]. We have tried to provide a treatment of the proof whose organization and brevity make it suitable for expository purposes. These results were first discussed in 1982, and almost all the details worked out by a year later. We would like to thank Peter Clote for his later interest and involvement in this web of ideas.},
    Author = {Kanamori, Akihiro and McAloon, Kenneth},
    File = {On Gödel incompleteness and finite combinatorics- 1-s2.0-0168007287900741-main - a - y.pdf},
    ISSN = {0168-0072},
    Journal = {Annals of Pure and Applied Logic},
    Pages = {23 - 41},
    Title = {On G{\"o}del incompleteness and finite combinatorics},
    URL = {http://www.sciencedirect.com/science/article/pii/0168007287900741},
    Volume = {33},
    Year = {1987},
    bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/0168007287900741},
    bdsk-url-2 = {https://doi.org/10.1016/0168-0072(87)90074-1},
    date-added = {2020-06-06 15:08:58 +0200},
    date-modified = {2020-06-06 15:08:58 +0200},
    doi = {10.1016/0168-0072(87)90074-1}
}

@article{KANAMORI198723, Abstract = {G{\"o}del's paper on formally undecidable propositions [3] raised the possibility that finite combinatorial theorems could be discovered which are independent of powerful axiomatic systems such as first-order Peano Arithmetic. An important advance was made by J. Paris in the late 1970's; building on joint work with L. Kirby, he used model-theoretic techniques to investigate arithmetic incompleteness and proved theorems of finite combinatorics which were unprovable in Peano Arithmetic [11]. The Paris-Harrington paper [13] gives a self-contained presentation of the proof that a straightforward variant of the familiar finite Ramsey Theorem is independent of Peano Arithmetic. In this paper, we consider a simple finite corollary of a theorem of infinite combinatorics of Erd{\"o}s and Rado [1] and show it to be independent of Peano Arithmetic. This formulation avoids the Paris-Harrington notion of relatively large finite set and deals with a generalized notion of partition. This shift of focus also provides for simplifications in the proofs and directly yields a level-by-level analysis for subsystems of Peano Arithmetic analogous to that in [12]. We have tried to provide a treatment of the proof whose organization and brevity make it suitable for expository purposes. These results were first discussed in 1982, and almost all the details worked out by a year later. We would like to thank Peter Clote for his later interest and involvement in this web of ideas.}, Author = {Kanamori, Akihiro and McAloon, Kenneth}, File = {On Gödel incompleteness and finite combinatorics- 1-s2.0-0168007287900741-main - a - y.pdf}, ISSN = {0168-0072}, Journal = {Annals of Pure and Applied Logic}, Pages = {23 - 41}, Title = {On G{\"o}del incompleteness and finite combinatorics}, URL = {http://www.sciencedirect.com/science/article/pii/0168007287900741}, Volume = {33}, Year = {1987}, bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/0168007287900741}, bdsk-url-2 = {https://doi.org/10.1016/0168-0072(87)90074-1}, date-added = {2020-06-06 15:08:58 +0200}, date-modified = {2020-06-06 15:08:58 +0200}, doi = {10.1016/0168-0072(87)90074-1} }