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Using Ramsey’s theorem once

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Abstract

We show that \(\mathsf {RT} (2,4)\) cannot be proved with one typical application of \(\mathsf {RT} (2,2)\) in an intuitionistic extension of \({\mathsf {RCA}}_{0}\) to higher types, but that this does not remain true when the law of the excluded middle is added. The argument uses Kohlenbach’s axiomatization of higher order reverse mathematics, results related to modified reducibility, and a formalization of Weihrauch reducibility.

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References

  1. Brattka, V., Dzhafarov, D., Marcone, A., Pauly, A.: Measuring the complexity of computational content: from combinatorial problems to analysis (Dagstuhl Seminar 18361). Dagstuhl Rep. 8 (2018) (to appear)

  2. Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symb. Log. 17(1), 73–117 (2011). https://doi.org/10.2178/bsl/1294186663

    Article  MathSciNet  MATH  Google Scholar 

  3. Brattka, V., Kawamura, A., Marcone, A., Pauly, A.: Measuring the complexity of computational content (Dagstuhl Seminar 15392). Dagstuhl Rep. 5(9), 77–104 (2016). https://doi.org/10.4230/DagRep.5.9.77

    Google Scholar 

  4. Brattka, V., Rakotoniaina, T.: On the uniform computational content of Ramsey’s theorem. J. Symb. Log. 82(4), 1278–1316 (2017). https://doi.org/10.1017/jsl.2017.43

    Article  MathSciNet  MATH  Google Scholar 

  5. Dorais, F.G.: Classical consequences of continuous choice principles from intuitionistic analysis. Notre Dame J. Form. Log. 55(1), 25–39 (2014). https://doi.org/10.1215/00294527-2377860

    Article  MathSciNet  MATH  Google Scholar 

  6. Dorais, F.G., Dzhafarov, D.D., Hirst, J.L., Mileti, J.R., Shafer, P.: On uniform relationships between combinatorial problems. Trans. Am. Math. Soc. 368(2), 1321–1359 (2016). https://doi.org/10.1090/tran/6465

    Article  MathSciNet  MATH  Google Scholar 

  7. Feferman, S.: Theories of finite type related to mathematical practice. In: Barwise, J. (ed.) Handbook of Mathematical Logic, Stud. Logic Found. Math., vol. 90, pp. 913–971. North-Holland, Amsterdam (1977). https://doi.org/10.1016/S0049-237X(08)71126-1

    Chapter  Google Scholar 

  8. Hirschfeldt, D.R., Jockusch Jr., C.G.: On notions of computability-theoretic reduction between \({\varPi }_2^1\) principles. J. Math. Log. 16(1), 1650002, 59 (2016). https://doi.org/10.1142/S0219061316500021

  9. Hirst, J.L., Mummert, C.: Reverse mathematics and uniformity in proofs without excluded middle. Notre Dame J. Form. Log. 52(2), 149–162 (2011). https://doi.org/10.1215/00294527-1306163

    Article  MathSciNet  MATH  Google Scholar 

  10. Kleene, S.C.: Introduction to Metamathematics. D. Van Nostrand Co., Inc., New York (1952)

    MATH  Google Scholar 

  11. Kohlenbach, U.: Higher order reverse mathematics. In: Simpson, S. (ed.) Reverse Mathematics 2001, Lect. Notes Log., vol. 21, pp. 281–295. Assoc. Symbol. Logic, La Jolla (2005). https://doi.org/10.1017/9781316755846.018

    Google Scholar 

  12. Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer Monographs in Mathematics. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-77533-1

    MATH  Google Scholar 

  13. Kuyper, R.: On Weihrauch reducibility and intuitionistic reverse mathematics. J. Symb. Log. 82(4), 1438–1458 (2017). https://doi.org/10.1017/jsl.2016.61

    Article  MathSciNet  MATH  Google Scholar 

  14. Patey, L.: The weakness of being cohesive, thin or free in reverse mathematics. Israel J. Math. 216(2), 905–955 (2016). https://doi.org/10.1007/s11856-016-1433-3

    Article  MathSciNet  MATH  Google Scholar 

  15. Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Logic, 2nd edn. Cambridge University Press, Cambridge (2009). https://doi.org/10.1017/CBO9780511581007

    Book  Google Scholar 

  16. Troelstra, A.S.: Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lect. Notes in Math., vol. 344. Springer, Berlin (1973). https://doi.org/10.1007/BFb0066739

    Book  Google Scholar 

  17. Uftring, P.: Proof-theoretic characterization of Weihrauch reducibility. A talk presented at Dagstuhl seminar 18361 (2018). https://materials.dagstuhl.de/files/18/18361/18361.PatrickUftring.Slides.pdf

  18. Uftring, P.: Proof-theoretic characterization of Weihrauch reducibility. Master’s thesis, Technische Universität Darmstadt (2018) (forthcoming)

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Authors and Affiliations

  1. Department of Mathematical Sciences, Appalachian State University, Walker Hall, Boone, NC, 28608, USA

    Jeffry L. Hirst

  2. Department of Mathematics, Marshall University, 1 John Marshall Drive, Huntington, WV, 25755, USA

    Carl Mummert

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  1. Jeffry L. Hirst
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  2. Carl Mummert
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Correspondence to Jeffry L. Hirst.

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Hirst, J.L., Mummert, C. Using Ramsey’s theorem once. Arch. Math. Logic 58, 857–866 (2019). https://doi.org/10.1007/s00153-019-00664-z

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  • DOI: https://doi.org/10.1007/s00153-019-00664-z

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